Some Global Correspondences for General Linear Groups and their Metaplectic Coverings

Paul Mezo

Abstract

Let A be the adele ring of a number field containing the nth roots of unity, and G̃L(r, A) be an n-fold metaplectic covering of GL(r, A). Under an assumption on n, we prove identities between all of the terms in Arthur’s invariant trace formulas for G̃L(r, A) and GL(r, A). We then establish a correspondence between the automorphic representations of these groups.

Contents

1 Introduction 2 2 Preliminaries 7 3 The local metaplectic correspondence 11 4 Comparison of orbital integrals at singular elements 18 5 Normalization of intertwining operators and the Plancherel

formula 25 6 The distribution I_{M}^{Σ}(γ) 29 7 The distribution I_{M}^{M}(γ) 36 8 Geometric vanishing properties 37 9 The invariant trace formula 45 10 Statement of Theorem A 47 11 Comparison of I_{M}^{M}(γ, f̃) and I_{M}^{Σ}(γ, f̃) 51 12 Comparison of germs 59 13 The distributions IM̃(π̃, X) and I_{M}^{Σ}(π̃, X) 70 14 Statement of Theorem B 73 15 Comparison of IM̃(π̃, X, f̃) and Î_{M}^{Σ}(π̃, X, f̃) 75 16 More on normalizing factors 82 17 A formula for I_{t}^{Σ} 85 18 The map ε_{M} 90 19 Comparison for f̃ ∈ H(G̃(A), M) 95 20 Completion of the proofs of Theorems A and B 100 21 A global correspondence 107 22 Appendix. Tensor products of metaplectic representations 114

1 Introduction

Correspondences of metaplectic covering groups have their origin in the work of Shimura ([Shi73]). Shimura constructed a correspondence between modular forms of half-integral weight and cusp forms of even weight, which preserves L-functions. He suggested that this correspondence be studied further by using the representation-theoretic techniques developed by Jacquet and Langlands ([JL70]). This approach was explored by Flicker ([Fli80]), who gave a complete description of a correspondence between the automorphic representations of the n-fold covering of GL(2) and the automorphic representations of GL(2). Flicker’s correspondence was proved using the Selberg trace formula and followed Langlands’ proof of base change for GL(2). [Lan80]. Trace formula methods were also exploited by Flicker, Kazhdan and Patterson ([FK86], [KP86]) in the proof of some additional correspondences between the automorphic representations of n-fold metaplectic coverings of GL(r), r ≥ 2, and automorphic representations of GL(r). We prove correspondences of automorphic representations for the same groups under some assumptions on n, the order of the covering. Our approach is novel in that we use the invariant trace formula of Arthur ([Art88b]) and follow Arthur and Clozel’s proof of base change for GL(r) ([AC89]). We refer the reader to the introductions of [Fli80], [KP84], [KP86] and [FK86] for the ramifications of metaplectic correspondences.

In §§2-4 we recapitulate and expand upon the local results of [FK86]. For a number field F containing the nth roots of unity, we define the orbit map,

γ ↦ γ′,

which is a map from the general linear group over a local completion F_{v} to its n-fold metaplectic covering G̃L(r, F_{v}). This map preserves conjugacy classes and is the foundation of all our correspondences. There is a dual map

f̃ ↦ f̃′,

from the Hecke space of G̃L(r, F_{v}) to the Paley-Wiener space of GL(r, F_{v}), which matches orbital integrals compared under the orbit map. The local metaplectic correspondence amounts to a map,

π̃ ↦ π̃′,

from certain tempered representations of G̃L(r, F_{v}) to tempered representations of GL(r, F_{v}), which satisfies the character identity,

tr (π̃(f̃)) = f̃′(π̃′).

We make two assumptions in these sections. The first is that the local metaplectic correspondence commutes with parabolic induction as claimed in Proposition 26.2 [FK86]. The second is that the trace Paley-Wiener theorem holds for G̃L(r, F_{v}) (cf. [BDK86]). Metaplectic coverings of the general linear group over the real numbers are not considered although they are well-understood ([AH97]).

The local terms in Arthur’s invariant trace formula depend on the normalization of intertwining operators between induced representations. This is the subject of §5. We show that the intertwining operators for the metaplectic coverings can be suitably normalized by following §2 II [AC89]. The argument depends on the Plancherel formula for G̃L(r, F_{v}). The proof of this formula has not been written out, but can be seen to follow from the proof for GL(r, F_{v}) once the observations at the beginning of §5 are taken into consideration.

The local terms of the geometric side of the invariant trace formula are considered in §§6-8. These terms are distributions parameterized by conjugacy classes. They are grouped according to the orbit map in §6, and the groupings I_{M}^{Σ}(γ) are shown to preserve properties of descent and splitting. In §7, we assume that the construction and properties of the local geometric terms passes to metaplectic coverings . From these terms we define the distributions I_{M}^{M}(γ) that are expected to match I_{M}^{Σ}(γ). Descent and splitting properties are proven for I_{M}^{M}(γ) as well. In order to compare I_{M}^{M}(γ) and I_{M}^{Σ}(γ) using the invariant trace formula, we must show that the distributions which occur in the trace formula of the metaplectic coverings, but are not of the form I_{M}^{M}(γ), are inconsequential. Ideally, every such distribution can be shown to vanish. In §8 we show that the ideal situation holds only if n is relatively prime to the positive integers less than or equal to r. Conversely, we show that if n is relatively prime to an additional integer, which depends on the metaplectic covering, then the ideal situation holds.

The conjectural invariant trace formula for a metaplectic covering of the general linear group is introduced in §9. This formula is conjectural not because of any expected complications in its proof; rather, because its proof is very long. The most formidable hurdle in its proof is quite possibly the trace Paley-Wiener theorem for metaplectic coverings (cf. [BDK86]). Due to the volume of the material, the reader is assumed to be familiar with the basic notions of the invariant trace formula for reductive groups.

The topic of §§10-20 is the comparison of invariant trace formulas. This comparison is made under an assumption on the order of the covering. We first describe the content of these sections and then consider the assumption. Let A be the adele ring of F. The invariant trace formula for G̃L(r, A) is placed into the form

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{F,S}/μ_{n}^{M}} aM̃(S, γ′)I_{M}^{M}(γ, f̃)

= ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{t≥0} ∫_{Π(M̃,t)} aM̃(π̃)IM̃(π̃, f̃)dπ̃,

and the invariant trace formula for GL(r, A) is placed into the form

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{F,S}/μ_{n}^{M}} a^{M}(S, γ)I_{M}^{Σ}(γ, f̃)

= ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{t≥0} ∫_{Π^{Σ}(M,t)} a^{M,Σ}(π̃)I_{M}^{Σ}(π̃, f̃)dπ̃.

The function f̃ in these equations is taken to be arbitrary in the global Hecke space of G̃L(r, A). The principal results of §§10-20 are Theorems A and B which match the global terms, aM̃(S, γ′) with a^{M}(S, γ) and aM̃(π̃) with a^{M,Σ}(π̃); and the local distributions, I_{M}^{M}(γ) with I_{M}^{Σ}(γ) and IM̃(π̃) with I_{M}^{Σ}(π̃). The structure of the proof of Theorems A and B follow II [AC89] very closely. There are however two notable deviations. The first is our use of the local invariant trace formula ([Art89c]) in §§11, 20. We assume that this formula holds for metaplectic coverings. The second is the use of strong approximation in §20. Otherwise, the reader familiar with [AC89] should have no difficulties in relating the ideas of §§10-20 to II [AC89]. To make this relation more transparent, the results of these sections, which have counterparts in II [AC89], have references to these counterparts in parentheses immediately following their own numbering.

Let us now consider assumption on the order of the covering, which is labeled as (†) in §10. Under this assumption n is relatively prime to the positive integers less than or equal to r and is also relatively prime to i(1 + 2m) − 1, where 1 < i ≤ r and 0 ≤ m ≤ n − 1 is a fixed integer. The reader may find it helpful to consider assumption (†) with m = 0, in which case the assumption is greatly simplified, but the covering groups are still non-trivial. We list below the obstructions that are removed under this assumption. The list is given in increasing order of the author’s perception of their difficulty.

Assumption (†) excludes even coverings. If n is even the orbit map is defined only on a proper subset of G(F_{v}) and contains a term defined in terms of K-theory. The former circumstance would require the proof of a vanishing property for the geometric side of the trace formula of GL(r, A), or restrictions on f̃. The latter would make computations involving the orbit map more complicated.

Proposition 26.2 [FK86] and the Appendix ensure that the local metaplectic correspondence commutes with parabolic induction under the assumption that n is relatively prime to i(1 + 2m) − 1, 1 < i ≤ r. It is expected that this assumption can be lifted, but to date there is no proof of this expectation.

Assumption (†) implies that n ≥ 3. In this case the archimedean completions of F are all complex. Since metaplectic coverings of GL(r, C) are trivial, the representation theory at the archimedean valuations is very straightforward. Jordan canonical form may also be used to simplify the comparison at the archimedean valuations. In order to complete the comparison in the case n = 2, one would presumably have to prove identities of differential operators on G̃L(r, R) parallel to those in [Mezb].

The assumption that n is relatively prime to the positive integers less than or equal to r simplifies the comparison of γ and γ^{n}, and of their centralizers in G(F_{v}). Many terms in the invariant trace formulas are expressed in terms of these objects (see §§11, 12). Under the above assumption, the discrepancies in these terms may be described purely in terms of nth roots of unity.

As already mentioned, the vanishing property for the geometric distributions of G̃L(r, F_{v}) does not follow for general n. This obstacle might be circumvented by making restrictions on f̃ or by showing that the sum of the undesirable distributions vanish.

Last, but certainly not least, the matching of weighted orbital integrals required for Lemma 11.4 and the approximation arguments of §20 is only proved in the case that n is relatively prime to certain integers which are included in (†) ([Meza]). Even under the assumption that the local trace formula ([Art89c]) holds for metaplectic coverings, this matching is not immediate.

Theorem B entails some global metaplectic correspondences which are listed in Theorem 21.1. In broad terms, there is a correspondence between unitary automorphic representations of G̃L(r, A) and unitary automorphic representations of Levi subgroups of GL(r, A). This correspondence preserves a character relation at almost every valuation of F. If a representation π̃ of G̃L(r, A) corresponds as above to a cuspidal representation π of GL(r, A) then a character relation is preserved at every valuation of F and π̃ is the only representation of G̃L(r, A) which corresponds to π. This implies the multiplicity one and strong multiplicity one properties for π̃.

Aside from the elimination of the rather vexing assumption (†), there remains much more work to be done in obtaining a general metaplectic correspondence. The paper is concluded with two conjectures concerning the general correspondence and suggestions for their possible solutions.

Portions of this paper appeared in the author’s thesis [Mez98]. The author would like to thank J. Arthur for his encouragement and the Max-Planck-Institut für Mathematik in Bonn for their generous support.

2 Preliminaries

We begin by briefly describing metaplectic coverings of the general linear group as in §2 [FK86]. We then fix Haar measures and define some vector spaces which are fundamental to the representation theory of our groups.

Let n be a positive integer and let F be a totally imaginary number field containing the group μ_{n} of nth roots of unity. If n ≥ 3 then the condition that F be totally imaginary is superfluous. The completion of F at a valuation v is denoted by F_{v}, and its absolute value, which is determined by Haar measure, is denoted by | · |_{v}. Throughout, S denotes a finite set of valuations of F. Let F_{S} be the ring ∏_{v∈S} F_{v} and set

|γ|_{S} = ∏_{v∈S} |γ_{v}|_{v}, γ = ∏_{v∈S} γ_{v} ∈ F_{S}.

We write A for the adele ring of F.

We take r ≥ 2 to be an integer and write G for the algebraic group GL(r). Clearly, GL(r) is defined over F. Thus we may form the group of F - or F_{v}-valued points of G. They are G(F) = GL(r, F) and G(F_{v}) = GL(r, F_{v}) respectively. F also embeds diagonally into F_{S} and so we may form the group of F_{S}-valued points of G. This is simply G(F_{S}) = ∏_{v∈S} GL(r, F_{v}). The adele group of GL(r, F) is denoted by G(A).

We now recall the construction of the metaplectic coverings as in §2 [FK86]. For each valuation v, there are two-cocycles,

σ_{mv} : G(F_{v}) × G(F_{v}) → μ_{n}, 0 ≤ m ≤ n − 1.

If v is archimedean, that is F_{v} = C, then σ_{mv} is the trivial cocycle.

Suppose v is nonarchimedean. Then these cocycles are “twists” of Matsumoto’s cocycle σ_{0v}. In this case there exists a compact open subgroup K_{v}^{0} and a Borel function,

κ_{v} : K_{v}^{0} → μ_{n},

such that

σ_{mv}(k_{1}, k_{2}) = \frac{κ_{v}(k_{1}k_{2})}{κ_{v}(k_{1})κ_{v}(k_{2})}, k_{1}, k_{2} ∈ K_{v}^{0}.

We extend κ_{v} to a Borel function on G(F_{v}) such that for any upper-triangular unipotent element u ∈ G(F_{v}) we have

κ_{v}(γu) = κ_{v}(γ), γ ∈ G(F_{v}).

This can be achieved, for example, by setting κ_{v} equal to one outside of K_{v}^{0} ((2.2) [FK86]).

Define the cocycle τ_{mv} by

τ_{mv}(γ_{1}, γ_{2}) = σ_{mv}(γ_{1}, γ_{2})\frac{κ_{v}(γ_{1})κ_{v}(γ_{2})}{κ_{v}(γ_{1}γ_{2})}, γ_{1}, γ_{2} ∈ G(F_{v}),

if v is nonarchimedean, and by τ_{mv} = σ_{mv} otherwise. By definition, τ_{mv} is cohomologous to σ_{mv}. It follows from our choice of κ_{v} that for any uppertriangular unipotent elements u_{1}, u_{2} ∈ G(F_{v}) we have

τ_{mv}(u_{1}γ_{1}, γ_{2}u_{2}) = τ_{mv}(γ_{1}, γ_{2}), γ_{1}, γ_{2} ∈ G(F_{v}). (1)

Furthermore, τ_{mv} is trivial when restricted to K_{v}^{0} × K_{v}^{0}.

The cocycle τ_{mv} yields a central extension,

1 → μ_{n} → G̃_{m}(F_{v}) → G(F_{v}) → 1,

called an n-fold metaplectic covering of G(F_{v}). Note that G̃_{m}(F_{v}) is not identical to the extension investigated in [KP84], [KP86] and [FK86], which is defined from just σ_{mv}. Nevertheless, G̃_{m}(F_{v}) shares all of the relevant properties of the extension used in these works. We fix m and write G̃(F_{v}) in place of G̃_{m}(F_{v}). The n-fold metaplectic covering,

1 → μ_{n} →^{i} G̃(F_{S}) \underrightarrow{p} ; \overleftarrow{s} G(F_{S}) → 1,

is defined by way of the two-cocycle τ_{S} = ∏_{v∈S} τ_{mv}. It is comforting to keep in mind that, as a set, G̃(F_{S}) is equal to G(F_{S}) × μ_{n}. The maps of the above sequence may therefore be expressed more concretely by

i(ζ) = (1, ζ), p(γ, ζ) = γ, and s(γ) = (γ, 1),

for all γ ∈ G(F_{S}) and ζ ∈ μ_{n}. Multiplication is in given by

(γ_{1}, ζ_{1})(γ_{2}, ζ_{2}) = (γ_{1}γ_{2}, ζ_{1}ζ_{2} τ_{S}(γ_{1}, γ_{2})), γ_{1}, γ_{2} ∈ G(F_{S}), ζ_{1}, ζ_{2} ∈ μ_{n}.

Given a subgroup H of G(F_{S}), we write H̃ for p^{−1}(H). We say that G̃(F_{S}) splits over H if H̃ is isomorphic to H × μ_{n} as a group. The group G̃(C) splits over G(C).

One can also form the product, τ = ∏_{v} τ_{mv}, over all valuations of F to obtain a two-cocycle, which gives rise to an n-fold metaplectic covering G̃(A) of G(A). The previous notions for G̃(F_{S}) have obvious parallels for G̃(A). It is important to realize that G̃(A) splits over G(F). The splitting homomorphism,

s_{0} : G(F) → G̃(A),

is given by

s_{0}(γ) = s(γ)i (∏_{v} κ_{v}(γ)^{−1}) , γ ∈ G(F). (2)

Suppose v is nonarchimedean. We define the orbit map,

G_{0}(F_{v}) →^{′} G̃(F_{v}), (3)

as the map given in §4 [FK86]. If n is odd then G_{0}(F_{v}) = G(F_{v}) and the orbit map is given by

γ′ = s(γ)^{n} = (γ, 1)^{n}, γ ∈ G(F_{v}).

If n is even then G_{0}(F_{v}) is a dense open subset of G(F_{v}) and

γ′ = u(γ)s(γ)^{n} = (1, ±1)(γ, 1)^{n}, γ ∈ G_{0}(F_{v}),

where u is a class function taking values in i(μ_{2}). At the archimedean places, we define the orbit map by

γ ↦ γ′ = s(γ)^{n} = s(γ^{n}), γ ∈ G(C).

The orbit map is so called, because it preserves conjugacy classes, i.e. orbits under the action of conjugation. The orbit map extends to maps on dense subsets of G(F_{S}) and G(A) by acting on each factor. We abuse notation by denoting these maps by ′ and referring to them as “the orbit map” as well.

We shall concern ourselves primarily with the regular semisimple elements and conjugacy classes of G̃(F_{S}). We therefore define G_{reg} to be the set of regular and semisimple elements in G. The sets G_{reg}(F) and G_{reg}(F_{S}) are the respective subsets of F - and F_{S}-valued points of G_{reg}. Set G̃_{reg}(F_{S}) = p^{−1}(G_{reg}(F_{S})). Taking the orbit map into consideration we define the set,

G_{oreg}(F_{S}) = {γ ∈ G(F_{S}) : γ^{n} ∈ G_{reg}(F_{S})}.

In the terminology of §1.i [Vig81], the image of G_{oreg}(F_{S}) under the orbit map is contained in the set of ordinary regular elements of G̃(F_{S}) (Proposition 3 [FK86]). Flicker and Kazhdan use the word good in place of ordinary (§8 [FK86]).

Let M_{0} be the diagonal subgroup of G.The set of Levi subgroups of G containing the minimal Levi subgroup M_{0} is denoted by L. We reserve the letter M for an element of L. As usual, the center of M is denoted by A_{M}. The set of Levi subgroups of some element L ∈ L, which also contain M ⊂ L, is written as L^{L}(M). Similarly, the set of parabolic subgroups of L with Levi component M ⊂ L is written as P^{L}(M). Given a parabolic subgroup P ∈ P(M), we write U_{P} for its unipotent radical. Let P_{0} be the subgroup of upper-triangular matrices in G. Its unipotent radical U_{P_{0}} is the subgroup of upper-triangular unipotent matrices. Clearly, we may adelize or take the F_{S}-valued points of these groups. We therefore write M(F_{S}), M(A), etc. without further comment.

The following decomposition of M will be useful in inductive arguments made later on. Recall that there exist positive integers, r_{1}, …, r_{ℓ}, and subgroups, M(1) ≅ GL(r_{1}), …, M(ℓ) ≅ GL(r_{ℓ}), of M such that ∑_{i=1}^{ℓ} r_{i} = r and M = M(1) × ⋯ × M(ℓ). (4)

For each valuation v we fix a maximal compact subgroup K_{v} of G(F_{v}) as follows. If v is nonarchimedean set K_{v} = GL(r, R_{v}), where R_{v} is the ring of integers of F_{v}. If F_{v} = C then K_{v} is the unitary group U(r, C). Set K_{S} = ∏_{v∈S} K_{v}.

In order to perform harmonic analysis on these groups without any ambiguity, we endow them with measures. Fix measures on M(F_{S}) and U_{P} (F_{S}), where P ∈ P^{L}(M), L ∈ L(M) as in §1 [Art81]. If H is a subgroup of G(F_{S}) with a fixed Haar measure dh, we choose dh̃ to be the unique Haar measure on H̃ satisfying

dh̃ ∘ p^{−1} = n · dh.

We define quotient measures as follows. Given subgroups H_{1} ⊂ H_{2} of G(F_{S}) with a fixed measure on H_{1}\H_{2}, we define the measure on H̃_{1}\H̃_{2} to be the pull-back of the measure on H_{1}\H_{2} via the obvious homeomorphism,

H̃_{1}\H̃_{2} → H_{1}\H_{2}.

We close this section with a description of some vector spaces which are basic to representation theory. Let X(M) be the group of rational characters of M. Set 𝔞_{M} = Hom_{Z}(X(M), R), 𝔞_{M}^{∗} = Hom_{R}(𝔞_{M}, R) and 𝔞_{M,C}^{∗} = Hom_{R}(𝔞_{M}, C). The group X(M) embeds into 𝔞_{M,C}^{∗} as a lattice. Fix a Euclidean norm on 𝔞_{M_{0}}, which is invariant under W_{0}^{G}, the Weyl group of (G, A_{M_{0}}). There is an obvious embedding of 𝔞_{M} into 𝔞_{M_{0}} and we endow 𝔞_{M} with the Euclidean norm obtained from this embedding by restriction. This norm provides a measure on 𝔞_{M}.

As usual, we define the homomorphism,

H_{M} : M(F_{S}) → 𝔞_{M},

by the equation,

e^{⟨H_{M} (γ),ξ⟩} = ∏_{v∈S} |ξ(γ_{v})|_{v}, γ = ∏_{v∈S} γ_{v} ∈ M(F_{S}), ξ ∈ X(M).

The composition of H_{M} with p is evidently a homomorphism of M̃(F_{S}) into 𝔞_{M}. The product formula allows us to generalize the above homomorphisms to apply to M(A) and M̃(A) as well. These maps provide the spaces M(A)^{1} = ker(H_{M}) and M̃(A)^{1} = ker(H_{M} ∘ p) with measures.

Suppose that γ belongs to either M(F_{S}) or M(A). Then

H_{M}(p(γ′)) = H_{M}(γ^{n}) = nH_{M}(γ). (5)

This equation motivates the definition of the isomorphism of 𝔞_{M} defined by

X ↦ X′ = n^{−1}X, X ∈ 𝔞_{M},

and the adjoint isomorphism of 𝔞_{M,C}^{∗} defined by

λ ↦ λ′ = nλ.

3 The local metaplectic correspondence

The local metaplectic correspondence is a map from certain representations of G̃(F_{S}) to representations of G(F_{S}), which also preserves a specific character relation. It also has dual characterizations in terms of a map of Hecke spaces or Paley-Wiener spaces. In the case of Hecke spaces, the map preserves an identity of orbital integrals. We examine each of these characterizations, giving appropriate definitions along the way.

A representation π̃ of M̃(F_{S}) or M̃(A) is said to be genuine if

π̃(γ, ζ) = ζπ̃(γ, 1),

for all ζ ∈ μ_{n} and γ in M(F_{S}) or M(A). Let Π(M̃(F_{S})) be the set of (equivalence classes of) genuine, irreducible and admissible representations of M̃(F_{S}). Let Π_{temp}(M̃(F_{S})) and Π_{unit}(M̃(F_{S})) be the subsets of Π(M̃(F_{S})) which consist respectively of tempered and unitary representations.

Given M_{1} ∈ L^{M}, P ∈ P^{M}(M_{1}) and π̃_{1} ∈ Π(M̃_{1}(F_{S})), we denote the unitarily induced representation Ind_{P}^{M} π̃_{1} by π̃M̃ ; 1. If λ ∈ 𝔞_{M_{1},C}^{∗} then the representation π̃_{1,λ}, given by

π̃_{1}(γ̃) = π̃_{1}(γ̃)e^{λ(H_{M_{1}}(p(γ̃)))}, γ̃ ∈ M̃_{1}(F_{S}),

belongs to Π(M̃_{1}(F_{S})) as well.

Given π̃ ∈ Π(M̃(F_{S})), we denote its central character by ωπ̃ and its distribution character by Θπ̃. For γ = ∏_{v∈S} γ_{v} ∈ M(F_{S}), the Weyl discriminant is defined as

D^{M}(γ) = ∏_{v∈S} det(1 − Ad(γ_{v}))_{𝔪/𝔪_{σv}} ∈ F_{S}.

Here σ_{v} is the semisimple component in the Jordan decomposition of γ_{v}, and 𝔪 and 𝔪_{σ_{v}} are the respective Lie algebras of M(F_{v}) and the centralizer of σ_{v} in M(F_{v}).

Suppose v is nonarchimedean, π̃ ∈ Π(G̃(F_{v})) and π ∈ Π(G(F_{v})). Let d = gcd(n, r − 1 + 2rm). Following (26.1) [FK86], π̃ is said to correspond (or lift) to π if

ωπ̃(γ′) = ω_{π}(γ), γ ∈ A_{G}(F_{v}),

and for any γ ∈ G_{oreg}(F_{v}),

|D^{G}(γ^{n})|_{v}^{1/2}Θπ̃(γ′) = \frac{n|d|_{v}^{1/2}}{d|n|_{v}^{r/2}} ∑_{\{γ_{1},γ_{2}:γ′_{1}γ_{2}=γ′\}} |D^{G}(γ)|_{v}^{1/2}Θ_{π}(γ_{1})ωπ̃(γ_{2}). (6)

The sum on the right is parameterized by γ_{1} ∈ G_{γ}(F_{v})/A_{G}(F_{v}) and γ_{2} ∈ Ã^{n/d}(F_{v})/A′_{G}(F_{v}). The group G_{γ}(F_{v}) is the set of F_{v}-valued points of G_{γ}, the centralizer of γ in G. The set A′_{G}(F_{v}) is equal to {γ′_{0} : γ_{0} ∈ A_{G}(F_{v})}.

Theorem 27.3 [FK86] implies that there is a map,

(7) Π_{temp}(G̃(F_{v})) →^{′} Π_{temp}(G(F_{v})),

such that π̃ corresponds to π̃′. Proposition 26.2 implies that (7) can be generalized to a map,

(8) Π_{temp}(M̃(F_{v})) →^{′} Π_{temp}(M(F_{v})),

which preserves a character relation analogous to (6). In addition, map (8) commutes with parabolic induction. These results depend on the method of induction delineated in §26.2 [FK86]. Unfortunately, a counterexample in [Sun97] shows that this method is faulty. Despite this circumstance, we shall assume that the above results are true. We refer the reader to the Appendix for further analysis of this issue.

It is simple to show (§3 [Mezb]) the existence of a map,

(9) Π_{temp}(M̃(C)) →^{′} Π_{temp}(M(C)),

which preserves a character relation analogous to (6), and commutes with parabolic induction.

Combining (8) and (9), we define the local metaplectic correspondence (of tempered representations) by

⊗_{v∈S}π̃_{v} ↦^{′} ⊗_{v∈S}π̃′_{v}, ⊗_{v∈S}π̃_{v} ∈ Π_{temp}(M̃(F_{S})). (10)

The set of genuine standard representations Σ(M̃(F_{S})) is defined to be the set of (equivalence classes of) representations of the form ⊗_{v∈S}π̃M̃ ; v,λ_{v}, where π̃_{v} ∈ Π_{temp}(M̃_{v}(F_{v})), λ_{v} ∈ 𝔞_{M_{v}}^{∗} is regular and M_{v} ∈ L^{M}, for v ∈ S. We generalize the local metaplectic correspondence to Σ(M̃(F_{S})) by

⊗_{v∈S}π̃M̃ ; v,λ_{v} ↦^{′} ⊗_{v∈S}(π̃′_{v,λ′_{v}})^{M}.

It follows from equation (5) and some basic properties of distribution characters that this map preserves a character relation analogous to that of the local metaplectic correspondence.

We now extend the local metaplectic correspondence to Π(M̃(F_{S})). By using the Jacquet modules introduced in §14 [FK86], and following the proof of [Sil78], it can be shown that the Langlands quotient theorem holds for M̃(F_{S}). More precisely, every π̃ ∈ Π(M̃(F_{v})) may be written uniquely as the quotient of some induced representation π̃M̃ ; 1,λ, where π̃_{1} ∈ Π_{temp}(M̃_{1}(F_{v})), λ is in a fixed positive Weyl chamber of 𝔞∗ ; M̃_{1} and M_{1} ∈ L^{M}. With the Langlands quotient theorem in place, we may extend map (10) by assigning the Langlands quotient of the representation π̃M̃ ; 1,λ to the Langlands quotient of (π̃′_{1,λ′})^{M}. This prescribes a map

Π(M̃(F_{S})) →^{′} Π(M(F_{S})). (11)

The reader is warned that, unlike the local metaplectic correspondence on Π_{temp}(M̃(F_{S})) or Σ(M̃(F_{S})), this map does not necessarily preserve any character relations.

The dual formulation of the local metaplectic correspondence is defined in terms of Hecke spaces or Paley-Wiener spaces. A function f̃: M̃(F_{S}) → C is said to be antigenuine if

f̃(γ, ζ) = ζ^{−1}f̃(γ, 1), γ ∈ M(F_{S}), ζ ∈ μ_{n}.

Let C_{c}^{∞}(M̃(F_{S})) be the space of antigenuine, smooth and compactly supported functions on M̃(F_{S}). The Hecke space H(M̃(F_{S})) is defined to be the subspace of functions in C_{c}^{∞}(M̃(F_{S})) which are (K̃_{S} ∩ M̃(F_{S}))-finite under left and right translation.

Given M_{1} ∈ L^{M} and f̃ ∈ H(M̃(F_{S})), we define the map, f̃M̃_{1} on the set Π_{temp}(M̃_{1}(F_{S})) by

f̃M̃_{1}(π̃) = tr (π̃M̃(f̃)) = tr (∫_{M̃(F_{S})} π̃M̃(x)f̃(x)dx) . (12)

This definition is justified by the well-known fact that π̃M̃(f̃) is a trace-class operator. Indeed, π̃ is admissible and f̃ has compact support.

The Paley-Wiener theorem holds for M̃(C) by [Art83] and the fact that M̃(C) splits over M(C). We assume that the trace Paley-Wiener theorem of [BDK86] generalizes to M̃(F_{v}) for nonarchimedean F_{v}. Under this assumption we may take the Paley-Wiener space of M̃(F_{S}) to be

I(M̃(F_{S})) = {f̃M̃ : f̃ ∈ H(M̃(F_{S}))}.

We recall some ideas presented on p. 79 [AC89]. Suppose θ is a continuous linear map from H(M̃(F_{S})) to a topological vector space V. Then θ is said to be supported on characters if it vanishes on any function f̃ with f̃M̃ = 0. If θ is supported on characters then there exists a unique continuous map,

θ̂: I(M̃(F_{S})) → V,

such that

θ̂(f̃M̃) = θ(f), f̃ ∈ H(M̃(F_{S})).

The map f̃ ↦ f̃M̃_{1} defined by (12) is supported on characters and factors through a map ϕ ↦ ϕM̃_{1} from I(M̃(F_{S})) to I(M̃_{1}(F_{S})).

The set of valuations S is said to have the closure property if

𝔞_{M,S} = {H_{M}(γ) : γ ∈ M(F_{S})}

is a closed subgroup of 𝔞_{M}. If S contains an archimedean valuation, it has the closure property. If not, S has the closure property if and only if it is comprised entirely of valuations which divide a fixed rational prime. For the remainder of this paper, S is assumed to have the closure property unless otherwise specified. Put i𝔞_{M,S}^{∗} = i𝔞_{M}^{∗}/iHom(𝔞_{M,S}, Z). The group i𝔞_{M,S}^{∗} inherits a measure from the Euclidean measure defined previously on 𝔞_{M}. We identify ϕ ∈ I(M̃(F_{S})) with its Fourier transform,

ϕ(π̃, X) = ∫_{i𝔞_{M,S}^{∗}} ϕ(π̃_{λ})e^{−λ(X)}dλ.

Define a map f̃ ↦ f̃′ from H(M̃(F_{S})) to functions on Π_{temp}(M(F_{S})) by

(13)  f̃′(π) = \{ \begin{matrix}tr (π̃(f̃)) , & if π̃′ = π \\ 0, & otherwise \end{matrix} \} , π ∈ Π_{temp}(M(F_{S})).

It is simple to check that f̃′ satisfies the axioms defining the Paley-Wiener space listed in [BDK86] and [Art83]. Thus f̃′ belongs to I(M(F_{S})). We define a map,

I(M̃(F_{S})) →^{′} I(M(F_{S})),

which is compatible with (13), by

ϕ′(π, X′) = \{ \begin{matrix}n^{dim(A_{M})}ϕ(π̃, X), & if π̃′ = π \\ 0, & otherwise \end{matrix} \} , π ∈ Π_{temp}(M(F_{S})).

Once again, it is a simple matter to check that the image of this map lies in I(M(F_{S})). To see that this map is compatible with (13), suppose f̃ ∈ H(M̃(F_{S})), π̃ ∈ Π_{temp}(M̃(F_{S})) and X ∈ 𝔞_{M,S}. Then we have

(f̃M̃)′(π̃′, X,) = n^{dim(A_{M})}f̃M̃(π̃, X)
= n^{dim(A_{M})} ∫_{i𝔞_{M,S}^{∗}} tr (π̃_{λ}(f̃)) e^{−λ(X)}dλ
= n^{dim(A_{M})} ∫_{i𝔞_{M,S}^{∗}} f̃′((π̃_{λ})′)e^{−λ(X)}dλ
= ∫_{i𝔞_{M,S}^{∗}} f̃′(π̃′_{λ′})e^{−λ′(X′)}dλ′
= f̃′(π̃′, X′)

as to be desired.

One would hope that map (13) commutes with the map (12). This is indeed the case, but in order to prove this fact we must describe map (13) purely in terms of Hecke spaces. The trace Paley-Wiener theorem tells us that for each f̃ ∈ H(M̃(F_{S})) there exists f ∈ H(M(F_{S})) such that f_{M} = f̃′. The relationship between the Hecke functions, f̃ and f, is expressed in terms of orbital integrals.

Suppose γ̃ ∈ M̃(F_{S}) such that p(γ̃) is semisimple. Let M̃γ̃(F_{S}) denote the centralizer of γ̃ in M̃(F_{S}). We define

IM̃ ; M̃(γ̃, h̃) = |D^{M}(p(γ̃))|_{S}^{1/2} ∫_{M̃γ̃(F_{S})\M̃(F_{S})} h̃(x^{−1}γ̃x)dx, h̃ ∈ C_{c}^{∞}(M̃(F_{S})).

These integrals converge (§6 [FK86], §2 [Art88c]) and are called orbital integrals. Now suppose that γ̃ has Jordan decomposition γ̃ = σ̃s(u), where u = ∏_{v∈S} u_{v} is a unipotent element of M(F_{S}). A general orbital integral is given by

IM̃ ; M̃(γ̃, h̃) = lim_{a→1} IM̃ ; M̃(σ̃s(a^{n}), h̃),

where a = ∏_{v∈S} a_{v} and a_{v} is regular, lying in the center of a parabolic subgroup for which u_{v} is a Richardson element. It is obvious that IM̃ ; M̃(γ̃, h̃) depends only on the conjugacy class of γ̃ in M̃(F_{S}).

For any valuation v set

Λ_{v}^{M}(γ) = |n|_{v}^{dim(M_{γ})/2}, γ ∈ M(F_{v}).

and

Λ^{M}(γ) = ∏_{v∈S} Λ_{v}^{M}(γ_{v}), γ = ∏_{v∈S} γ_{v} ∈ M(F_{S}).

It follows from Proposition 27.3 [FK86], §24 [FK86], §4 [Mezb] and the normalization of our measures that

I_{M}^{M}(γ, f) = Λ^{M}(γ)IM̃ ; M̃(γ′, f̃), γ ∈ M(F_{S}) ∩ G_{oreg}(F_{S}), (14)

where f̃ ∈ H(M̃(F_{S})) and f ∈ H(M(F_{S})) are as above. Any two functions, f̃ ∈ H(M̃(F_{S})) and f ∈ H(M(F_{S})), satisfying (14) are said to match.

Using this terminology, we can characterize the image of the local metaplectic correspondence on Π_{temp}(M̃(F_{S})). An admissible representation π of M(F_{S}) is called metic if there exist matching functions f̃ ∈ H(M̃(F_{S})) and f ∈ H(M(F_{S})) such that tr (π_{1}(f)) ≠ 0 for every subquotient π_{1} of π. Theorem 27.3 [FK86] and Proposition 4.1 [Mezb] imply that the image of the local metaplectic correspondence on Π_{temp}(M̃(F_{S})) is the subset of metic representations in Π_{temp}(M(F_{S})).

With this characterization in hand, we can prove that map (12) and map (13) commute. Suppose f̃ ∈ H(M̃(F_{S})), π ∈ Π_{temp}(M_{1}(F_{S})) and M_{1} ∈ L^{M}. Then, by definition

(f̃M̃_{1})′(π) = \{ \begin{matrix}f̃M̃(π̃M̃), & if π = π̃′, π̃ ∈ Π_{temp}(M̃_{1}(F_{S})) \\ 0, & otherwise \end{matrix} ,

and

(f̃′)_{M_{1}}(π) = \{ \begin{matrix}f̃M̃(π̃), & if π^{M} = π̃′, π̃ ∈ Π_{temp}(M̃(F_{S})) \\ 0, & otherwise \end{matrix}

Thus, in order to show that (f̃M̃_{1})′ = (f̃′)_{M_{1}} we need only show that π is metic if and only if π^{M} is metic. If π is metic then π^{M} is metic since the local metaplectic correspondence commutes with parabolic induction. Conversely, if π^{M} is metic then there exist matching f̃_{1} ∈ H(M̃(F_{S})) and f_{1} ∈ H(M(F_{S})) such that tr (π^{M}(f_{1})) ≠ 0. Applying a well-known descent property for orbital integrals (§7 [FK86]), there exist matching functions f_{1,P̃} ∈ H(M̃_{1}(F_{S})) and f_{1,P} ∈ H(M_{1}(F_{S})) for P ∈ P^{M}(M_{1}) such that

tr (π(f_{1,P})) = tr (π^{M}(f_{1})) ≠ 0.

This means that π is metic.

4 Comparison of orbital integrals at singular elements

In this section we show how the identity of orbital integrals (14) may be generalized to some singular elements of G(F_{S}).

Lemma 4.1 Suppose E is a field and u is a unipotent element of GL(k, E). Then u is conjugate to u^{n} in GL(k, E).

Proof. Set u_{s} to be the s × s matrix of the form

( \begin{matrix}1 & 1 & 0 & 0 \\ 0 & ... & ... & 0 \\ ⋮ & ... & ... & 1 \\ 0 & ⋯ & 0 & 1 \end{matrix} ) .

Jordan canonical form tells us that any unipotent element u of GL(k, E) is conjugate to

( \begin{matrix}u_{s_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & u_{s_{j}} \end{matrix} ) ,

for some positive integers, s_{1}, … s_{j}, satisfying ∑_{i=1}^{j} s_{i} = k. It therefore suffices to prove the lemma in the case that j = 1 and u = u_{k}.

Observe that u − 1 is a nilpotent matrix of order k − 1. Now

u^{n} − 1 = (1 + (u − 1))^{n} − 1 = ∑_{i=0}^{n} \binom{n}{i}(u − 1)^{i} − 1 = ∑_{i=1}^{n} \binom{n}{i}(u − 1)^{i}.

From this it is apparent that u^{n} − 1 is a nilpotent matrix of order k − 1. Another application of Jordan canonical form tells us that u^{n} is conjugate to

( \begin{matrix}u_{b_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & u_{b_{t}} \end{matrix} ) ,

where ∑_{i=1}^{t} b_{i} = k. Clearly,

( \begin{matrix}u_{b_{1}} & & & & 0 \\ & . & . & . \\ 0 & & & & u_{b_{t}} \end{matrix} ) − ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ 0 & & & & 1 \end{matrix} )

is a nilpotent matrix of order max_{1≤i≤t}(b_{i} − 1). This implies that t = 1 and b_{1} = k, whence the lemma. □

Given a nonarchimedean valuation v and a semisimple element σ ∈ M(F_{v}), let U_{M_{σ}}(F_{v}) be the set of unipotent elements in M_{σ}(F_{v}) and let (U_{M_{σ}}(F_{v})) be the corresponding set of M_{σ}(F_{v})-conjugacy classes. The latter set is the parameter set in the Shalika germ expansion of an orbital integral at σ. Such germ expansions exist for M̃(F_{v}) as well ([Vig81]). We shall use these germ expansions to compare orbital integrals at elements which are not regular. As preparation for this, we show that (U_{M_{σ}}(F_{v})) parameterizes the germs in the Shalika germ expansion for M̃(F_{v}).

Lemma 4.2 Suppose v is nonarchimedean, γ̃ ∈ G̃(F_{v}), p(γ̃) has Jordan decomposition σu and γ̃ = σ̃s(u). Suppose further that σu_{0} belongs to the closure of the G(F_{v})-conjugacy class of p(γ̃). Then σ̃s(u_{0}) belongs to the closure of the G̃(F_{v})-conjugacy class of γ̃.

Proof. We prove the lemma in a special case and leave the details of the general case to the interested reader. Suppose r = 2, σ_{1} ∈ F_{v}^{×}, ζ ∈ μ_{n},

σ = ( \begin{matrix}σ_{1} & 0 \\ 0 & σ_{1} \end{matrix} ), σ̃ = i(ζ)s(σ), u = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) and u_{0} is the identity element.

Let x ∈ F_{v}^{×} such that |x|_{v} < 1. Denote the nth Hilbert symbol of F_{v} by (·, ·)_{F_{v}}. Using equation (1) we can show that

s ( \begin{matrix}x^{kn} & 0 \\ 0 & 1 \end{matrix} ) s(u)s ( \begin{matrix}x^{kn} & 0 \\ 0 & 1 \end{matrix} )^{−1} = s ( \begin{matrix}1 & x^{kn} \\ 0 & 1 \end{matrix} ) .

Together with Proposition 0.1.5 [KP86], this implies that

lim_{k→∞} s ( \begin{matrix}x^{kn} & 0 \\ 0 & 1 \end{matrix} ) σ̃s(u)s ( \begin{matrix}x^{kn} & 0 \\ 0 & 1 \end{matrix} )^{−1}

= i(ζ)s(σ) lim_{k→∞}(σ_{1}^{2}, x^{kn})_{F_{v}}^{(1+2m)}(σ_{1}, x^{kn})_{F_{v}}^{−1}(σ_{1}, 1)_{F_{v}}^{−1}s ( \begin{matrix}1 & x^{kn} \\ 0 & 1 \end{matrix} )
= σ̃.

In the general case G_{σ}(F_{v}) is isomorphic to ∏_{i=1}^{k} GL(m_{i}, F_{i}) where F_{1}, …, F_{k} are extensions of F_{v} (§1 [KP86]). One can imitate the above computation in GL(m_{i}, F_{i}) for each 1 ≤ i ≤ k to obtain the lemma.□

In the notation of the previous lemma, §1.j [Vig81] tells us that if the closure of the G(F_{v})-conjugacy class of σu is equal to the disjoint union of the G(F_{v})-conjugacy classes of σu_{1}, …, σu_{k} then there exist ζ_{1}, …, ζ_{k} ∈ μ_{n} such that the closure of the G̃(F_{v})-conjugacy class of σ̃s(u) is equal to the disjoint union of the G̃(F_{v})-conjugacy classes of i(ζ_{1})s(σ)s(u_{1}), …, i(ζ_{k})s(σ)s(u_{k}). Lemma 4.2 implies that ζ_{1} = ⋯ = ζ_{k} and σ̃ = i(ζ_{1})s(σ). In particular, the closure of the two conjugacy classes may be identified by way of the map s acting on the unipotent classes. Bearing in mind decomposition (4), it follows that, (U_{M_{σ}}(F_{v})) is an index set for the germs in the Shalika germ expansion for M̃(F_{v}) via the map s.

Lemma 4.3 Suppose v is nonarchimedean, and σ ∈ G(F_{v}) is a semisimple element which lies in the domain of the orbit map and satisfies G_{σ} = G_{σ^{n}}. Suppose further that f̃ ∈ H(G̃(F_{v})) and f ∈ H(G(F_{v})) match. Then

I_{G}(σ, f) = Λ^{G}(σ)IG̃(σ′, f̃).

Proof. According to Corollaire 2.4 [Vig81], §3.3 [Vig81] and the remarks preceding this lemma, we have the following two germ expansions:

IG̃(γ′, f̃) = ∑_{u∈(U_{Gσ} (F_{v}))} I_{G}(γ^{n}, h_{u})IG̃(σ′s(u), f̃),

I_{G}(γ, f) = ∑_{u∈(U_{Gσ} (F_{v}))} I_{G}(γ, f_{u})I_{G}(σu, f).

In these equations, the element γ belongs to G_{oreg}(F_{v}) and is taken to be in a very small open neighborhood of σ depending on f̃ and f. The functions f_{u}, h_{u} ∈ C_{c}^{∞}(G(F_{v})) have the following properties:

I_{G}(σ^{n}u_{1}, h_{u}) = \{ \begin{matrix}1, & if u_{1} = u \\ 0, & otherwise \end{matrix} \} ,

I_{G}(σu_{1}, f_{u}) = \{ \begin{matrix}1, & if u_{1} = u \\ 0, & otherwise \end{matrix} \} , u, u_{1} ∈ (U_{G_{σ}}(F_{v})).

We may assume that

f_{u}(γ_{1}) = |\frac{D^{G}(σ^{n})}{D^{G}(σ)}|_{v}^{1/2} h_{u}(γ_{1}^{n}) = |n|_{v}^{(r^{2}−dim(G_{σ}))/2}h_{u}(γ_{1}^{n}), γ_{1} ∈ G(F_{v})

( Application 1 §4.1 [Vig81]). Indeed, for u_{1} ∈ (U_{G_{σ}}(F_{v})), we have

I_{G}(σu_{1}, f_{u}) = |D^{G}(σ)|_{v}^{1/2} ∫_{G_{σ}(F_{v})\G(F_{v})} f_{u}(x^{−1}σu_{1}x)dx
= |D^{G}(σ^{n})|_{v}^{1/2} ∫_{G_{σ^{n}}(F_{v})\G(F_{v})} h_{u}(x^{−1}σ^{n}u_{1}^{n}x)dx
= I_{G}(σ^{n}u_{1}^{n}, h_{u})
= \{ \begin{matrix}1, & if u_{1}^{n} = u \\ 0, & otherwise \end{matrix} ,

and by Lemma 4.1, u_{1}^{n} = u_{1}, so f_{u} satisfies the required property. As a consequence I_{G}(γ, f_{u}) equals

|D^{G}(γ)|_{v}^{1/2} ∫_{G_{γ}(F_{v})\G(F_{v})} f_{u}(x^{−1}γx)dx

= |D^{G}(γ^{n})|_{v}^{1/2} |\frac{D^{G}(γ)}{D^{G}(γ^{n})}|_{v}^{1/2} |n|_{v}^{(r^{2}−dim(G_{σ}))/2} ∫_{G_{γ^{n}}(F_{v})\G(F_{v})} h_{u}(x^{−1}γ^{n}x)dx
= |n|_{v}^{(−r^{2}+r+r^{2}−dim(G_{σ}))/2}|D^{G}(γ^{n})|_{v}^{1/2} ∫_{G_{γ^{n}}(F_{v})\G(F_{v})} h_{u}(x^{−1}γ^{n}x)dx
= |n|_{v}^{−(dim(G_{σ})−r)/2}I_{G}(γ^{n}, h_{u}).

The germ expansion for G is thus

I_{G}(γ, f) = ∑_{u∈(U_{Gσ} (F_{v}))} I_{G}(γ^{n}, h_{u}) (|n|_{v}^{−(dim(G_{σ})−r)/2}I_{G}(σu, f)) .

After applying Lemma 4.1 to the germ expansion for G̃, we obtain

IG̃(γ′, f̃) = ∑_{u∈(U_{Gσ} (F_{v}))} I_{G}(γ^{n}, h_{u})IG̃(σ′s(u), f̃).

Let us examine the germ I_{G}(γ^{n}, h_{u}) in more detail. According to §2.5 [Vig81],

I_{G}(γ^{n}, h_{u}) = |\frac{D^{G}(σ^{n}u)D^{G}(γ^{n})}{D^{G_{σ}}(σ^{n}u)D^{G_{σ}}(γ^{n})}|_{v}^{1/2} I_{G_{σ}}(γ^{n}, h_{u}). (15)

Since σ is central in G_{σ}(F_{v}), it follows that I_{G_{σ}}(γ, h_{u}) satisfies a homogeneity property (Application 2 §4.1 [Vig81]), namely

I_{G_{σ}}(γ^{k^{2}}, h_{u}) = |k|_{v}^{dim(G_{σu})−r}I_{G_{σ}}(γ, h_{u}), k ∈ Z.

Together with (15), this implies that

I_{G}(γ^{nk^{2}}, h_{u}) = |k|_{v}^{((r^{2}−dim G_{σ})/2)+dim(G_{σu})−r}I_{G}(γ^{n}, h_{u}).

Hence the germs, I_{G}(γ^{n}, h_{u}), for u ∈ (U_{G_{σ}}(F_{v})), may be said to be homogeneous of degree dim(G_{σu}) respectively. It is clear that nonzero germs of different degree are linearly independent. Now suppose γ ∈ G_{oreg}(F_{v}) is F_{v}-elliptic in G_{σ}(F_{v}) and is close to σ. The difference,

Λ_{v}^{G}(γ)IG̃(γ′, f̃) − I_{G}(γ, f)

= ∑_{u∈(U_{Gσ} (F_{v}))} (Λ_{v}^{G}(γ)IG̃(σ′s(u), f̃) − |n|_{v}^{−(dim(G_{σ})−r)/2}I_{G}(σu, f)) I_{G}(γ^{n}, h_{u}),

vanishes since f̃ matches f. It follows from Proposition 2.1 [Rog83] and Theorem 2.2 [Rog83] that the germ I_{G}(γ^{n}, h_{1}) is not zero. It is also the only germ in this linear combination of degree dim(G_{σ}). Its coefficient must vanish by linear independence. This proves the lemma.□

Lemma 4.4 Suppose σ ∈ G(C) is semisimple and G_{σ} = G_{σ^{n}}. Suppose further that f̃ ∈ H(G̃(C)) matches f ∈ H(G(C)). Then

I_{G}(σ, f) = Λ^{G}(σ)IG̃(σ′, f̃).

Proof. We first consider the case σ = 1. In this case we must show that

f(1) = |n|_{C}^{r^{2}/2}f̃(1).

This equation was shown in [Mezb] for a specific choice of f matching f̃. However, it follows from the Weyl integration formula and the Plancherel formula that the value at the identity of any function matching f̃ is equal to f(1). Now suppose that σ ∈ G(C) is any semisimple element satisfying G_{σ} = G_{σ^{n}}. Since we will only be considering orbital integrals at semisimple elements, Jordan canonical form allows us to assume that our orbital integrals taken are taken at elements in M_{0}(C). Clearly, we may assume that σ ∈ A_{M}(C) for some M ∈ L, so that G_{σ} = M. Fix P ∈ P(M) and set f̃ σ^{n} ; P̃(γ) equal to

|det(1 − Ad(γσ^{n}))_{|𝔤/𝔪}|_{C}^{1/2} ∫_{U_{P} (C)} ∫_{K_{C}} f̃(s(k)^{−1}s(u)^{−1}s(γσ^{n})s(u)s(k))dudk, where γ ∈ M(C) and 𝔪 is the (real) Lie algebra of M(C). For γ ∈ M_{0}(C) ∩ G_{oreg}(C) in a sufficiently small neighborhood V_{σ} ⊂ M_{0}(C) of the identity, we have γσ ∈ M_{0}(C) ∩ G_{oreg}(C). It then follows from a standard descent formula (§7 [FK86]) that

Λ^{G}(γσ)IG̃((γσ)′, f̃) = Λ^{M}(γ)IM̃ ; M̃(γ′, f̃ σ^{n} ; P̃), γ ∈ V_{σ} ∩ G_{oreg}(C).

After making similar definitions, we find that

I_{G}(γσ, f) = I_{M}^{M}(γ, f_{P}^{σ}), γ ∈ V_{σ} ∩ G_{oreg}(C).

Combining the last two equations with the matching property, we obtain

Λ^{M}(γ)IM̃ ; M̃(γ, f̃ σ^{n} ; P̃) = I_{M}^{M}(γ, f_{P}^{σ}), γ ∈ V_{σ} ∩ G_{oreg}(C). (16)

The functions f̃ σ^{n} ; P̃ and f_{P}^{σ} also satisfy the identities,

Λ^{M}(1)f̃ σ^{n} ; P̃(1) = Λ^{G}(σ^{n}) |det(1 − Ad(σ^{n}))_{|𝔤/𝔤_{σ^{n}}} |_{C}^{1/2} ∫_{G̃_{σ′}(C)\G̃(C)} f̃(x^{−1}σ′x)dx
= Λ^{G}(σ^{n})IG̃(σ′, f̃),

and f_{P}^{σ}(1) = I_{G}(σ, f). The lemma therefore follows if we show that

f_{P}^{σ}(1) = Λ^{M}(1)f̃ σ^{n} ; P̃(1).

According to [Mezb], there exists a function h ∈ H(M(C)) matching f̃ σ^{n} ; P̃ ∈ H(M̃(C)). From our proof of the case σ = 1 it follows that

h(1) = Λ^{M}(1)f̃ σ^{n} ; P̃(1).

Thus, if h(1) = f_{P}^{σ}(1) the lemma is complete. This last equation follows from the fact that h(1) is determined by the values of I_{M}^{M}(·, h) restricted to any neighborhood of the identity in M_{0}(C). More precisely, there exists a differential operator ∂(ϖ) on M_{0}(C) such that h(1) = ∂(ϖ)I_{M}^{M}(1, h) (Theorem 11.17 [Kna86]). Notice that equation (16) and the matching property imply

I_{M}^{M}(γ, h) = I_{M}^{M}(γ, f_{P}^{σ}), γ ∈ V_{σ} ∩ G_{oreg}(C).

Since V_{σ} ∩G_{oreg}(C) contains an open subset of V_{σ}, the previous two equations imply that

h(1) = ∂(ϖ)I_{M}^{M}(1, h) = ∂(ϖ)I_{M}^{M}(1, f_{P}^{σ}) = f_{P}^{σ}(1).□

The reader might wonder whether one might be able to remove the hypothesis G_{σ} = G_{σ^{n}} from the previous two lemmas. The following example shows that, at least for Lemma 4.4, this hypothesis cannot be removed. Suppose F_{v} = C, n = r = 2 and f̃ ∈ H(G̃(C)) such that f̃(1) ≠ 0. It is a simple exercise to show that the function given by,

f(γ) = |\frac{D^{G}(γ^{n})}{D^{G}(γ)}|_{C}^{1/2} f̃(γ′), γ ∈ G_{oreg}(C),

extends to a smooth function on G(C). More precisely, if γ ∈ G(C) has eigenvalues γ_{1}, γ_{2} ∈ C then

f(γ) = |(γ_{1} + γ_{2})^{2}/γ_{1}γ_{2}|_{C}^{1/2} f̃(γ′).

It is not difficult to show that f ∈ H(G(C)) and that f̃ matches f. However, for the semisimple element σ = diag(1, −1) ∈ G(C) the equation of Lemma 4.4 does not hold. Indeed, the equation,

f(x^{−1}σx) = 0, x ∈ G_{σ}(C)\G(C),

implies that I_{G}(σ, f) = 0, but Λ^{G}(σ)IG̃(σ′, f̃) is equal to n^{4}f̃(1), which by assumption is not zero.

This example suggests that we should restrict ourselves to the comparison of orbital integrals at elements in the subset M_{comp}(F_{S}) of M(F_{S}), defined by

M_{comp}(F_{S}) = {γ = ∏_{v∈S} γ_{v} ∈ M(F_{S}) : M_{σ_{v}} = M_{σ_{v}^{n}}, σ_{v} semisimple part of γ_{v}}.

The set M_{comp}(F_{S}) is a dense open subset of M(F_{S}).

Lemma 4.5 Suppose γ ∈ G(F_{S}) lies in the domain of the orbit map and has Jordan decomposition σu. Suppose further that that f̃ ∈ H(G̃(F_{S})) matches f ∈ H(G(F_{S})). Then

I_{G}(γ, f) = Λ^{G}(γ)IG̃(γ′, f̃), γ ∈ G_{comp}(F_{S}).

Proof. It suffices to prove the lemma in the case that S consists of a single valuation v. The unipotent element u is contained in the Richardson orbit of some parabolic subgroup P_{σ^{n}} = M_{σ}U_{σ} of G_{σ}. Let A_{σ} be the center of M_{σ}. By equation (1) and definition (1.3) II [AC89], we have

IG̃(γ′, f̃) = IG̃(σ′s(u^{n}), f̃) = lim_{a→1} IG̃(s(a^{n})σ′, f̃),

where a is regular in A_{σ}(F_{v}). We also have the parallel definition,

I_{G}(γ, f) = I_{G}(σu, f) = lim_{a→1} I_{G}(aσ, f).

The lemma now follows from these two definitions and the matching of f̃ and f.□

5 Normalization of intertwining operators and the Plancherel formula

Our goal here is to normalize the intertwining operators between induced representations. This is necessary for the definition of the distributions occurring in the invariant trace formula of Arthur ([Art89a] and [Art88a]).

Our method of normalization follows §2 II [AC89]. In particular, we make use of the Plancherel formula ([HC84]). The proof of this formula has not been carried out for G̃(F_{v}). We nevertheless assert that the Plancherel formula holds in this case as well. In what follows, we justify this assertion by listing the properties of reductive algebraic groups used in Harish-Chandra’s proof of the Plancherel formula and show that they also hold for G̃(F_{v}). In this justification we assume that v is nonarchimedean.

Suppose P ∈ P(M) such that P ⊃ P_{0}. Then G̃(F_{v}) splits over U_{P} (F_{v}) (§2 [FK86]). The splitting homomorphism is s. In other words,

s(U_{P} (F_{v})) = {(u, 1) : u ∈ U_{P} (F_{v})}

is a subgroup of G̃(F_{v}). Every parabolic subgroup in P(M) is of the form P^{w} = w^{−1}P w for some representative w of the Weyl group W_{0}^{G}. It is easy to check that G̃(F_{v}) splits over U_{P^{w}}(F_{v}) = w^{−1}U_{P} (F_{v})w with splitting homomorphism s_{w} defined by

w^{−1}uw ↦ s(w)^{−1}s(u)s(w).

Clearly P̃^{w}(F_{v}) = M̃(F_{v})s_{w}(U_{P^{w}}(F_{v})) as M̃(F_{v}) is stable under conjugation by s(w).

We define the Jacquet module of an admissible representation (π̃, V) of finite length with respect to U_{P^{w}}(F_{v}) in the following way. Let V_{U_{P^{w}}} be the linear span of

{π̃(s_{w}(w^{−1}uw))v − v : u ∈ U_{P} (F_{v}), v ∈ V }.

It is a consequence of equation (1) that M̃(F_{v}) normalizes s_{w}(U_{P^{w}}(F_{v})). Thus M̃(F_{v}) stabilizes V_{U_{P^{w}}}. We define the Jacquet module of π̃ with respect to U_{P^{w}} to be the representation obtained by twisting the quotient representation V/V_{U_{P^{w}}} with the modular function δ_{P^{w}}^{−1/2}. We denote this representation by π̃_{U_{P^{w}}}. This is a mild generalization of §14 [FK86]. It is left to the reader to check that this definition yields the expected properties of Jacquet modules.

Another consequence of the splitting of G̃(F_{v}) over U_{P^{w}}(F_{v}) is the decomposition,

G̃(F_{v}) = M̃(F_{v})s_{w}(U_{P^{w}}(F_{v}))K_{v} = M̃(F_{v})s(w)^{−1}s(U_{P} (F_{v}))s(w)K_{v}.

The associated integration formula follows in the usual fashion.

Suppose, for this paragraph only, that w is the representative of W_{0}^{G} such that the opposite parabolic subgroup P̄ is equal to P^{w}. Then we obtain the Gelfand-Naimark decomposition, which is a decomposition of an open dense subset of G̃(F_{v}) as

s_{w}(U_{P^{w}}(F_{v}))M̃(F_{v})s(U_{P} (F_{v})).

Its associated integration formula is given by

∫_{G̃(F_{v})} f̃(x)dx = γM̃ ∫_{s_{w}(U_{P^{w}} (F_{v}))} ∫_{M̃(F_{v})} ∫_{s(U_{P} (F_{v}))} f̃(ūm̃u)dudm̃dū,

where γM̃ = ∫_{s_{w}(U_{P^{w}} (F_{v}))} δ_{P} (m_{P} (p(ū)))dū = γ_{M} (17)

ū = u_{P} (ū)m_{P} (ū)k_{P} (ū), u_{P} (ū) ∈ U_{P} (F_{v}), m_{P} (ū) ∈ M(F_{v}) and k_{P} (ū) ∈ K_{v}. The results of this paragraph do not rely on the assumption that P ⊃ P_{0}.

The abelian group,

Ã_{M}^{n/d}(F_{v}) = {γ̃ ∈ M̃(F_{v}) : p(γ̃) = γ^{n/d}, γ ∈ A_{M}(F_{v})},

lies in the center of M̃(F_{v}) (Proposition 0.1.1 [KP84]) and provides a decomposition of s_{w}(U_{P^{w}}(F_{v})) by way of the adjoint action. Using this decomposi-

tion we may define a subset Ã_{M}^{n/d}^{−}(F_{v}) of Ã_{M}^{n/d}(F_{v}) as in §1.4 [Cas93]. This subset may be used to examine the asymptotic behavior of matrix coefficients (cf. §4 [Cas93]). The only other ingredient needed to prove the results concerning the asymptotic behavior of matrix coefficients for G̃(F_{v}) is the Iwahori factorization for arbitrarily small compact open subgroups. These Iwahori factorizations exist in G̃(F_{v}), because there exist arbitrarily small compact open subgroups of G(F_{v}) over which G̃(F_{v}) splits (§2 [FK86]), and Iwahori factorizations hold in G(F_{v}) (Proposition 1.4.4 [Cas93]).

The only decomposition which still needs to be addressed for G̃(F_{v}) is the Cartan decomposition. This may be recast as

G̃(F_{v}) = ⋃_{γ} K_{v}γÃ_{M_{0}}^{n/d}(F_{v})K_{v},

where γ runs over a set of representatives of Ã_{M_{0}}(F_{v})/Ã_{M_{0}}^{n/d}(F_{v}). This union is finite and disjoint. It is the finiteness which allows us to restrict our attention

to K_{v}Ã_{M_{0}}^{n/d}(F_{v})K_{v} when proving the convergence of integrals or bounds of certain functions.

We make one further remark concerning bounding functions on G̃(F_{v}). If f̃ is a genuine or antigenuine function on G̃(F_{v}) then clearly

sup_{γ̃∈G̃(F_{v})} |f̃(γ̃)| = sup_{γ̃∈G̃(F_{v})} |f̃(p(γ̃), 1)| = sup_{γ∈G(F_{v})} |f̃(s(γ))|.

Therefore, in cases where one is interested in finding uniform bounds of such functions, the techniques of the non-metaplectic groups may be used.

This concludes the discussion of the properties necessary for the proof of the Plancherel formula. The proof may now be imitated after making some apparent definitions.

The normalization of intertwining operators amounts to the definition of functions,

r_{Q̃|P̃} : Π(M̃(F_{S})) × 𝔞_{M,C}^{∗} → C, Q, P ∈ P(M),

which satisfy the conditions of Theorem 2.1 [Art89a]. These functions are called normalizing factors. Such normalizing factors exist for general linear groups ([Sha84], §4 [Art89a]). Since we may take the normalizing factors for Π(M̃(F_{S})) to be the product of the normalizing factors for Π(M̃(F_{v})), v ∈ S, it suffices to consider the case that S consists of a single valuation. As mentioned above, we already have normalizing factors of the form r_{Q|P}. We define candidates for normalizing factors of metaplectic coverings by setting

r_{Q̃|P̃}(π̃_{λ}) = Λ^{M}(1)^{1/2}Λ^{G}(1)^{−1/2}r_{Q|P} (π̃′_{λ′}),

for all π̃ ∈ Π(M̃(F_{S})) and λ ∈ 𝔞_{M,C}^{∗}. As indicated in §2 II [AC89] and §§2, 4 [Art89a], there is only one required property which r_{Q̃|P̃} does not obviously satisfy. Let Π_{disc}(M̃(F_{v})) be the subset of representations in Π_{temp}(M̃(F_{v})) which are square-integrable modulo the center of M̃(F_{v}). The property which remains to be verified is

r_{P̃| ˜P̄}(π̃_{λ})r_{˜P̄|P̃}(π̃_{λ}) = J_{P̃| ˜P̄}(π̃_{λ})J_{˜P̄|P̃}(π̃_{λ}), π̃ ∈ Π_{disc}(M̃(F_{v})), λ ∈ 𝔞_{M,C}^{∗}.

The right-hand side is, by definition, the inverse of the Harish-Chandra μfunction μM̃ (not to be confused with the group μ_{n}^{M}).

Lemma 5.1 Suppose π̃ ∈ Π_{disc}(M̃(F_{v})) and λ ∈ 𝔞_{M,C}^{∗}. Then

r_{P̃| ˜P̄}(π̃_{λ})r_{˜P̄|P̃}(π̃_{λ}) = μM̃(π̃_{λ})^{−1}.

Proof. The normalizing factors for P, as defined in [Sha84], already satisfy the condition of the lemma. Furthermore, μM̃ is meromorphic in 𝔞_{M,C}^{∗}. Therefore it suffices to show that

μM̃(π̃) = Λ^{M}(1)Λ^{G}(1)^{−1}μ_{M}(π̃′), π̃ ∈ Π_{disc}(M̃(F_{v})).

Suppose f̃ ∈ H(G̃(F_{v})) such that

f̃M̃_{1}(π̃) = 0, π̃ ∈ Π_{disc}(M̃_{1}(F_{v})), M_{1} ∈ L, M_{1} ≠ M.

It follows from the Plancherel formula for G̃(F_{v}) that

f̃(1) = γM̃ ∑_{Π_{disc}(M̃(F_{v}))/i𝔞_{M,{v}}^{∗}} ∫_{i𝔞_{M,{v}}^{∗}} dM̃(π̃_{λ})μM̃(π̃_{λ})f̃M̃(π̃_{λ})dλ,

where dM̃(π̃_{λ}) is the formal degree of π̃_{λ}. Suppose f ∈ H(G(F_{v})) such that f_{G} = f̃′ (Proposition 27.3 [FK86]). Then, by Lemma 4.3 and Lemma 4.4, we have

f̃(1) = IG̃(1, f̃) = Λ^{G}(1)^{−1}I_{G}(1, f).

Combined with the Plancherel formula and equation (17), this equation implies that f̃(1) is equal to the product of Λ^{G}(1)^{−1} with

γ_{M} ∑_{Π_{disc}(M(F_{v}))/i𝔞_{M,{v}}^{∗}} ∫_{i𝔞_{M,{v}}^{∗}} d^{M}(π_{λ′})μ_{M}(π_{λ′})f̃′_{M}(π_{λ′})dλ′

= γM̃ ∑_{Π_{disc}(M̃(F_{v}))/i𝔞_{M,{v}}^{∗}} ∫_{i𝔞_{M,{v}}^{∗}} d^{M}((π̃_{λ})′)μ_{M}((π̃_{λ})′)f̃M̃(π̃_{λ})dλ′

= n^{dim(A_{M})}γM̃ ∑_{Π_{disc}(M̃(F_{v}))/i𝔞_{M,{v}}^{∗}} ∫_{i𝔞_{M,{v}}^{∗}} d^{M}((π̃_{λ})′)μ_{M}((π̃_{λ})′)f̃M̃(π̃_{λ})dλ

Taking the difference of this equation from the earlier Plancherel expansion of f̃(1) we find that

∫_{i𝔞_{M,{v}}^{∗}} (dM̃(π̃)μM̃(π̃_{λ}) − d^{M}(π̃′)n^{dim(A_{M})}Λ^{G}(1)^{−1}μ_{M}(π̃′_{λ′})) f̃M̃(π̃_{λ})dλ

vanishes for all π̃ ∈ Π_{disc}(M̃(F_{v})). An application of the trace Paley-Wiener formula (for metaplectic coverings) to this equation implies that

dM̃(π̃)μM̃(π̃) = d^{M}(π̃′)n^{dim(A_{M})}Λ^{G}(1)^{−1}μ_{M}(π̃′), π̃ ∈ Π_{disc}(M̃(F_{v})).

Notice that μG̃ = μ_{G} = 1 and so

dG̃(π̃_{0}) = nΛ^{G}(1)^{−1}d^{G}(π̃′_{0}), π̃_{0} ∈ Π_{disc}(G̃(F_{v})).

It follows in general that

dM̃(π̃) = ∏_{i=1}^{ℓ} d^{M̃(i)}(π̃_{i}) = ∏_{i=1}^{ℓ} nΛ^{M(i)}(1)^{−1}d^{M(i)}(π̃′_{i}) = n^{dim(A_{M})}Λ^{M}(1)^{−1}d^{M}(π̃′),

where M = M(1) × ⋯ × M(ℓ) is decomposition (4) and π̃_{1}, …, π̃_{ℓ} are as in the Appendix. The lemma is now complete.□

6 The distribution I_{M}^{Σ}(γ)

The invariant trace formula for G contains invariant distributions of the form I_{M}(γ) parameterized by γ ∈ M(F_{S}). These distributions were introduced in §2 [Art88a]. The reader is assumed to be familiar with their definitions and properties. Our goal is to compare these distributions with the analogous distributions on G̃(F_{S}) using the orbit map. The orbit map is not injective. Consequently we must group the distributions I_{M}(γ) in terms of the fibers of the orbit map. Since the orbit map raises elements of G̃(F_{S}) to the nth power, a good guess would be that we should group the distributions in terms of the nth roots of unity μ_{n}. According to decomposition (4),

M(F_{S}) ≅ GL(r_{1}, F_{S}) × ⋯ × GL(r_{ℓ}, F_{S}).

The center of GL(r_{i}, F_{S}) is isomorphic to GL(1, F_{S}). In consequence,

A_{M}(F_{S}) ≅ ∏_{i=1}^{ℓ} GL(1, F_{S}).

Let μ_{n}^{M} be the finite subgroup of A_{M}(F_{S}) corresponding to the subgroup generated by the diagonal embedding of ∏_{i=1}^{ℓ} μ_{n} into ∏_{i=1}^{ℓ} GL(1, F_{S}). Suppose γ ∈ M(F_{S}), η ∈ μ_{n}^{M} and that (ηγ)′ is defined. Then by Theorem 4.1 (iii)

[KP86] we have (ηγ)′ = s((ηγ)^{n})κ_{v}((ηγ)^{n})^{−1} = γ′. (18)

This equation suggests that we ought to group our distributions into μ_{n}^{M}- invariant sums. The most obvious method is to take the sum,

∑_{η∈μ_{n}^{M}} I_{M}(ηγ, f), f ∈ H(M(F_{S})).

This grouping has the shortcoming that if M = G and f̃ ∈ H(G̃(F_{S})) then

∑_{η∈μ_{n}^{G}} Î_{G}(ηγ, f̃′) = nΛ^{G}(γ)IG̃(γ′, f̃) ≠ Λ^{G}(γ)IG̃(γ′, f̃), γ ∈ ∩G_{oreg}(F_{S}).

It would be convenient to correct this shortcoming by replacing the sum over μ_{n}^{M} with the sum over μ_{n}^{M}/μ_{n}^{G}. The following two lemmas show that this is indeed possible. We assume the reader is familiar with the spaces H_{ac}(M(F_{S})) and I_{ac}(M(F_{S})) of §11 [Art89a] and their respective subspaces of moderate functions.

Lemma 6.1 Suppose L ∈ L(M) and ϕ is a moderate function in I_{ac}(M(F_{S})) such that ϕ(π) vanishes for all π ∈ Π_{temp}(M(F_{S})) whose central character is not trivial on μ_{n}^{L}. Then there exists a moderate function f ∈ H_{ac}(M(F_{S})) such that f_{M} = ϕ and f is invariant under translation by μ_{n}^{L}.

Proof. By Lemma 6.1 [Art88a] there exists a moderate function h ∈ H_{ac}(M(F_{S})) such that h_{M} = ϕ. Given η ∈ μ_{n}^{L}, set

h^{η}(γ) = h(ηγ), γ ∈ M(F_{S}).

Since μ_{n}^{L} is contained in the center of M(F_{S}) and the nonzero matrix entries of η ∈ μ_{n}^{L} have absolute value one with respect to any valuation in S, the K_{S}-finiteness, support and growth conditions of h^{η} are identical with those of h. Thus, the function given by

f(γ) = |μ_{n}^{L}|^{−1} ∑_{η∈μ_{n}^{L}} h^{η}(γ), γ ∈ M(F_{S}),

is a moderate function in H_{ac}(M(F_{S})). It is simple to see that f satisfies the remaining properties asserted by the lemma.□

Lemma 6.2 Suppose L ∈ L(M), η ∈ μ_{n}^{L} and f̃ ∈ H(L̃(F_{S})). Then

Î_{M}^{L}(ηγ, f̃′) = Î_{M}^{L}(γ, f̃′), γ ∈ M(F_{S}).

Proof. Suppose γ ∈ M(F_{S}) such that M_{γ}(F_{S}) = G_{γ}(F_{S}). By Lemma 6.1 we may choose f ∈ H(L(F_{S})) such that f_{L} = f̃′ and f is invariant under μ_{n}^{L}. If L = M then the lemma follows immediately from the μ_{n}^{L}-invariance of f and μ_{n}^{L} ⊂ A_{L}(F_{S}). Assume inductively that the lemma holds if L is replaced by L_{1} ∈ L^{L}(M) such that L_{1} ⫋ L. By definition

Î_{M}^{L}(ηγ, f̃′) = I_{M}^{L}(ηγ, f)
= J_{M}^{L}(ηγ, f) − ∑_{{L_{1}∈L^{L}(M):L_{1}⫋L}} Î_{M}^{L_{1}}(ηγ, ϕ_{L_{1}}^{L}(f)).

The first term on the right-hand side satisfies

J_{M}^{L}(ηγ, f) = |D^{L}(ηγ)|_{S}^{1/2} ∫_{L_{ηγ}(F_{S})\L(F_{S})} f(x^{−1}ηγx)v_{M}^{L}(x)dx = J_{M}^{L}(γ, f),

as f is μ_{n}^{L}-invariant and μ_{n}^{L} ⊂ A_{L}(F_{S}). As for the remaining terms on the right-hand side, we know by Theorem 12.1 [Art89a] that ϕ_{L_{1}}^{L}(f) is a moderate function in I_{ac}(L_{1}(F_{S})). Furthermore ϕ_{L_{1}}^{L}(f) is defined by

ϕ_{L_{1}}^{L}(f)(π) = tr (R_{L_{1}}^{L}(π)π^{L}(f)) , π ∈ Π_{temp}(L_{1}(F_{S})),

for a certain linear operator R_{L_{1}}^{L}(π). Obviously,

ϕ_{L_{1}}^{L}(f)(π) = tr (R_{L_{1}}^{L}(π) ∫_{μ_{n}^{L}\L(F_{S})} f(x) ∑_{η∈μ_{n}^{L}} π^{L}(ηx)dx) , π ∈ Π_{temp}(L_{1}(F_{S})).

It follows that ϕ_{L_{1}}^{L}(f) satisfies the hypotheses of Lemma 6.1 and so there exists a moderate function h ∈ H_{ac}(L_{1}(F_{S})) such that h_{L_{1}} = ϕ_{L_{1}}^{L}(f) and h is invariant under μ_{n}^{L}. Apparently, μ_{n}^{L} ⊂ μ_{n}^{L_{1}}, so by induction

Î_{M}^{L_{1}}(ηγ, ϕ_{L_{1}}^{L}(f)) = I_{M}^{L_{1}}(ηγ, h) = I_{M}^{L_{1}}(γ, h) = Î_{M}^{L_{1}}(γ, ϕ_{L_{1}}^{L}(f)).

In consequence, the lemma holds if M_{γ}(F_{S}) = G_{γ}(F_{S}). For arbitrary γ ∈ M(F_{S}) we have ((2.2) [Art88a])

I_{M}^{L}(ηγ, f) = lim_{{a→1:a∈A_{M,reg}(F_{S})}} ∑_{L_{1}∈L^{L}(M)} r_{M}^{L_{1}}(ηγ, a)I_{L_{1}}^{L}(aηγ, f)
= lim_{{a→1:a∈A_{M,reg}(F_{S})}} ∑_{L_{1}∈L^{L}(M)} r_{M}^{L_{1}}(ηγ, a)I_{L_{1}}^{L}(aγ, f),

where A_{M,reg}(F_{S}) is the subset of elements in A_{M}(F_{S}) whose centralizer in G is contained in M. The function r_{M}^{L}(γ, a), defined in §5 [Art88c], is easily seen to be invariant under translation by A_{L}(F_{S}) in the first variable. In particular we have r_{M}^{L}(ηγ, a) = r_{M}^{L}(γ, a). The lemma now follows.□

Suppose γ ∈ M(F_{S}). By Lemma 6.2, it makes sense to define the distribution I_{M}^{Σ}(γ) on G̃(F_{S}) by

I_{M}^{Σ}(γ, f̃) = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} Î_{M}(ηγ, f̃′), f̃ ∈ H(G̃(F_{S})).

More generally, if L ∈ L(M) then

I_{M}^{L,Σ}(γ, f̃) = ∑_{η∈μ_{n}^{M} /μ_{n}^{L}} Î_{M}^{L}(ηγ, f̃′), f̃ ∈ H(L̃(F_{S})).

Two important properties that I_{M}(γ) satisfies are descent (§8 [Art88a]) and splitting (§9 [Art88a]). We show that I_{M}^{Σ}(γ) satisfies these properties as well.

Lemma 6.3 Suppose M and M_{1} belong to L and M_{1} ⊂ M. Suppose further that L ∈ L(M_{1}) such that d_{M_{1}}^{G}(M, L) ≠ 0. Then the map,

μ_{n}^{M}/μ_{n}^{G} × μ_{n}^{L}/μ_{n}^{G} → μ_{n}^{M_{1}}/μ_{n}^{G},

given by (η_{1}μ_{n}^{G}, η_{2}μ_{n}^{G}) ↦ η_{1}η_{2}μ_{n}^{G}, is an isomorphism.

Proof. If d_{M_{1}}^{G}(M, L) ≠ 0 as above, then by definition, 𝔞_{M_{1}}^{M} ⊕ 𝔞_{M_{1}}^{L} ≅ 𝔞_{M_{1}}^{G} (§7 [Art88a]). The vector spaces 𝔞_{M}^{G} and 𝔞_{L}^{G} may be regarded as the respective orthogonal complements of 𝔞_{M_{1}}^{M} and 𝔞_{M_{1}}^{L} in 𝔞_{M_{1}}^{G}. As a consequence we also have 𝔞_{M}^{G} ⊕ 𝔞_{L}^{G} ≅ 𝔞_{M_{1}}^{G}. Consider the homomorphism

H_{M_{1}} : M_{1}(F_{S}) → 𝔞_{M_{1}}.

It is readily verified that it passes to a homomorphism

H_{A_{M_{1}}} : A_{M_{1}}(F_{S})/A_{G}(F_{S}) → 𝔞_{M_{1}}^{G}

such that H_{A_{M_{1}}}(A_{M}(F_{S})/A_{G}(F_{S})) ⊂ 𝔞_{M}^{G} and H_{A_{M_{1}}}(A_{L}(F_{S})/A_{G}(F_{S})) ⊂ 𝔞_{L}^{G}. Accordingly

H_{A_{M_{1}}}((A_{M}(F_{S}) ∩ A_{L}(F_{S}))/A_{G}(F_{S})) ⊂ 𝔞_{M}^{G} ∩ 𝔞_{L}^{G} = 0.

In other words, |ξ(γ)| = 1 for all γ belonging to the split torus A_{M}(F_{S}) ∩ A_{L}(F_{S}), and all characters ξ ∈ X(M_{1}) which are trivial when restricted to G. This implies that A_{L}(F_{S})∩A_{M}(F_{S}) ⊂ A_{G}(F_{S}). As a result, the multiplication map

A_{M}(F_{S})/A_{G}(F_{S}) × A_{L}(F_{S})/A_{G}(F_{S}) → A_{M_{1}}(F_{S})/A_{G}(F_{S})

is injective. It now follows from the commutative diagram,

μ_{n}^{M}/μ_{n}^{G} × μ_{n}^{L}/μ_{n}^{G} ↪ A_{M}(F_{S})/A_{G}(F_{S}) × A_{L}(F_{S})/A_{G}(F_{S}) ↓ μ_{n}^{M_{1}}/μ_{n}^{G} ↪ A_{M_{1}}(F_{S})/A_{G}(F_{S}) ↓

that the map of the lemma is injective. The surjectivity of the map can be seen from the following equalities.

|μ_{n}^{M}/μ_{n}^{G} × μ_{n}^{L}/μ_{n}^{G}| = n^{dim(𝔞_{M}^{G})}n^{dim(𝔞_{L}^{G})} = n^{dim(𝔞_{M_{1}}^{G})} = |μ_{n}^{M_{1}}/μ_{n}^{G}|.□

Lemma 6.4 Let M, M_{1} and L be as in Lemma 6.3. The map

μ_{n}^{M}/μ_{n}^{G} → μ_{n}^{M_{1}}/μ_{n}^{L},

given by ημ_{n}^{G} ↦ ημ_{n}^{L} for η ∈ μ_{n}^{M}, is an isomorphism.

Proof. The proof of this lemma follows from arguments similar to those of Lemma 6.3.□

The following proposition proves a descent property for I_{M}^{Σ}(γ). For this we need the notion of an induced space of orbits. Given γ ∈ M(F_{S}) define the induced space γ^{G} to be the union of the G(F_{S})-conjugacy classes which intersect γU_{P} (F_{S}) in an open set. Here P ∈ P(M) is arbitrary (§6 [Art88c]). Since γ^{G} is contained in a single geometric conjugacy class and GL(r) is stable, the induced space γ^{G} is a single conjugacy class which we call the induced conjugacy class of γ.

Proposition 6.1 Suppose M and M_{1} belong to L with M_{1} ⊂ M. Moreover suppose γ ∈ M_{1}(F_{S}). Then

I_{M}^{Σ}(γ^{M}, f̃) = ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L)Î_{M_{1}}^{L,Σ}(γ, f̃L̃), f̃ ∈ H(G̃(F_{S})).

Proof. Suppose η ∈ μ_{n}^{M}. Since η lies in the center of M(F_{S}),

ηγ^{M} = {ηγ_{1} : γ_{1} ∈ γ^{M}}

is a conjugacy class in M(F_{S}). Suppose P ∈ P^{M}(M_{1}). Clearly, left multiplication by η is a homeomorphism between γU_{P} (F_{S}) and ηγU_{P} (F_{S}). It follows that ηγ^{M} is a conjugacy class of M(F_{S}) which intersects ηγU_{P} (F_{S}) in an open set. In other words ηγ^{M} = (ηγ)^{M}. The descent property for I_{M}(γ) (Theorem 8.1 [Art88a]) together with Lemma 6.4 imply that

I_{M}^{Σ}(γ^{M}, f̃) = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} Î_{M}(ηγ^{M}, f̃′)
= ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} Î_{M}((ηγ)^{M}, f̃′)
= ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L) ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} Î_{M_{1}}^{L}(ηγ, f̃′_{L})
= ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L) ∑_{η∈μ_{n}^{M_{1}}/μ_{n}^{L}} Î_{M_{1}}^{L}(ηγ, f̃′_{L})

= ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L)Î_{M_{1}}^{L,Σ}(γ, f̃L̃).□

Corollary 6.1 Suppose M and M_{1} belong to L with M_{1} ⊂ M. Moreover suppose γ ∈ M_{1}(F_{S}) and M_{1,γ} = M_{γ}. Then

I_{M}^{Σ}(γ, f̃) = ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L)Î_{M_{1}}^{L,Σ}(γ, f̃L̃), f̃ ∈ H(G̃(F_{S})).

Proposition 6.2 Suppose S is the disjoint union of nonempty sets S_{1} and S_{2}, and that f̃ = f̃_{1}f̃_{2} ∈ H(G̃(F_{S})), γ = γ_{1}γ_{2} ∈ M(F_{S}) are corresponding decompositions. Then

I_{M}^{Σ}(γ, f̃) = ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2})Î_{M}^{L_{1},Σ}(γ_{1}, f̃_{1,L̃_{1}})Î_{M}^{L_{2},Σ}(γ_{2}, f̃_{2,L̃_{2}}).

Proof. We begin by applying the splitting property to the summands of I_{M}^{Σ}(γ).

I_{M}^{Σ}(γ, f̃) = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} Î_{M}(ηγ, f̃′)
= ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2}) ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} Î_{M}^{L_{1}}(ηγ_{1}, f̃′_{1,L_{1}})Î_{M}^{L_{2}}(ηγ_{2}, f̃′_{2,L_{2}}).

Suppose d_{M}^{G}(L_{1}, L_{2}) is not zero. By using arguments similar to those of the proof of Lemma 6.3, it may be established that

μ_{n}^{M}/μ_{n}^{G} ≅ μ_{n}^{M}/μ_{n}^{L_{1}} × μ_{n}^{M}/μ_{n}^{L_{2}},
μ_{n}^{M}/μ_{n}^{G} ≅ μ_{n}^{L_{1}}/μ_{n}^{G} × μ_{n}^{L_{2}}/μ_{n}^{G},

and μ_{n}^{L_{1}} ∩ μ_{n}^{L_{2}} = μ_{n}^{G}. From μ_{n}^{L_{1}} ∩ μ_{n}^{L_{2}} = μ_{n}^{G} it follows that the homomorphism

μ_{n}^{L_{1}}/μ_{n}^{G} → μ_{n}^{M}/μ_{n}^{L_{2}},

given by ημ_{n}^{G} ↦ ημ_{n}^{L_{2}}, is injective. This homomorphism is also surjective as

|μ_{n}^{L_{1}}/μ_{n}^{G}| = n^{dim(𝔞_{M}^{L_{1})}−dim(𝔞_{M}^{G})} = n^{dim(𝔞_{M}^{L_{2}})} = |μ_{n}^{M}/μ_{n}^{L_{2}}|.

It may be deduced in the same manner that μ_{n}^{L_{2}}/μ_{n}^{G} ≅ μ_{n}^{M}/μ_{n}^{L_{1}}. Thus the previous sum is equal to

∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2}) ∑_{η_{1}∈μ_{n}^{M} /μ_{n}^{L_{1}}} ∑_{η_{2}∈μ_{n}^{M} /μ_{n}^{L_{2}}} Î_{M}^{L_{1}}(η_{1}η_{2}γ_{1}, f̃′_{1,L_{1}})Î_{M}^{L_{2}}(η_{1}η_{2}γ_{2}, f̃′_{2,L_{2}})

= ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2}) ∑_{η_{1}∈μ_{n}^{L_{2}}/μ_{n}^{G}} ∑_{η_{2}∈μ_{n}^{L_{1}}/μ_{n}^{G}} Î_{M}^{L_{1}}(η_{1}η_{2}γ_{1}, f̃′_{1,L_{1}})Î_{M}^{L_{2}}(η_{1}η_{2}γ_{2}, f̃′_{2,L_{2}})

= ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2}) ∑_{η_{1}∈μ_{n}^{L_{2}}/μ_{n}^{G}} Î_{M}^{L_{1}}(η_{1}η_{2}γ_{1}, f̃′_{1,L_{1}}) ∑_{η_{2}∈μ_{n}^{L_{1}}/μ_{n}^{G}} Î_{M}^{L_{2}}(η_{1}η_{2}γ_{2}, f̃′_{2,L_{2}})

= ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2}) ∑_{η_{1}∈μ_{n}^{M} /μ_{n}^{L_{1}}} Î_{M}^{L_{1}}(η_{1}γ_{1}, f̃′_{1,L_{1}}) ∑_{η_{2}∈μ_{n}^{M} /μ_{n}^{L_{2}}} Î_{M}^{L_{2}}(η_{2}γ_{2}, f̃′_{2,L_{2}})

= ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2})Î_{M}^{L_{1},Σ}(γ_{1}, f̃_{1,L̃_{1}})Î_{M,Σ}^{L_{2}}(γ_{2}, f̃_{2,L̃_{2}}).□

7 The distribution I_{M}^{M}(γ)

From here on we assume that the invariant trace formula of [Art88b] is valid for metaplectic coverings of the general linear group. In particular, we are assuming that there exist invariant distributions IM̃(γ̃) parameterized by γ̃ ∈ M̃(F_{S}), which are defined in the same fashion as I_{M}(γ) of the previous section, and also satisfy splitting and descent properties. We would like to compare these distributions to I_{M}^{Σ}(γ). To this end, we define

I_{M}^{M}(γ, f̃) = Λ^{M}(γ)IM̃(γ′, f̃), f̃ ∈ H(G̃(F_{S})),

for all γ ∈ M(F_{S}) in the domain of the orbit map such that M_{γ}(F_{S}) = G_{γ}(F_{S}). This definition is very similar to the definition of I_{M}^{E}(γ) of (3.8) II [AC89]. Indeed I_{M}^{M}(γ) shares most of the properties of I_{M}^{E}(γ) listed in §3 II [AC89]. We list them below and invite the reader to verify that the proofs of that section apply to I_{M}^{M}(γ). From now on we shall tacitly assume that γ belongs to the domain of the orbit map whenever γ′ appears.

A mild paraphrase of Lemma 3.1 and Corollary 3.2 II [AC89] implies that

lim_{{a→1:a∈A_{M,reg}(F_{S})}} ∑_{L∈L(M)} r_{M}^{L}(γ^{n}, a)I_{L}^{M}(aγ, f̃), γ ∈ M(F_{S}) (19)

exists and is equal to

Λ^{M}(γ) ∑_{L∈L(M)} c_{M}^{L}(γ^{n}, n)IL̃((γ^{L})′, f̃). (20)

The functions c_{M}^{L}(γ, n) are derived from the (G, M) family (§6 [Art81]),

c_{P} (ν, γ, n) = ∏_{v∈S} ∏_{β} |n|_{v}^{−ρ(β,u_{v})ν(β^{∨})/2}, P ∈ P(M),

where γ = ∏_{v∈S} γ_{v} has Jordan decomposition ∏_{v∈S} σ_{v}u_{v}, and the remaining terms stem from the (G, M) family r_{P} (ν, γ, a) defined in §5 [Art88c]. We define I_{M}^{M}(γ, f̃) to be equal to (19) for general γ ∈ M(F_{S}). It is clear from the proof of Corollary 3.4 II [AC89] that if γ ∈ M(F) is embedded diagonally into M(F_{S}) and S contains {v : |n|_{v} ≠ 1} then

I_{M}^{M}(γ, f̃) = IM̃(γ′, f̃), f̃ ∈ H(G̃(F_{S})). (21)

The descent property,

I_{M}^{M}(γ, f̃) = ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L)Î_{M_{1}}^{L,M}(γ, f̃L̃), (22)

for γ ∈ M_{1}(F_{S}) with M_{1} ∈ L^{M} and M_{1,γ} = M_{γ}; and the splitting property

I_{M}^{M}(γ, f̃) = ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2})Î_{M}^{L_{1},M}(γ_{1}, f̃_{1,L̃_{1}})Î_{M}^{L_{2},M}(γ_{2}, f̃_{2,L̃_{2}}), (23)

for γ = γ_{1}γ_{2} ∈ M(F_{S}) and f̃ = f̃_{1}f̃_{2} as in Proposition 6.2 also hold. This is immediate for γ ∈ M(F_{S}) ∩ G_{oreg}(F_{S}) and can be established for general γ ∈ M(F_{S}) by using Corollary 3.2 [Art88a] (see also the end of the proof of Theorem 8.1 [Art88a]).

8 Geometric vanishing properties

As alluded to earlier, we wish to compare I_{M}^{M}(γ) with I_{M}^{Σ}(γ). At least in the case M = G, our wish is partially fulfilled as

I_{G}^{Σ}(γ, f̃) = Î_{G}(γ, f̃′) = Λ^{G}(γ)IG̃(γ′, f̃) = I_{G}^{M}(γ, f̃)

for γ ∈ G_{oreg}(F_{S}) and f̃ ∈ H(G̃(F_{S})). One might wonder whether, in our definition of I_{G}^{M}(γ), we are neglecting to take into account the distributions IG̃(γ̃) for which γ̃ ∈ G̃_{reg}(F_{S}) does not lie in the image of the orbit map. In this regard, Proposition 3 [FK86] tells us that if γ̃ ∈ G̃_{reg}(F_{S}) then IG̃(γ̃) is zero unless p(γ̃) = γ_{0}^{n/r}γ^{n} for some γ_{0} ∈ A_{G}(F_{S}) and γ ∈ G_{oreg}(F_{S}). This is a local vanishing property for IG̃(γ̃). We shall extend this local vanishing property to the case M ≠ G and arbitrary γ̃ ∈ M̃(F_{S}). However, our real interest lies in proving a global vanishing property, for which the local vanishing property is just a prelude.

Lemma 8.1 Suppose v is nonarchimedean, σ̃ is an element of M̃(F_{v}) such that σ = p(σ̃) is semisimple, and IM̃(σ̃, f̃) ≠ 0 for some f̃ ∈ H(G̃(F_{v})). Then there exist elements γ_{0} ∈ A_{G}(F_{v}) and γ ∈ M(F_{v}) such that σ = γ_{0}^{n/d}γ^{n}.

Proof. We will first prove the lemma for M = G. After inverting the germ expansion for orbital integrals at σ̃ (Corollaire 2.4, §3.3 [Vig81]), we find that IG̃(σ̃, f̃) is a finite linear combination of nonzero orbital integrals of f̃ at elements in G̃σ̃(F_{v}) ∩ G̃_{reg}(F_{v}) close to σ̃. Let δ̃ be one of these elements. By Proposition 3 [FK86], p(δ̃) is equal to γ_{0}^{n/d}δ^{n} for some γ_{0} ∈ A_{G}(F_{v}) and δ ∈ G_{oreg}(F_{v}). By a standard argument (§1 [KP86]) there exist extensions E_{1}, …, E_{k} of F_{v} such that G_{δ}(F_{v}) is isomorphic to ∏_{i=1}^{k} E_{i}^{×}. Let σ_{i} and δ_{i}, 1 ≤ i ≤ k, be the respective ith components of σ and δ with respect to this isomorphism and regard γ_{0} as an element of F_{v}^{×} ⊂ E_{i}^{×}. Applying Hensel’s lemma to the polynomial X^{n} − γ_{0}^{−n/d}σ_{i} ∈ E_{i}[X], we obtain an element γ_{i} ∈ E_{i} such that |γ_{i} − δ_{i}|_{E_{i}} < 1 and γ_{0}^{n/d}γ_{i}^{n} = σ_{i}, 1 ≤ i ≤ k. It is now clear that the element γ ∈ G_{δ}(F_{v}) which maps to (γ_{1}, …, γ_{k}) under the above isomorphism and γ_{0} are the desired elements of the lemma in the case M = G. For the general case, note that we are assuming that the metaplectic version of (2.3) [Art88a] holds. Namely, if M̃σ̃(F_{v}) = G̃σ̃(F_{v}) then there exists h̃ ∈ C_{c}^{∞}(M̃(F_{v})) such that for all γ̃ ∈ M̃σ̃(F_{v}) close to σ̃

IM̃(γ̃, f̃) = IM̃ ; M̃(γ̃, h̃).

Since the K_{v}-finiteness of f̃ did not play role in the proof of the case M = G, that case together with the substitution γ̃ = σ̃ in this equation imply the lemma whenever M̃σ̃(F_{v}) = G̃σ̃(F_{v}). The lemma now follows for general σ̃ from the definition ((2.2*) [Art88a]),

IM̃(σ̃, f̃) = lim_{{a→1:a∈A_{M,reg}(F_{v})}} ∑_{L∈L(M)} r_{M}^{L}(p(σ̃), a^{n})IL̃(s(a^{n})σ̃, f̃),

and the fact that

L̃_{s(a^{n})σ̃}(F_{v}) = M̃_{s(a^{n})σ̃}(F_{v}) = G̃_{s(a^{n})σ̃}(F_{v})

for a ∈ A_{M,reg}(F_{v}) close to the identity.□

It is unfortunate that Lemma 8.1 can not be extended to general elements in M̃(F_{v}) without some additional hypotheses. To see this, suppose F_{v} is nonarchimedean, n = r = 2, m = 0, x, y ∈ F_{v}^{×}, x ∉ F_{v}^{×2} and γ̃ = s ( \begin{matrix}x & y \\ 0 & x \end{matrix} ). Using Proposition 0.1.5 [KP84] and equation (1), one can

compute G̃γ̃(F_{v}) and show that it is equal to

p^{−1} (G_{p(γ̃)}(F_{v})) = p^{−1} (\{( \begin{matrix}z_{1} & z_{2} \\ 0 & z_{1} \end{matrix} ) : z_{1} ∈ F_{v}^{×}, z_{2} ∈ F_{v}\}) .

It follows from §1.i [Vig81] that IG̃(γ̃, f̃) ≠ 0 for some f̃ ∈ H(G̃(F_{v})). On the other hand, it is simple to show that if p(γ̃) = γ^{2} for some γ ∈ G(F_{v}) then γ must lie in P_{0}(F_{v}) and in turn x must lie in F_{v}^{×2}. Our assumption on x therefore excludes this possibility.

The pathology in this example stems from the fact that r divides n. In fact, one can construct similar examples for any n and r for which gcd(i, n) ≠ 1, for some 1 < i ≤ r. Thus, n must be relatively prime to the positive integers less than or equal to r if Lemma 8.1 is to be generalized. The following lemma converts this condition on n into properties which are compatible with the structure of G(F_{v}).

Lemma 8.2 Suppose that v is nonarchimedean and that E/F_{v} is a finite extension such that n is relatively prime to [E : F_{v}]. If x ∈ E^{×} such that N_{E/F_{v}}(x) ∈ F_{v}^{×n} then x ∈ E^{×n}.

Proof. We first show that E^{×}/E^{×n} ≅ F_{v}^{×}/F_{v}^{×n} and that we may take coset representatives of E^{×}/E^{×n} to lie in F_{v}^{×}. The homomorphism, F_{v}^{×}/F_{v}^{×n} → E^{×}/E^{×n}, given by

zF_{v}^{×n} ↦ zE^{×n}, z ∈ F_{v}^{×},

is injective. Indeed, suppose z does not belong to F^{×n}, but does belong to E^{×n}. By Theorem 10 (b) VIII §6 [Lan84], [F (z^{1/n}) : F_{v}] divides n. Moreover z^{1/n} ∈ E, so

[E : F_{v}] = [E : F_{v}(z^{1/n})][F_{v}(z^{1/n}) : F_{v}].

This contradicts gcd(n, [E : F_{v}]) = 1. The surjectivity of this map follows at once from the fact that (Lemma 0.3.2 [KP84])

|E^{×}/E^{×n}| = |F_{v}^{×}/F_{v}^{×n}| = n^{2}/|n|_{v}. (24)

The enunciation of the lemma amounts to showing the injectivity of the homomorphism given by

xE^{×n} ↦ (N_{E/F_{v}}(x))F_{v}^{×n} = x^{[E:F_{v}]}F_{v}^{×n}, x ∈ F_{v}^{×}.

If x^{[E:F_{v}]}F_{v}^{×n} = F_{v}^{×n} and x ∉ F_{v}^{×n}, then [E : F_{v}] must divide n^{2}/|n|_{v} by (24), thereby contradicting gcd([E : F_{v}], n) = 1. Thus this map is injective. □

Lemma 8.3 Suppose v is nonarchimedean and n is relatively prime to r − 1 + 2rm and all positive integers less than or equal to r. Moreover suppose that γ̃ ∈ M̃(F_{v}) such that IM̃(γ̃, f̃) ≠ 0 for some f̃ ∈ H(G̃(F_{v})). Then there exists γ ∈ M(F_{v}) such that p(γ̃) = γ^{n}.

Proof. Let σu be the Jordan decomposition of p(γ̃). Suppose M = G. In this case, by the arguments of §1 [KP86], we may take σ to be a block diagonal matrix of the form

( \begin{matrix}σ_{1} & & & & & & & & & & & & 0 \\ & . & . & . \\ & & & & σ_{1} \\ & & & & & . & . & . \\ & & & & & & & & σ_{k} \\ & & & & & & & & & . & . & . \\ 0 & & & & & & & & & & & & σ_{k} \end{matrix} ) ,

where σ_{i} ∈ GL(m_{i}, F_{v}), 1 ≤ i ≤ k are pairwise distinct, generate respective extensions F_{i}/F_{v} of degree m_{i} and each appear b_{i} times. We proceed by showing that σ_{i} is an nth power in GL(m_{i}, F_{v}). Our hypothesis on γ̃ is equivalent to G̃γ̃(F_{v}) = p^{−1}(G_{p(γ̃)})(F_{v}) (cf. 1.i [Vig81]). The set p^{−1}(G_{p(γ̃)})(F_{v}) contains the diagonal matrices of the form

δ̃ = s ( \begin{matrix}δ_{1} & & & & & & & & & & & & 0 \\ & . & . & . \\ & & & & δ_{1} \\ & & & & & . & . & . \\ & & & & & & & & δ_{k} \\ & & & & & & & & & . & . & . \\ 0 & & & & & & & & & & & & δ_{k} \end{matrix} ) ,

where δ_{i} ∈ F_{v}^{×} appears m_{i}b_{i} times, 1 ≤ i ≤ k. In particular, δ̃ commutes with s(σ). Let (·, ·)_{F_{v}} and (·, ·)_{F_{i}} be the nth Hilbert symbol of F_{v} and F_{i} respectively, 1 ≤ i ≤ k. Suppose that δ_{1} = ⋯ = δ_{k}. Then Proposition 0.1.5 [KP84], the fact that δ̃ commutes with s(σ) and the properties of the Hilbert symbol imply that

1 = (det(σ), δ_{1}^{r})_{F_{v}}^{1+2m} ∏_{i=1}^{k}(σ_{i}, δ_{1}^{b_{i}})_{F_{i}}^{−1}

= (det(σ)^{r(1+2m)}, δ_{1})_{F_{v}} (∏_{i=1}^{k} N_{F_{i}/F_{v}}(σ_{i})^{b_{i}}, δ_{1})_{F_{v}}^{−1}
= (det(σ)^{r−1+2rm}, δ_{1})_{F_{v}}.

As δ_{1} ∈ F_{v}^{×} is arbitrary, det(σ)^{r−1+2rm} must lie in F_{v}^{×n}. It follows from gcd(n, r − 1 + 2rm) = 1 and (24) that det(σ) ∈ F_{v}^{×n}. Now fix 1 ≤ j ≤ k and let δ̃ be such that δ_{i} = 1 for 1 ≤ i ≤ k and i ≠ j. Then, repeating the above procedure and noting that n is relatively prime to b_{j}, we find that

1 = (det(σ), det(p(δ̃)))_{F_{v}}^{1+2m} (σ_{j}, δ_{j}^{b_{j}})_{F_{j}}^{−1}
= (N_{F_{j}/F_{v}}(σ_{j}), δ_{j})_{F_{v}}^{−b_{j}}
= (N_{F_{j}/F_{v}}(σ_{j}), δ_{j})_{F_{v}}.

Consequently N_{F_{j}/F_{v}}(σ_{j}) ∈ F_{v}^{×n}. By Lemma 8.2 and our hypotheses on n we have σ_{j} ∈ F_{j}^{×n}. Therefore there exists σ_{0j} ∈ GL(m_{j}, F_{v}) such that σ_{0j}^{n} = σ_{j}, 1 ≤ j ≤ k. Set σ_{0} to be the obvious block diagonal matrix composed of the blocks σ_{01}, …, σ_{0k} such that σ_{0}^{n} = σ. It is evident from the inclusion G_{σ_{0}} ⊂ G_{σ} that we have

G_{σ_{0}}(F_{v}) = G_{σ}(F_{v}) ≅ ∏_{i=1}^{k} GL(b_{i}, F_{i}).

Now set u_{0} = exp (\frac{1}{n} log(u)). By construction, u_{0}^{n} = u and u_{0} ∈ G_{σ}(F_{v}) = G_{σ_{0}}(F_{v}). The lemma now follows in the case M = G by setting γ = σ_{0}u_{0}. The lemma follows in the general case by using the arguments near the end of the proof of Lemma 8.1.□ The assumption placed on n in Lemma 8.3 will remain to be a restriction in the comparison of the trace formulas later on.

We can use the local vanishing properties to prove global vanishing properties by way of some local-global results on nth roots in F and G(F).

Lemma 8.4 Suppose x ∈ F^{×} such that x ∈ F_{v}^{×n} for almost all valuations v. Then x ∈ F^{×n}.

Proof. We will prove this lemma by contradiction. To this end, suppose that σ is a non-trivial element in the Galois group of the abelian extension F (x^{1/n}). (Observe that F (x^{1/n}) is well-defined since μ_{n} ⊂ F.) It is immediate from the hypothesis of the lemma that the density of the valuations for which σ is the Frobenius automorphism is zero. However, the Tchebotarev density theorem (VIII, §4 Theorem 10 [Lan70]) tells us that this density is 1/[F (x^{1/n}) : F].□

Lemma 8.5 Suppose that γ ∈ M(F) and that for almost every valuation v of F, there exists γ_{v} ∈ M(F_{v}) such that γ_{v}^{n} = γ. Then there exists γ_{0} ∈ M(F) such that γ_{0}^{n} = γ.

Proof. By decomposition (4) of M it suffices to prove the case M = G. Let γ = σu be the Jordan decomposition of γ. By §1 [KP86], we may take the semisimple element σ to be a block diagonal matrix of the form

( \begin{matrix}σ_{1} & & & & & & & & & & & & 0 \\ & . & . & . \\ & & & & σ_{1} \\ & & & & & . & . & . \\ & & & & & & & & σ_{k} \\ & & & & & & & & & . & . & . \\ 0 & & & & & & & & & & & & σ_{k} \end{matrix} ) ,

where σ_{i} ∈ GL(m_{i}, F), 1 ≤ i ≤ k, are pairwise distinct, generate respective extensions F_{i}/F of degree m_{i} and each appear b_{i} times. Clearly, G_{σ}(F_{v}) may be identified with a subgroup of ∏_{i=1}^{k} GL(m_{i}b_{i}, F_{v}). Since γ_{v} ∈ G_{σ}(F_{v}), we may assume that k = 1 without any loss of generality. Let σ_{v}u_{v} be the Jordan decomposition of γ_{v}. We obviously have σ_{v}^{n} = σ and u_{v}^{n} = u. The latter equation implies that u_{v} = u_{0} = exp (\frac{1}{n} log(u)) ∈ G(F) so we need only produce an appropriate semisimple factor. We first consider the case u ≠ 1. Using Jordan canonical form in G_{σ}(F) ≅ GL(b_{1}, F_{1}) we may assume that γ is of the form

( \begin{matrix}σ_{1} & 1 & 0 & 0 \\ 0 & σ_{1} & ∗ & 0 \\ ⋮ & ... & ... & ∗ \\ 0 & ⋯ & 0 & σ_{1} \end{matrix} ) ∈ G(F),

where the 1 in the (1, 2) entry represents the m_{1} × m_{1} identity matrix. Since σ_{v} commutes with u_{v} = u_{0}, it also commutes with u. This implies that σ_{v} is of the form the form

( \begin{matrix}σ_{v0} & ∗ & ∗ \\ 0 & σ_{v0} & ∗ \\ 0 & ∗ & ∗ \end{matrix} ) ∈ G(F_{v}),

for some σ_{v0} ∈ GL(m_{1}, F_{v}). Consequently, σ_{v0}^{n} = σ_{1} for almost every v. If F_{1} = F then m_{1} = 1 and we may apply Lemma 8.4 to σ_{1} to conclude that there exists σ_{0} ∈ F^{×} such that the nth power of γ_{0} = σ_{0}u_{0} is equal to γ. Otherwise, suppose w_{1}, …, w_{t} are the valuations of F_{1} which divide v. Then σ_{1} is GL(m_{1}, F_{v})-conjugate to

( \begin{matrix}σ_{11} & & & & 0 \\ & . & . & . \\ 0 & & & & σ_{1t} \end{matrix} ) ,

where σ_{1i} generates the completion F_{1,w_{i}} over F_{v}, 1 ≤ i ≤ t. This implies that the centralizer of σ_{1} in GL(m_{1}, F_{v}) is isomorphic to ∏_{i=1}^{t} F_{1,w_{i}}^{×}. In accordance with this last isomorphism, we decompose σ_{v0} as (σ_{v1}, …, σ_{vt}), where σ_{vi} ∈ F_{1,w_{i}}^{×}, 1 ≤ i ≤ t. Observe that the map σ_{1} ↦ σ_{1i} corresponds to the embedding F_{1} ↪ F_{1,w_{i}}, 1 ≤ i ≤ t. Therefore σ_{1}, regarded as an element of F_{1}, has an nth root in almost every completion of F_{1}. Applying Lemma 8.4 (with F replaced by F_{1}) to σ_{1} proves the case u ≠ 1. Now suppose u = 1 and F_{1} = F. Then

σ_{1}^{r} = det(σ) = det(σ_{v}^{n}) ∈ F_{v}^{×n},

for almost every v. Set c = n/ gcd(n, r). Since F_{v} contains the gcd(n, r)th roots of unity, we have σ_{1}^{r/ gcd(n,r)} ∈ F_{v}^{×c}. Furthermore, since r/ gcd(n, r) is relatively prime to c^{2}/|c|_{v}, the order of F_{v}^{×}/F_{v}^{×c} (cf. equation (24)), we have σ_{1} ∈ F_{v}^{×c}. Lemma 8.4 (with n replaced by c) implies that σ_{1} ∈ F^{×c}. The lemma now follows in this case by setting γ_{0} to be the block diagonal matrix in G(F) with blocks of the form

( \begin{matrix}0 & 0 & 0 & σ_{1}^{1/c} \\ 1 & ... & ... & 0 \\ 0 & ... & ... & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ) ∈ GL(gcd(n, r), F).

The remaining case that u = 1 and F_{1} ≠ F, is proven similarly, but with the added complication of using the embeddings F_{1} ↪ F_{1,w_{i}}, 1 ≤ i ≤ t, mentioned earlier. We leave the details to the interested reader.□

Corollary 8.1 (Kazhdan-Patterson) Suppose σ ∈ G(F) is semisimple and for almost every valuation v there exist elements δ_{v} ∈ A_{G}(F_{v}) and σ_{v} ∈ G(F_{v}) such that σ = δ_{v}^{n/d}σ_{v}^{n}. Then there exist δ_{0} ∈ A_{G}(F) and σ_{0} ∈ G(F) such that σ = δ_{0}^{n/d}σ_{0}^{n}.

Proof. Our proof follows the sketch on p. 226 [KP86]. Let x = det(σ), y_{v} = det(σ_{v}) and z_{v} ∈ F_{v}^{×} be the scalar corresponding to δ_{v} for almost every v. Clearly x = z_{v}^{rn/d}y_{v}^{n} ∈ F_{v}^{n/d}. Applying Lemma 8.4 (with n/d in place of n) and recalling that F_{v} ⊃ μ_{n} ⊃ μ_{n/d}, we find that there exists x_{0} ∈ F^{×} such that x_{0} = z_{v}^{r}y_{v}^{d} for almost every v. It is simple to verify that r and d are relatively prime. In turn, r and d^{2}, the order of F_{v}^{×}/F_{v}^{×d} (cf. equation (24)) whenever |d|_{v} = 1, are relatively prime. Combining this fact with the equality x_{0}F_{v}^{×d} = z_{v}^{r}F_{v}^{×d}, we conclude that z_{v} = x_{0}^{t}w_{v}^{d} for some t ∈ Z and w_{v} ∈ F_{v}^{×} for almost every v. We may therefore assume that δ_{v} = δ_{0} ∈ A_{G}(F). The application of Lemma 8.5 to δ_{0}^{−1}σ completes the proof.□

The following two lemmas are global vanishing properties (cf. Proposition 8.1 [Art88b]) which follow from the local vanishing properties, Lemma 8.1 and Lemma 8.3.

Proposition 8.1 Suppose σ̃ ∈ G̃(F) is embedded diagonally into G̃(F_{S}), for some large finite set of valuations S. Suppose further that p(σ̃) is semisimple. Then

IG̃(σ̃, f̃) = 0, f̃ ∈ H(G̃(F_{S})),

unless p(σ̃) = γ_{0}^{n/d}γ^{n} for some γ_{0} ∈ A_{G}(F) and γ ∈ G(F).

Proof. Suppose p(σ̃) is not of the form γ_{0}^{n/d}γ^{n} for any γ_{0} ∈ A_{G}(F) and γ ∈ G(F). Then by Corollary 8.1 we may assume that there exists a nonarchimedean valuation v_{1} ∈ S such that p(γ̃) is not of the form δ_{v_{1}}^{n/d}σ_{v_{1}}^{n} for any δ_{v_{1}} ∈ A_{G}(F_{v_{1}}) and σ_{v_{1}} ∈ G(F_{v_{1}}). The proposition now follows from Lemma 8.1.□

Proposition 8.2 Suppose γ̃ ∈ M̃(F) is embedded diagonally into M̃(F_{S}), for some large finite set of valuations S. Suppose further that the assumption of Lemma 8.3 on n holds. Then

IM̃(γ̃, f̃) = 0, f̃ ∈ H(G̃(F_{S})),

unless p(γ̃) is an nth power in M(F).

Proof. Suppose p(γ̃) is not an nth power in M(F). Then by Lemma 8.5 we may assume that there exists a nonarchimedean valuation v_{1} ∈ S such that p(γ̃) is not an nth power in M(F_{v_{1}}). Set S_{1} = {v_{1}}, S_{2} = S − {v_{1}} and let f̃ = f̃_{1}f̃_{2} be the corresponding decomposition of f̃ ∈ H(G̃(F_{S})). It follows from Lemma 8.1 that

ÎL̃ ; M̃(γ̃, f̃_{1,L̃}) = 0, L ∈ L(M).

The proposition follows by combining this equation with the splitting property (Proposition 9.1 [Art88a]),

IM̃(γ̃, f̃) = ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2})ÎL̃_{1} ; M̃(γ̃, f̃_{1,L̃_{1}})ÎL̃_{2} ; M̃(s(p(γ̃)), f̃_{2,L̃_{2}}).□

9 The invariant trace formula

We set forth the invariant trace formula of a function f̃ ∈ H(G̃(A)) as the equality of

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ̃∈(s_{0}(M(F)))_{M̃,S}} aM̃(S, γ̃)IM̃(γ̃, f̃) (25)

with

∑_{t} ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∫_{Π(M̃,t)} aM̃(π̃)IM̃(π̃, f̃)dπ̃. (26)

It will be convenient to denote (25) as I(f̃). This trace formula is extrapolated from the trace formula given in [Art88b] for reductive algebraic groups. Since non-trivial metaplectic coverings of algebraic groups are not algebraic, one ought to verify the results of Arthur for metaplectic groups in order to rigorously assert the existence of a trace formula as we have done above. There are, lamentably, too many results to be checked to be included here. Those results which have been checked (§5 for example) follow in a straightforward manner. There is no reason to doubt that the other results do not follow in the same way. We therefore assume that the invariant trace formula is correct as stated.

Expansion (26) is known as the spectral side of the trace formula, as its terms depend on representations of M̃(A). It will be further elaborated upon in §14. Expansion (25) is known as the geometric side of the trace formula, as its terms depend (in our case) on conjugacy classes in M̃(F_{S}). Both sides of the trace formula contain terms which are local, i.e. determined by M̃(F_{S}), and global, i.e. determined by the subgroup s_{0}(M(F)) of M̃(A).

The local terms of the geometric side have already been introduced in §§6-7. In order to introduce some of the global terms of the geometric side we perform a familiar calculation. Recall that the map s_{0} given by (2) is a splitting homomorphism for G̃(A) over G(F). Let L^{2}(s_{0}(G(F))\G̃(A)) be the space of genuine square-integrable functions on G̃(A), which are left-invariant under s_{0}(G(F)). This space admits a theory of automorphic representations through the decomposition of the (right-)regular representation R of G̃(A). The metaplectic version of the geometric side of the trace formula originates from the following calculation. Let φ ∈ L^{2}(s_{0}(G(F))\G̃(A)). We then have

(R(f̃)φ)(y) = ∫_{G̃(A)} f̃(x)φ(yx)dx
= ∫_{s_{0}(G(F))\G̃(A)} ∑_{γ∈s_{0}(G(F))} f̃(y^{−1}γx)φ(γx)dx

= ∫_{s_{0}(G(F))\G̃(A)} ( ∑_{γ∈s_{0}(G(F))} f̃(y^{−1}γx)) φ(x)dx.

Roughly speaking, the trace of the operator R(f̃) is obtained by integrating the integral kernel,

(x, y) ↦ ∑_{γ∈s_{0}(G(F))} f̃(y^{−1}γx), x, y ∈ s_{0}(G(F))\G̃(A),

over the diagonal.

Suppose S contains {v : |n|_{v} ≠ 1} and f̃ ∈ H(G̃(F_{S})). For nonarchimedean valuations v define f̃_{v}^{0} by

(27)  f̃_{v}^{0}(γ, ζ) = \{ \begin{matrix}ζ^{−1}, & if γ ∈ K_{v} \\ 0, & otherwise \end{matrix} \} , γ ∈ G(F_{v}), ζ ∈ μ_{n}.

We embed f̃ into H(G̃(A)) by taking its product with ∏_{v∉S} f̃_{v}^{0}. If S satisfies some additional properties, which are given in §3 [Art88b], then I(f̃) equals

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ̃∈(s_{0}(M(F)))_{M̃,S}} aM̃(S, γ̃)IM̃(γ̃, f̃).

The set (s_{0}(M(F)))_{M̃,S} denotes the set of (M̃, S)-equivalence classes of elements in s_{0}(M(F)) (§8 [Art86]). In the present case (s_{0}(M(F)))_{M̃,S} is simply the set of M̃(F_{S})-conjugacy classes of elements in s_{0}(M(F)). Since conjugacy classes play such an prominent role we shall, for the sake of convenience, often abuse notation by identifying an element γ ∈ M(F_{S}) with its conjugacy class.

The coefficient aM̃(S, γ̃) requires more explanation. Let σu be the Jordan decomposition of p(γ̃) ∈ M(F). Set i^{M}(S, σ) = 1 if σ is F -elliptic in M(F), and the M(F_{v})-orbit of σ meets K_{v} ∩ M(F_{v}) for every valuation v ∉ S. Otherwise set i^{M}(S, σ) = 0. It follows from the nature of the conjugacy classes of M(F_{S}) and (3.2) [Art88b] that

aM̃(S, γ) = i^{M}(S, σ)aM̃_{s_{0}(σ)}(S, u). (28)

For a description of a^{M_{σ}}(S, u) see §7 [Art86]. The term aM̃_{s_{0}(σ)}(S, u) is defined analogously.

10 Statement of Theorem A

Henceforth, we assume that

(†) n is relatively prime to i(1 + 2m) − 1 and i for all 1 < i ≤ r.

We shall cite each use of this assumption in §§10-22 with (†). Assumption (†) is more severe than the assumption needed for Lemma 8.3 and Proposition 8.2. The additional restrictions of this assumption are required for the arguments of the Appendix, which ensure that parabolic induction commutes with the local metaplectic correspondence.

We begin by comparing the geometric sides of the trace formulas. Suppose that S is as in §9 and consider the summand of (25) indexed by M = G, namely

∑_{γ̃∈(s_{0}(G(F)))_{G̃,S}} aG̃(S, γ̃)IG̃(γ̃, f̃).

If we are to have any hope in comparing this term with its counterpart,

∑_{γ∈(G(F))_{G,S}} a^{G}(S, γ)IG̃(γ, f̃),

by using the orbit map, then we must eliminate those γ̃ from the former sum such that γ̃ ≠ γ′ for all γ ∈ G(F). The global vanishing properties, Proposition 8.1 and Proposition 8.2, were proven with this in mind. However, there is also the additional concern that the orbit map is not injective. The following lemma determines the extent to which it is not injective on the elliptic set.

Lemma 10.1 Suppose γ_{1} and γ_{2} are F -elliptic in M(F). Then γ_{1}^{n} is M(F_{v})- conjugate to γ_{2}^{n} if and only if there exists η ∈ μ_{n}^{M} such that γ_{1} is M(F_{v})- conjugate to ηγ_{2} .

Proof. The “if” direction is trivial to prove. Suppose therefore that γ_{1}^{n} is M(F_{v})-conjugate to γ_{2}^{n}. Using decomposition (4) it suffices to prove the lemma in the case that M = G (cf. (4)) and γ_{1}^{n} is actually equal to γ_{2}^{n}. In this case, there exist extensions E_{1}, E_{2} of F such that the elliptic torus of G(F) containing γ_{i} is isomorphic to E_{i}^{×}, 1 = 1, 2. Let E = E_{1} ∩ E_{2}. Regarding γ_{1} and γ_{2} as nth roots of the elements γ_{1}^{n}, γ_{2}^{n} ∈ E, it is obvious that E(γ_{1}) is isomorphic to E(γ_{2}) over E. In particular these fields are isomorphic over F and so any two embeddings of these fields into G(F) must be conjugate over G(F). Thus a G(F)-conjugate of γ_{2} lies in the torus isomorphic to E(γ_{1}). Since this element may also be regarded as a nth root of γ_{1} it must be of the form ηγ_{1} for some η ∈ μ_{n}^{G} ≅ μ_{n}.□

Proposition 10.1 (5.1) Suppose f̃ ∈ H(G̃(A)). Then I(f̃) is equal to

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{M,S}/μ_{n}^{M}} aM̃(S, γ′)IM ; M̃(γ, f̃)

for a sufficiently large finite set of valuations S.

Proof. According to Proposition 8.2 (which contains (†) in its hypotheses), the distribution IM̃(s_{0}(γ)), γ ∈ M(F), vanishes unless γ is an nth power in M(F). From Lemma 10.1, we see that the map,

(M(F))_{M,S}/μ_{n}^{M} →^{′} (s_{0}(M(F)))_{M̃,S},

given by (equation (18), equation (2))

γμ_{n}^{M} ↦ γ′ = s(γ)^{n}i (∏_{v} κ_{v}(γ)^{−1})^{n} = s_{0}(γ)^{n} = s_{0}(γ^{n}),

is injective when restricted to the conjugacy classes of F -elliptic elements in M(F). The proposition now follows from equation (21) and the fact that aM̃(S, s_{0}(γ)) vanishes unless γ is F -elliptic in M(F) (equation (28)).□

The trace formula for G(A), which we expect to match I(f̃), is

I(f̃′) = ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{M,S}} a^{M}(S, γ)Î_{M}(γ, f̃′)
= ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{M,S}/μ_{n}^{M}} ∑_{η∈μ_{n}^{M}} a^{M}(S, ηγ)Î_{M}(ηγ, f̃′).

Equation (28) specializes to

a^{M}(S, γ) = i^{M}(S, σ)a^{M_{σ}}(S, u), (29)

for the Jordan decomposition σu of γ ∈ M(F). Since every element of μ_{n}^{M} ⊂ A_{M}(F) is F -elliptic in M(F) and also lies in K_{v} ∩ M(F_{v}) for all valuations v, it is immediate that i^{M}(S, ησ) = i^{M}(S, σ) for all η ∈ μ_{n}^{M} and semisimple σ ∈ M(F). This implies that a^{M}(S, ηγ) = a^{M}(S, γ) in the previous sum. Explicitly, we have

I(f̃′) = ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{M,S}/μ_{n}^{M}} a^{M}(S, γ) ∑_{η∈μ_{n}^{M}} Î_{M}(ηγ, f̃′)
= n ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{M,S}/μ_{n}^{M}} a^{M}(S, γ)I_{M}^{Σ}(γ, f̃).

We shall shall carry the coefficient n over to the spectral side of I(f̃′) in §§13-14 and set I^{Σ}(f̃) = n^{−1}I(f̃′). The present forms of the two trace formulas and Theorem A II [AC89] motivate the first main theorem of the comparison. Theorem A (i) Suppose that S is a finite set of valuations with the closure property. Then

I_{M}^{Σ}(γ, f̃) = I_{M}^{M}(γ, f̃), γ ∈ M_{comp}(F_{S}), f̃ ∈ H(G̃(F_{S})).

(ii) Suppose γ ∈ M(F). Then

aM̃(S, γ′) = a^{M}(S, γ)

for any suitably large finite set S.

The proof of Theorem A will be completed in §20. Observe that by the splitting properties ((23), Proposition 6.2), Theorem A (i) holds if and only if it holds in the case that S consists of a single valuation.

The proof of Theorem A consists of several induction arguments. Suppose σ is a semisimple element of G(F). We make the induction hypothesis that Theorem A holds if G is replaced by G_{σ} and dim(G_{σ}) < dim(G). Every Levi subgroup in L is of the form G_{σ}. Furthermore, it follows from Proposition 3 §2 II [Lan70] (Krasner’s Lemma) and §1 [KP86] that for any nonarchimedean valuation v and semisimple element σ_{1} ∈ G(F_{v}) there exists a semisimple element σ ∈ G(F) such that G_{σ_{1}}(F_{v}) = G_{σ}(F_{v}). This fact allows us to apply the induction hypothesis in the local nonarchimedean context. Let us consider some immediate implications of our induction hypothesis. By combining the induction hypothesis with the descent properties, we obtain the following lemma.

Lemma 10.2 Suppose M_{1} and M are in L such that M_{1} ⫋ M. Suppose further that γ ∈ M_{1}(F_{S}) ∩ M_{comp}(F_{S}) satisfies M_{1,γ} = M_{γ}. Then

I_{M}^{M}(γ, f̃) = I_{M}^{Σ}(γ, f̃), f̃ ∈ H(G̃(F_{S})).

In particular, this equation holds for γ ∈ M_{1}(F_{S}) ∩ G_{oreg}(F_{S}).

Proof. It follows from Jordan canonical form and exercises 33-34 Chapter 5 [AW92] that M_{1,γ} = M_{γ} if and only if M_{1,σ} = M_{σ}. We therefore have

M_{1,σ} ⊂ M_{1,σ^{n}} ⊂ M_{σ^{n}} = M_{σ} = M_{1,σ},

and so M_{1,σ} = M_{1,σ^{n}}. That is, γ ∈ M_{1,comp}(F_{S}). By the descent properties ((22), Corollary 6.1), we have

I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃) = ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L) (Î_{M_{1}}^{L,M}(γ, f̃L̃) − Î_{M_{1}}^{L,Σ}(γ, f̃L̃)) .

The induction hypothesis and the fact that d_{M_{1}}^{G}(M, G) = 0 (§7 [Art88a]) imply that the right-hand side vanishes.□

Applying our induction hypothesis to the geometric sides of the trace formulas and also to expansions (28) and (29), we obtain the following lemma whose proof can be gleaned from the proof of Lemma 5.2 II [AC89].

Lemma 10.3 (5.2) The distribution,

f̃ ↦ I(f̃) − I^{Σ}(f̃′), f̃ ∈ H(G̃(F_{S})),

is the sum of

∑_{M∈L,M≠G} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{γ∈(M(F))_{M,S}/μ_{n}^{M}} a^{M}(S, γ) (I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃))

and

∑_{δ∈A_{G}(F)\μ_{n}^{G}} ∑_{u∈(U_{G}(F))_{G,S}} (aG̃(S, u) − a^{G}(S, u)) I_{G}^{M}(δu, f̃).

11 Comparison of I_{M}^{M}(γ, f̃) and I_{M}^{Σ}(γ, f̃)

Our aim here is to show that I_{M}^{Σ}(γ, f̃) is equal to I_{M}^{M}(γ, f̃) under various circumstances. We begin by showing that Theorem A (i) holds if it holds at certain regular elements. In order to do this, we need to be able to compare the functions r_{M}(γ, a), which appear in the definitions of our invariant distributions. The following two lemmas afford such a comparison.

Lemma 11.1 Suppose that γ ∈ M(F_{v}) is not G(F_{v})-conjugate to any element γ_{0} ∈ L(F_{v}), such that L ∈ L, L ⫋ M and L_{γ_{0}} = M_{γ_{0}}. Suppose further that γ = σu is the Jordan decomposition of γ in M(F_{v}). Then there exists η ∈ μ_{n}^{M} such that G_{ηγ} = G_{γ^{n}} and G_{ησ} = G_{σ^{n}}. Moreover, M_{σ} = M_{σ^{n}}.

Proof. We shall show that dim(G_{ηγ}) = dim(G_{γ^{n}}) for some η ∈ μ_{n}^{M}. The first assertion of the lemma then follows from the fact that centralizers in G are closed and connected. The remaining assertions shall follow easily from the proof of the first. Recall decomposition (4),

M = M(1) × ⋯ × M(ℓ) ≅ GL(r_{1}) × ⋯ × GL(r_{ℓ}).

Identify γ ∈ M(F_{v}) with (γ_{1}, …, γ_{ℓ}), where γ_{i} ∈ GL(r_{i}, F_{v}), 1 ≤ i ≤ ℓ. As explained at the beginning of §1 [KP86], there exist positive integers, k_{i}, r_{ij}, and commuting elements, σ_{ij}, u_{ij} ∈ GL(r_{ij}, F_{v}), for 1 ≤ i ≤ ℓ, 1 ≤ j ≤ k_{i}, such that the elements σ_{i1}, …, σ_{ik_{i}} are semisimple, pairwise distinct and generate field extensions, E_{i1}, …, E_{ik_{i}}, respectively; the elements u_{i1}, …, u_{ik_{i}} are unipotent and upper-triangular; and γ_{i} is GL(r_{i}, F_{v})-conjugate to the block diagonal matrix

( \begin{matrix}σ_{i1}u_{i1} & & & & 0 \\ & . & . & . \\ 0 & & & & σ_{ik_{i}}u_{ik_{i}} \end{matrix} ) ∈ GL(r_{i}, F_{v}) ≅ M(i)(F_{v}). (30)

This matrix determines a unique Levi subgroup,

L_{i} ≅ GL(r_{i1}) × ⋯ × GL(r_{ik_{i}}),

of M(i) ≅ GL(r_{i}). In turn, we obtain a unique Levi subgroup L ∈ L such that L ≅ L_{1}×⋯×L_{ℓ} and L ⊂ M. Suppose γ_{0} is the element of L(F_{v}) determined by (30). Then L_{γ_{0}} = M_{γ_{0}}. To see this, observe that for 1 ≤ j_{1} < j_{2} ≤ k_{i}, the eigenvalues of σ_{ij_{1}} are pairwise distinct from the eigenvalues of σ_{ij_{2}}. This implies that L_{i,γ_{i}} = M(i)_{γ_{i}} (exercise 33-34 Chapter 5 [AW92]). This last equality clearly implies L_{γ_{0}} = M_{γ_{0}}. By the hypothesis of the lemma we must have L = M. This means that k_{1} = ⋯ = k_{ℓ} = 1 and that γ_{i} is GL(r_{i}, F_{v})- conjugate to σ_{i}u_{i}, where σ_{i} = σ_{i1} and u_{i} = u_{i1}. An immediate consequence is that γ_{i}^{n} is GL(r_{i}, F_{v})-conjugate to σ_{i}^{n}u_{i}^{n}. The element σ_{i}^{n} generates a field F_{i} which lies between E_{i} = E_{i1} and F_{v}. Assumption (†) implies that E_{i} = F_{i}. Indeed, after identifying σ_{i} with an element of E_{i}, we have E_{i} = F_{i}(σ_{i}) = F_{i}((σ_{i}^{n})^{1/n}). According to Theorem 10 (b) VIII §6 [Lan84], the index [E_{i} : F_{i}] divides n. At the same time we have [E_{i} : F_{i}] ≤ r_{i} ≤ r and so (†) implies that [E_{i} : F_{i}] = 1. It is immediate that the degree of the minimal polynomial of σ_{i}, namely [E_{i} : F_{v}], is the same as that of σ_{i}^{n}. Furthermore, by applying Jordan canonical form (in GL(r_{i}/[E_{i} : F_{v}], E_{i})) and Lemma 4.1 to σ_{i}^{n}u_{i}^{n}, we find that it is GL(r_{i}, F_{v})-conjugate to σ_{i}^{n}u_{i}. Suppose that

(31) σ_{j_{1}}^{n} ≠ σ_{j_{2}}^{n}, whenever σ_{j_{1}} ≠ σ_{j_{2}}, 1 ≤ j_{1} < j_{2} ≤ ℓ.

Then the previous two observations imply that there is a degree-preserving bijection between the elementary divisors of γ and γ^{n}. Since the dimensions of G_{γ} and G_{γ^{n}} are determined by the degrees of the elementary divisors of γ and γ^{n} respectively (Theorem 5.15 [AW92]), we have dim(G_{γ}) = dim(G_{γ^{n}}). Now suppose that σ_{j_{1}}^{n} = σ_{j_{2}}^{n}, but σ_{j_{1}} ≠ σ_{j_{2}} for some 1 ≤ j_{1} < j_{2} ≤ ℓ. By regarding σ_{j_{1}} and σ_{j_{2}} as elements of E_{j_{1}}(= E_{j_{2}}), we find that σ_{j_{1}} = ζσ_{j_{2}} for some ζ ∈ μ_{n}. This implies the existence of an element η ∈ μ_{n}^{M}, such that if γ is replaced by ηγ in the earlier argument, then (31) holds. This proves the first assertion of the lemma. The second assertion is seen to follow easily from the above argument by taking u = 1. The final assertion of the lemma is a consequence of E_{i} = F_{i}.□

We already mentioned in the proof of Lemma 6.2 that r_{M}(γ, a) is invariant under translation by A_{G}(F_{S}) in the first variable. This justifies the appearance of the quotient in the index set of the sum in the following lemma.

Lemma 11.2 Suppose γ ∈ M(F_{v}) satisfies the hypotheses of Lemma 11.1 and a ∈ A_{M,reg}(F_{v}). Then

∑_{η∈μ_{n}^{M} /μ_{n}^{G}} r_{M}(ηγ, a) = r_{M}(γ^{n}, a).

Proof. By Lemma 11.1, we may assume that γ satisfies G_{γ} = G_{γ^{n}}, G_{σ} = G_{σ^{n}} and M_{σ} = M_{σ^{n}}. Suppose η ∈ μ_{n}^{M}. We would like to use Lemma 8.2 [Art88c] to show that

(32) r_{M}(ηγ, a) = \{ \begin{matrix}r_{M}^{G_{η}}(γ, a), & if 𝔞_{G_{η}} = 𝔞_{G} \\ 0, & otherwise \end{matrix} .

In order to be able to invoke Lemma 8.2 [Art88c] we must verify the three conditions listed on p. 262 [Art88c]. To satisfy the first condition, we must have η ∈ G(F). This is trivial. The second condition is satisfied if 𝔞_{M_{η}} = 𝔞_{M}. This is obvious as M_{η} = M. Finally, we must verify that G_{ηγ}(F_{v}) is contained in G_{γ}(F_{v}). This is simple to verify since,

G_{ηγ}(F_{v}) ⊂ G_{(ηγ)^{n}}(F_{v}) = G_{γ^{n}}(F_{v}) = G_{γ}(F_{v}).

We now have equation (32). A moments thought reveals that 𝔞_{G_{η}} = 𝔞_{G} if and only if η ∈ μ_{n}^{G}. Thus,

∑_{η∈μ_{n}^{M} /μ_{n}^{G}} r_{M}(ηγ, a) = r_{M}(γ, a).

Now r_{M}(γ, a) is defined in the usual fashion (cf. §6 [Art81]) from the (G, M) family

r_{P} (ν, γ, a) = r_{P} (ν, σu, a) = ∏_{β} |a^{β} − a^{−β}|^{ρ(β,u)ν(β^{∨})/2} , P ∈ P(M).

The product on the right is taken over the roots of (P ∩G_{σ}, A_{M_{σ}}), and ρ(β, u) is a real number (defined in §3 [Art88c]) which depends only on the conjugacy class of u in M_{σ}. Since u^{n} is conjugate to u in M_{σ} (Lemma 4.1) and

(P ∩ G_{σ}, A_{M_{σ}}) = (P ∩ G_{σ^{n}}, A_{M_{σ^{n}}}),

we have in turn that

r_{P} (ν, γ^{n}, a) = r_{P} (ν, σ^{n}u^{n}, a) = r_{P} (ν, σu, a) = r_{P} (ν, γ, a)

and

r_{M}(γ^{n}, a) = r_{M}(γ, a).

This completes the lemma.□

Lemma 11.3 (3.6) Suppose that f̃ is a function in H(G̃(F_{S})) such that

I_{M}^{Σ}(γ, f̃) = I_{M}^{M}(γ, f̃)

for every element γ ∈ M(F_{S}) ∩ G_{oreg}(F_{S}). Then the same formula holds for any element γ ∈ M_{comp}(F_{S}).

Proof. By the splitting properties ((23), Proposition 6.2), it suffices to prove the lemma in the case that S consists of a single valuation v. By Lemma 10.2, it suffices to prove the lemma under the assumption that γ ∈ M_{comp}(F_{v}) satisfies the hypotheses of Lemma 11.1. By following the argument of Lemma 3.6 II [AC89], we may conclude that

I_{M}^{Σ}(γ, f̃) = I_{M}^{M}(γ, f̃),

if the semisimple component σ of γ satisfies G_{σ} = M_{σ}. Now suppose that a ∈ A_{M,reg}(F_{v}) is close to the identity so that

I_{M}^{Σ}(aγ, f̃) = I_{M}^{M}(aγ, f̃)

holds. According to the definitions of our invariant distributions and Lemma 11.2, we have

I_{M}^{Σ}(γ, f̃) = lim_{a→1} ∑_{L∈L(M)} ∑_{η∈μ_{n}^{M} /μ_{n}^{L}} r_{M}^{L}(ηγ, a)I_{L}^{Σ}(aηγ, f̃)
= lim_{a→1} ∑_{L∈L(M)} r_{M}^{L}(γ^{n}, a)I_{L}^{M}(aγ, f̃)
= I_{M}^{M}(γ, f̃).□

Lemma 11.3 tells us that in order to prove Theorem A (i) it is enough to consider the case that γ belongs to M(F_{S}) ∩ G_{oreg}(F_{S}). This is an important observation which we shall, often implicitly, make use of in the rest of this paper.

We now consider a very different comparison. At almost every place v, a function in H(G̃(A)) is of the form f̃_{v}^{0} (cf. (27)). It is therefore valuable to compare I_{M}^{M}(γ, f̃_{v}^{0}) with I_{M}^{Σ}(γ, f̃_{v}^{0}).

Lemma 11.4 Suppose v is a valuation such that |n|_{v} = 1. Then

I_{M}^{M}(γ, f̃_{v}^{0}) = I_{M}^{Σ}(γ, f̃_{v}^{0}), γ ∈ M_{comp}(F_{v}).

Proof. By Lemma 11.3, it suffices to prove the lemma for γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}). Obviously, |n|_{v} = 1 implies

I_{M}^{M}(γ, f̃_{v}^{0}) = IM̃(γ′, f̃_{v}^{0}).

By Lemma 2.1 [Art88a] the term on the right is equal to the weighted orbital integral JM̃(γ′, f̃_{v}^{0}) (cf. §2 [Art88c]). Let f_{v}^{0} ∈ H(G(F_{v})) be the characteristic function of K_{v}. It may easily be verified that f_{v,G}^{0} = (f̃_{v}^{0})′. Under assumption (†) we may apply the Theorem of [Meza] to conclude that

JM̃(γ′, f̃_{v}^{0}) = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} J_{M}(ηγ, f_{v}^{0}).

Again, by Lemma 2.1 [Art88a], the sum on the right is equal to

∑_{η∈μ_{n}^{M} /μ_{n}^{G}} I_{M}(ηγ, f_{v}^{0}) = I_{M}^{Σ}(γ, f̃_{v}^{0}).□

We now turn to comparisons which are more obviously connected to Theorem A. Through the comparison of trace formulas in §20, we will prove Theorem A (i) in the special case that S contains {v : |n|_{v} ≠ 1}. The purpose of the next theorem and its corollary is to show that the general case of Theorem A (i) follows from this special case.

Theorem 11.1 (6.1) In the special case that S contains {v : |n|_{v} ≠ 1}, we suppose that

I_{L}^{M}(γ, f̃) = I_{L}^{Σ}(γ, f̃),

for any γ ∈ L_{comp}(F_{S}) and L ∈ L(M). Then there are unique constants

ε_{L}(S) = ε_{L}^{G}(S), L ∈ L(M),

such that

I_{M}^{M}(γ, f̃) = ∑_{L∈L(M)} Î_{M}^{L,Σ}(γ, ε_{L}(S)f̃L̃), γ ∈ M(F_{S}).

The constants have the descent property

ε_{M}(S) = ∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L)ε_{M_{1}}^{L}(S), M_{1} ⊂ M,

and the splitting property

ε_{M}(S) = ∑_{L_{1},L_{2}∈L(M)} d_{M}^{G}(L_{1}, L_{2})ε_{M}^{L_{1}}(S_{1})ε_{M}^{L_{2}}(S_{2}), S = S_{1} ∪ S_{2}.

Proof. This theorem follows from the proof of Theorem 6.1 II [AC89], with I_{M}^{E} replaced by I_{M}^{M} and I_{M} replaced by I_{M}^{Σ}.□

Corollary 11.1 (6.4) In the special case that S contains {v : |n|_{v} ≠ 1}, we suppose that

I_{L}^{M}(γ, f̃) = I_{L}^{Σ}(γ, f̃),

for any γ ∈ L_{comp}(F_{S}) and L ∈ L(M). Then

ε_{M}(S) = \{ \begin{matrix}1, & M = G \\ 0, & M ≠ G \end{matrix}

for any finite set S of valuations with the closure property.

Proof. Fix M ⫋ G. By the induction hypothesis following Theorem A we have that ε_{M}^{L}(S) = 0, if M ⫋ L ⫋ G. It follows from the the descent property of Theorem 11.1 that ε_{M}(S) = 0 unless M = M_{0}. It follows from the splitting property of Theorem 11.1 that

ε_{M_{0}}(S) = ∑_{v∈S} ε_{M_{0}}(v).

The corollary therefore follows if we show that ε_{M_{0}}(v) = 0. We assume that the invariant local trace formula of Arthur (Proposition 8.1 [Art89c]) holds for G̃(F_{v}) and shall compare local trace formulas to show that ε_{M_{0}}(v) = 0. Our assumption is that the geometric side of the local trace formula ((8.5) [Art89c]),

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1}(−1)^{dim(A_{M} /A_{G})} ∫_{(M̃(F_{v})_{ell})} IM̃(γ̃, f̃_{1} × f̃_{2})dγ̃,

is equal to the spectral side ((8.6) [Art89c]),

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1}(−1)^{dim(A_{M} /A_{G})}

× ∑_{π̃∈Π_{disc}(M̃)} aM̃ ; disc(π̃) ∫_{i𝔞_{M}^{∗}} rM̃(π̃_{λ}, P̃)tr (IndG̃ ; P̃(π̃_{λ}, f̃_{1} × f̃_{2})) dλ.

We shall make a few remarks concerning the notations. The set (M̃(F_{v})_{ell}) is the set of M̃(F_{v})-conjugacy classes of elements γ̃ in M(F_{v}) such that p(γ̃) is F_{v}-elliptic in M(F_{v}). It is a measure space by virtue of an identification with a set of anisotropic tori. The invariant distribution IM̃(γ̃, f̃_{1} × f̃_{2}) is defined analogously to the distribution IM̃(γ̃, f̃) of §7, the only differences being that f̃_{1} and f̃_{2} are Hecke functions on the same group and that f̃_{1} is genuine, not antigenuine. The set Π_{disc}(M̃) is, in our case (Proposition 27 [FK86]), a subset of (equivalence classes of) representations of the form

π̃ = π̃_{1}^{∨} ⊗ π̃_{1}, π̃_{1} ∈ Π_{temp}(M̃(F_{v})),

where π̃_{1}^{∨} denotes the contragredient of π̃_{1}. The term rM̃(π̃_{λ}, P̃) is obtained from the (G, M) family of normalizing factors given in §5. The reader is referred to [Art89c] for a description of the remaining notations. Applying the local vanishing property (Lemma 8.1), making the change of variable from γ to γ′, and making some obvious definitions, we find that the geometric side is equal to

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1}(−1)^{dim(A_{M} /A_{G})} ∫_{(M(F_{v})_{ell})/μ_{n}^{M}} |n|_{v}^{r}IM̃(γ′, f̃_{1} × f̃_{2})dγ

= ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1}(−1)^{dim(A_{M} /A_{G})} ∫_{(M(F_{v})_{ell})/μ_{n}^{M}} I_{M}^{M}(γ, f̃_{1} × f̃_{2})dγ.

The geometric side of the local trace formula for G(F_{v}) at f̃′_{1} × f̃′_{2} is equal to

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1}(−1)^{dim(A_{M} /A_{G})} ∫_{(M(F_{v})_{ell})} Î_{M}(γ, f̃′_{1} × f̃′_{2})dγ

= n ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1}(−1)^{dim(A_{M} /A_{G})} ∫_{(M(F_{v})_{ell})/μ_{n}^{M}} I_{M}^{Σ}(γ, f̃_{1} × f̃_{2})dγ.

The comparison of the spectral side of the local trace formula of G(F_{v}) follows in three steps. First, from a simple computation (cf. §16) and the definition of our normalizing factors (§5), it follows that

rM̃(π̃_{λ}, P̃) = n^{dim(A_{M} /A_{G})}r_{M}(π̃′_{λ′}, P), P ∈ P(M).

Second, the local metaplectic correspondence for tempered representations (8) implies that

tr (IndG̃ ; P̃(π̃_{λ}, f̃_{1} × f̃_{2})) = (f̃′_{1} × f̃′_{2}) (Ind_{P}^{G}(π̃′_{λ′})) .

Finally, we leave it to the interested reader to show that once the terms in the definition of aM̃ ; disc(π̃) and a_{disc}^{M}(π̃′) are unraveled, they are easily seen to be equal. Taking the forgoing into consideration and noting that dλ′ equals n^{dim(A_{M})}dλ, we conclude that the spectral side of the local trace formula for G(F_{v}) is n times the spectral side of the local trace formula for G̃(F_{v}). The coefficient n cancels the n appearing in front of the geometric side of the trace formula for G(F_{v}). Thus, taking the difference of our modified local trace formulas yields

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1}(−1)^{dim(A_{M} /A_{G})}

× ∫_{(M(F_{v})_{ell})/μ_{n}^{M}} I_{M}^{M}(γ, f̃_{1} × f̃_{2}) − I_{M}^{Σ}(γ, f̃_{1} × f̃_{2})dγ = 0.

An application the splitting properties of I_{M}^{M}(γ) and I_{M}^{Σ}(γ) (which follow as in (23) and Proposition 6.2) and the induction hypothesis to the integrand reduces this equation to

r^{−1}(−1)^{r−1} ∫_{M_{0}(F_{v})/μ_{n}^{M}} 2|n|_{v}^{r}ε_{M_{0}}(v)ÎM̃_{0} ; M̃_{0}(γ′, f̃_{1,M̃_{0}})ÎM̃_{0} ; M̃_{0}(γ′, f̃_{2,M̃_{0}})dγ = 0,

which in turn implies

∫_{M_{0}(F_{v})/μ_{n}^{M}} ε_{M_{0}}(v)IG̃(γ′, f̃_{1})IG̃(γ′, f̃_{2})dγ = 0. (33)

Choosing appropriate non-negative functions f̃_{1}, f̃_{2} in this equation implies that ε_{M_{0}}(v) = 0 and the corollary is complete.□

12 Comparison of germs

We shall establish germ expansions for I_{M}^{M}(γ) and I_{M}^{Σ}(γ) and then compare them. Before writing the germ expansion for IM̃(γ̃) we extend the notion of (M, σ)-equivalence (cf. §2 [Art88c]) to metaplectic coverings. Suppose that σ̃ is a semisimple element of M̃(F_{S}), and ϕ_{1}, ϕ_{2} are functions defined on an open subset Σ of M̃σ̃(F_{S}), whose closure contains and M̃σ̃-invariant neighborhood of σ̃. We say ϕ_{1} is (M̃, σ̃)-equivalent to ϕ_{2} and write

ϕ_{1}(γ̃) ∼^{(M̃,σ̃)} ϕ_{2}(γ̃), γ̃ ∈ Σ,

if there exists h̃ ∈ C_{c}^{∞}(M̃(F_{S})) and a neighborhood U of σ̃ in M̃(F_{S}) such that

ϕ_{1}(γ̃) − ϕ_{2}(γ̃) = IM̃ ; M̃(γ̃, h̃), γ̃ ∈ Σ ∩ U.

For the remainder of this section we assume that v is nonarchimedean and f̃ ∈ H(G̃(F_{v})). Suppose that γ̃ ∈ M̃(F_{v}) such that p(γ̃) has Jordan decomposition σu. Choose σ̃ such that p(σ̃) = σ and γ̃ = σ̃u. The germ expansion of IM̃(γ̃, f̃) is the extension of (2.5) [Art88a] to metaplectic coverings and reads as

IM̃(γ̃, f̃) ∼^{(M̃,σ̃)} ∑_{L∈L(M)} ∑_{δ̃∈σ̃(U_{Lσ} (F_{v}))} gL̃ ; M̃(γ̃, δ̃)IL̃(δ̃, f̃). (34)

As in the germ expansions of invariant orbital integrals (§3), the germs gL̃ ; M̃(γ̃, δ̃) are parameterized by classes in (U_{L_{σ}}(F_{v})) (by way of the orbit map s). Therefore in comparing germ expansions at γ = σu ∈ M(F_{v}) and γ′ it would be convenient to have a bijection between (U_{L_{σ}}(F_{v})) and (U_{L_{σ^{n}}}(F_{v})). The following lemma gives us such a bijection provided σ is F_{v}-elliptic in L.

Lemma 12.1 Suppose L ∈ L(M) and σ is a semisimple element of M(F_{v}) such that σ is F_{v}-elliptic in M and σ^{n} is F_{v}-elliptic in L. Then there exists an element η in μ_{n}^{M} such that L_{ησ}(F_{v}) = L_{σ^{n}}(F_{v}). In particular ησ is F_{v}-elliptic in L.

Proof. For the sake of convenience, we suppose that L = G. Recall decomposition (4),

M(F_{v}) = M(1)(F_{v}) × ⋯ × M(ℓ)(F_{v}) ≅ GL(r_{1}, F_{v}) × ⋯ × GL(r_{ℓ}, F_{v}).

We identify M(F_{v}) with this direct product of general linear groups. By assumption, σ is F_{v}-elliptic in M(F_{v}). It is therefore M(F_{v})-conjugate to

( \begin{matrix}( \begin{matrix}σ_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & σ_{1} \end{matrix} ) & & & & 0 \\ & . & . & . \\ 0 & & & & ( \begin{matrix}σ_{ℓ} & & & & 0 \\ & . & . & . \\ 0 & & & & σ_{ℓ} \end{matrix} ) \end{matrix} ) ,

where σ_{i} ∈ GL(m_{i}, F_{v}) generates an extension F_{i}/F_{v} of degree m_{i}, m_{i} divides r_{i}, and σ_{i} appears r_{i}/m_{i} times (cf. §1 [KP86]). Since σ^{n} is F_{v}-elliptic in G(F_{v}), it is M(F_{v})-conjugate to

( \begin{matrix}σ_{0} & & & & 0 \\ & . & . & . \\ 0 & & & & σ_{0} \end{matrix} ) ,

where σ_{0} ∈ GL(m_{0}, F_{v}) generates a field extension F_{0} of degree m_{0}, m_{0} divides r, and σ_{0} appears r/m_{0} times. The elements σ_{i} ∈ GL(m_{i}, F_{v}) may also be regarded as elements of the fields F_{i}, 0 ≤ i ≤ ℓ. From this perspective we have

F_{i} = F_{v}(σ_{i}) ⊃ F_{v}(σ_{i}^{n}) = F_{v}(σ_{0}) = F_{0}, 1 ≤ i ≤ ℓ.

Obviously [F_{i} : F_{0}] ≤ r and, by Theorem 10 (b) VIII §6 [Lan84], [F_{i} : F_{0}] divides n, 1 ≤ i ≤ ℓ. As we are assuming (†), we must have F_{0} = F_{1} = ⋯ = F_{ℓ}. As a result, m_{1} = ⋯ = m_{ℓ} = m_{0} and σ_{1}^{n} = ⋯ = σ_{ℓ}^{n} = σ_{0}. Since F_{0} contains μ_{n}, it follows that there exist ζ_{1}, …, ζ_{ℓ} ∈ μ_{n} such that ζ_{i}σ_{1} = σ_{i}. The elements ζ_{1}, …, ζ_{ℓ} determine an element η ∈ μ_{n}^{M} such that

ησ = ( \begin{matrix}σ_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & σ_{1} \end{matrix} ) .

The lemma now follows from the fact that

G_{ησ}(F_{v}) ≅ GL(r/m_{0}, F_{0}) ≅ G_{σ^{n}}(F_{v}).□

In the hypothesis of Lemma 12.1 we take σ to be F_{v}-elliptic in M(F_{v}). We shall see in a moment that this restriction does not pose a problem for the comparison of germs because the germs of (34) vanish for non-elliptic σ. We therefore assume for the rest of this section that σ is a semisimple element of M(F_{v}) which is F_{v}-elliptic in M(F_{v}).

Let us consider the ellipticity of σ^{n}. There exists a unique maximal Levi subgroup L_{0} ∈ L(M) such that σ^{n} is F_{v}-elliptic in L_{0}. Indeed, if σ^{n} is F_{v}- elliptic in L_{1}, L_{2} ∈ L(M), then the centralizer in G of the split torus A_{L_{1}}∩A_{L_{2}} is a Levi subgroup L ∈ L(M). It is then simple to verify that

A_{L}(F_{v}) = A_{L_{1}}(F_{v}) ∩ A_{L_{2}}(F_{v}) = A_{L_{1,σ^{n}}}(F_{v}) ∩ A_{L_{2,σ^{n}}}(F_{v}) = A_{L_{σ^{n}}}(F_{v}),

which is equivalent to σ^{n} being F_{v}-elliptic in L. By Lemma 12.1 we know that there exists η_{0} ∈ μ_{n}^{M} such that L_{0,η_{0}σ}(F_{v}) = L_{0,σ^{n}}(F_{v}). We shall assume that η_{0} is the identity for the rest of this section. As a result, σ is F_{v}-elliptic in L ∈ L(M) if and only if σ^{n} is F_{v}-elliptic in L.

Lemma 12.2 Suppose that L ∈ L(M), σ is F_{v}-elliptic in L, and η ∈ μ_{n}^{M}. Then ησ is F_{v}-elliptic in L if and only if η ∈ μ_{n}^{L}.

Proof. If η ∈ μ_{n}^{L} then L_{ησ}(F_{v}) = L_{σ}(F_{v}), so ησ is clearly seen to be F_{v}-elliptic in L. Conversely, suppose ησ is F_{v}-elliptic in L and, for the sake of simplicity, that L = G. Since σ is F_{v}-elliptic in G, we may assume by rational canonical form that

σ = ( \begin{matrix}σ_{0} & & & & 0 \\ & . & . & . \\ 0 & & & & σ_{0} \end{matrix} ) ,

where σ_{0} ∈ GL(m_{0}, F_{v}) generates a field extension F_{0} of degree m_{0} and σ_{0} appears r/m_{0} times. Consequently,

ησ = ( \begin{matrix}η_{1}σ_{0} & & & & 0 \\ & . & . & . \\ 0 & & & & η_{r/m_{0}}σ_{0} \end{matrix} ) ,

where the scalar matrices, η_{1}, …, η_{r/m_{0}}, are projections of η into the GL(m_{0}, F_{v})- blocks. In order for ησ to be F_{v}-elliptic in G, we must have η_{i}σ_{0} = η_{j}σ_{0} for 1 ≤ i, j ≤ r/m_{0}. This implies that η ∈ μ_{n}^{G}.□

We are now ready to give germ expansions for the local geometric terms of the trace formula.

Lemma 12.3 Suppose γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v}) and f̃ ∈ H(G̃(F_{v})). Then there exist functions γ ↦ g_{M}^{L}(γ, δ) such that

I_{M}^{Σ}(γ, f̃) ∼^{(M,σ)} ∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} g_{M}^{L}(γ, δ)I_{L}^{Σ}(δ, f̃).

Proof. Suppose f ∈ H(G(F_{v})). If η ∈ μ_{n}^{M} then ησ is a semisimple element and by (2.5) [Art88a], we have

I_{M}(γ, f) ∼^{(M,ησ)} ∑_{L∈L(M)} ∑_{δ∈(ησU_{Lησ} (F_{v}))} g_{M}^{L}(γ, δ)I_{L}(δ, f), γ ∈ M_{ησ}(F_{v})∩G_{oreg}(F_{v}).

As η lies in the center of M(F_{v}), it is easy to see that (M, ησ)-equivalence of a function in γ is the same as (M, σ)-equivalence of a function in ηγ. The above expansion may thus be written as

I_{M}(ηγ, f) ∼^{(M,σ)} ∑_{L∈L(M)} ∑_{δ∈(ησU_{Lησ} (F_{v}))} g_{M}^{L}(ηγ, δ)I_{L}(δ, f), γ ∈ M_{σ}(F_{v})∩G_{oreg}(F_{v}).

Consequently

I_{M}^{Σ}(γ, f̃) = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} Î_{M}(ηγ, f̃′)

∼^{(M,σ)} ∑_{L∈L(M)} ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ∑_{δ∈ησ(U_{Lησ} (F_{v}))} g_{M}^{L}(ηγ, δ)Î_{L}(δ, f̃′).

Now if γ = σγ_{1} and δ = ησu, where u ∈ U_{L_{ησ}}(F_{v}), then by Lemma 9.2 [Art88a] we have

g_{M}^{L}(ηγ, δ) = \{ \begin{matrix}g_{M_{σ}}^{L_{ησ}}(γ_{1}, u), & if ησ is F_{v}-elliptic in L \\ 0, & otherwise \end{matrix} .

If σ^{n} is not F_{v}-elliptic in L then neither is ησ for any η ∈ μ_{n}^{M}/μ_{n}^{G} and g_{M}^{L}(ηγ, δ) = 0. On the other hand, if σ^{n} is F_{v}-elliptic in L then, by our assumption on σ, Lemma 12.2 and Lemma 9.2 [Art88a],

g_{M}^{L}(ηγ, δ) = \{ \begin{matrix}g_{M_{σ}}^{L_{ησ}}(γ_{1}, u), & if η ∈ μ_{n}^{L}/μ_{n}^{G} \\ 0, & otherwise \end{matrix}
= \{ \begin{matrix}g_{M_{σ}}^{L_{σ}}(γ_{1}, u), & if η ∈ μ_{n}^{L}/μ_{n}^{G} \\ 0, & otherwise \end{matrix}
= \{ \begin{matrix}g_{M}^{L}(γ, δ), & if η ∈ μ_{n}^{L}/μ_{n}^{G} \\ 0, & otherwise \end{matrix} .

Combining these last two observations, we find that

I_{M}^{Σ}(γ, f̃) ∼^{(M,σ)} ∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} g_{M}^{L}(γ, δ) ∑_{η∈μ_{n}^{L}/μ_{n}^{G}} Î_{L}(ηδ, f̃′)
= ∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} g_{M}^{L}(γ, δ)I_{L}^{Σ}(δ, f̃).□

The next lemma shows that there is a similar germ expansion for the distribution I_{M}^{M}(γ, f). We shall abuse notation slightly by identifying the index set (U_{L_{σ}}(F_{v})) below with (s(U_{L_{σ}}(F_{v}))).

Lemma 12.4 Suppose γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v}) and f̃ ∈ H(G̃(F_{v})). Then there exist functions γ ↦ gL̃ ; M̃(s(γ), s(δ)) such that

I_{M}^{M}(γ, f̃) ∼^{(M,σ)} ∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} gL̃ ; M̃(s(γ), s(δ))I_{L}^{M}(δ, f̃).

Proof. Expansion (2.5) [Art88a] translates into the metaplectic context as

IM̃(γ̃, f̃) ∼^{(M̃,σ′)} ∑_{L∈L(M)} ∑_{δ̃∈σ′(U_{L_{σ^{n}}} (F_{v}))} gL̃ ; M̃(γ̃, δ̃)IL̃(δ̃, f̃)

for γ̃ ∈ M̃_{σ′}(F_{v}) ∩ G̃_{reg}(F_{v}). Lemma 9.2 [Art88c] translates as

gL̃ ; M̃(γ̃, δ̃) = \{ \begin{matrix}gL̃_{σ^{n}} ; M̃_{σ^{n}}(γ_{1}, s(u)), & if σ^{n} is F_{v}-elliptic in L \\ 0, & otherwise \end{matrix} ,

where γ̃ = σ′γ_{1}, δ̃ = σ′s(u) and u ∈ U_{L_{σ^{n}}}(F_{v}). If σ^{n} is F_{v}-elliptic in L, it follows from our assumptions on σ that L_{σ} = L_{σ^{n}}. Consequently, L̃_{σ} = L̃_{σ^{n}} and the previous equation becomes

(35) gL̃ ; M̃(γ̃, δ̃) = \{ \begin{matrix}gL̃_{σ} ; M̃_{σ}(γ_{1}, s(u)), & if σ is F_{v}-elliptic in L \\ 0, & otherwise \end{matrix} .

By taking these facts into consideration, we obtain the expansion,

IM̃(γ̃, f̃) ∼^{(M̃,σ′)} ∑_{L∈L(M)} ∑_{δ̃∈σ′(U_{Lσ} (F_{v}))} gL̃ ; M̃(γ̃, δ̃)IL̃(δ̃, f̃), γ̃ ∈ M̃_{σ′}(F_{v}) ∩ G̃_{reg}(F_{v}).

The local vanishing property, Proposition 8.3, tells us that IL̃(δ̃, f̃) vanishes unless p(δ̃) = δ^{n} for some δ ∈ G(F_{v}). The set σ(U_{L_{σ}}(F_{v})) maps bijectively onto the set σ′(U_{L_{σ}}(F_{v})) under the orbit map. This can be deduced from Lemma 4.1 and the identity of cocycles (1). Hence,

IM̃(γ̃, f̃) ∼^{(M̃,σ′)} ∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} gL̃ ; M̃(γ̃, δ′)IL̃(δ′, f̃), γ̃ ∈ M̃_{σ′}(F_{v}) ∩ G̃_{reg}(F_{v}).

Once again, by the local vanishing property, IM̃(γ̃, f̃) vanishes unless p(γ̃) = γ^{n} for some γ ∈ G(F_{v}) ∩ G_{oreg}(F_{v}). We claim that for each L ∈ L(M) and δ ∈ σ(U_{L_{σ}}(F_{v})), the function gL̃ ; M̃(γ̃, δ′) has the same vanishing property. We may assume inductively that this is true for L ≠ G. Fix δ_{1} ∈ σ(U_{L_{σ}}(F_{v})). There exists a function f̃_{1} ∈ H(G̃(F_{v})) such that

IG̃(δ̃, f̃_{1}) = \{ \begin{matrix}1, & if δ̃ = δ′_{1} \\ 0, & otherwise \end{matrix} \} , δ̃ ∈ σ′(U_{G_{σ}}(F_{v}))

(cf. §3.3 [Vig81]). In particular,

IG̃(δ′, f̃_{1}) = \{ \begin{matrix}1, & if δ′ = δ′_{1} \\ 0, & otherwise \end{matrix} \} , δ ∈ σ(U_{G_{σ}}(F_{v})).

It is easily shown using Lemma 4.1 in this instance that δ′ = δ′_{1} if and only if δ = δ_{1}. Thus

IG̃(δ′, f̃_{1}) = \{ \begin{matrix}1, & δ = δ_{1} \\ 0, & otherwise \end{matrix} \} , δ ∈ σ(U_{G_{σ}}(F_{v})).

If we substitute f̃_{1} into our last germ expansion, the desired vanishing property for gG̃ ; M̃(γ̃, δ′) follows. Our germ expansion now has the form

IM̃(γ′, f̃) ∼^{(M̃,σ′)} ∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} gL̃ ; M̃(γ′, δ′)IL̃(δ′, f̃), γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}).

As noted in §3, the orbital integral of any function in C_{c}^{∞}(M̃(F_{v})) is equal to the orbital integral of a function in C_{c}^{∞}(M(F_{v})). Therefore (M̃, σ′)- equivalence may be taken to be (M, σ)-equivalence and we may write

I_{M}^{M}(γ, f̃) ∼^{(M,σ)} |n|_{v}^{r/2} ∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} gL̃ ; M̃(γ′, δ′)IL̃(δ′, f̃), γ ∈ M(F_{v})∩G_{oreg}(F_{v}).

We now follow an argument on pp. 121-122 [AC89]. Suppose σ is F_{v}-elliptic in L. Then by equation (35) we have

gL̃ ; M̃(γ′, δ′) = gL̃_{σ} ; M̃_{σ}(γ′_{1}, u′) = gL̃_{σ} ; M̃_{σ}(s(γ_{1})^{n}, s(u)^{n}),

where γ = σγ_{1} and δ = σu. The metaplectic version of the homogeneity property of germs (Proposition 10.2 [Art88c]) implies that gL̃_{σ} ; M̃_{σ}(s(γ_{1})^{n}, s(u)^{n})

is equal to the product of |n|_{v}^{(dim(L_{σu})−r)/2} with

∑_{L_{1}∈L^{L}(M)} ∑_{u_{1}∈(U_{L_{1,σ}} (F_{v}))} gL̃_{1,σ} ; M̃_{σ}(s(γ_{1}), s(u_{1}))c_{L_{1,σ}}^{L_{σ}}(u_{1}, n)[u_{1}^{L_{σ}} : u],

where c_{L_{1,σ}}^{L_{σ}}(u_{1}, n) is defined as in (20) and

[u_{1}^{L_{σ}} : u] = \{ \begin{matrix}1, & if u = u_{1}^{L_{σ}} \\ 0, & otherwise \end{matrix} .

Suppose that u = u_{1}^{L_{σ}} and set δ_{1} = σu_{1}. Then

|n|_{v}^{r/2}|n|^{(dim(L_{σu})−r)/2} = |n|^{dim(L_{1,δ_{1}})/2} = Λ^{L_{1}}(δ_{1}).

Furthermore, by Lemma 3.5 II [AC89], we have

c_{L_{1,σ^{n}}}^{L_{σ^{n}}}(u_{1}, n) = c_{L_{1,σ^{n}}}^{L_{σ^{n}}}(u_{1}^{n}, n) = c_{L_{1}}^{L}(σ_{1}^{n}u, n) = c_{L_{1}}^{L}(δ_{1}^{n}, n).

Again, by equation (35) we have

gL̃_{1,σ} ; M̃_{σ}(s(γ_{1}), s(u_{1})) = gL̃_{1} ; M̃(s(γ), s(δ_{1})).

Putting these facts together, we see that

|n|_{v}^{r/2} ∑_{δ∈σ(U_{Lσ} (F_{v}))} gL̃ ; M̃(γ′, δ′)IL̃(δ′, f̃)

is (M, σ)-equivalent to

∑_{L_{1}∈L^{L}(M)} ∑_{δ_{1}∈σ(U_{L_{1,σ}} (F_{v}))} Λ^{L_{1}}(δ_{1})gL̃_{1} ; M̃(s(γ), s(δ_{1}))c_{L_{1}}^{L}(δ_{1}^{n}, n)IL̃((δ_{1}^{L})′, f̃),

if σ is F_{v}-elliptic in L. If σ is not F_{v}-elliptic in L then equation (35) and Lemma 3.5 II [AC89] imply that both of these expressions vanish, and are thus still equal. In consequence, I_{M}^{M}(γ, f̃) is (M, σ)-equivalent to the sum of the latter expression over L ∈ L(M). Interchanging the sums over L and L_{1} and applying (20) then completes the lemma.□

In what is left of this section we compare the germ expansions of Lemma 12.3 and Lemma 12.4.

Lemma 12.5 (7.1) Suppose Theorem A (i) holds for G. Then for each u ∈ U_{G}(F_{v}) we have

g_{M}^{G}(γ, u) ∼^{(M,1)} gG̃ ; M̃(s(γ), s(u)), γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}).

Proof. We may assume by induction that

g_{M}^{L}(γ, u) ∼^{(M,1)} gL̃ ; M̃(s(γ), s(u)), γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}),

for all L ∈ L(M) such that L ≠ G. We may equate the germ expansions of Lemmas 12.3 and 12.4 since we are assuming Theorem A to hold. Together with the induction assumption, this yields

∑_{u∈(U_{G}(F_{v}))} (gG̃ ; M̃(s(γ), s(u)) − g_{M}^{G}(γ, u)) Λ^{G}(u)IG̃(u′, f̃) ∼^{(M,1)} 0,

for γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}). As in Lemma 12.4, for a a fixed element u_{1} ∈ (U_{G}(F_{v})), we may choose f̃_{1} ∈ H(G̃(F_{v})) such that

IG̃(u, f̃_{1}) = \{ \begin{matrix}1, & u = u_{1} \\ 0, & otherwise \end{matrix} \} , u ∈ (U_{G}(F_{v})).

The lemma now follows by replacing f̃ with f̃_{1} in the last (M, 1)-equivalence. □

Lemma 12.6 (7.2) Suppose γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}). Then

gG̃ ; M̃(s(γ), 1) ∼^{(M,1)} g_{M}^{G}(γ, 1).

Proof. Fix a supercuspidal representation π̃ ∈ Π_{temp}(G̃(F_{v})) and let f̃ be a matrix coefficient of π̃. Although f̃ only has compact support modulo the center of G̃(F_{v}), the distributions I_{M}(γ) and I_{M}^{Σ}(γ) are still defined at f̃. Indeed, these distributions are defined on the space H_{ac}(G̃(F_{v})) of “almost compact” Hecke functions (cf. §11 [Art89a]). Furthermore, it is clear from §27.3 [FK86] that there exists a matrix coefficient f of π̃′ such that

Λ^{G}(γ)IG̃(γ′, f̃) = I_{G}(γ, f), γ ∈ G_{oreg}(F_{v}).

The local trace formula ((9.3) [Art89c]) implies the equations,

nIM̃(γ′, f̃) = (−1)^{dim(A_{M})}|D^{G}(γ^{n})|_{v}^{1/2}Θπ̃(γ′),
I_{M}(γ, f) = (−1)^{dim(A_{M})}|D^{G}(γ)|_{v}^{1/2}Θ_{}_{π̃′}(γ),

for γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}) which are F_{v}-elliptic in M (the coefficient n in the former equation arises from an invariant sum over i(μ_{n})). It is immediate from these equations that we also have

I_{M}^{M}(γ, f̃) = n^{−1}|n|_{v}^{r/2}(−1)^{dim(A_{M})}|D^{G}(γ^{n})|_{v}^{1/2}Θπ̃(γ′),

I_{M}^{Σ}(γ, f̃) = (−1)^{dim(A_{M})} ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} |D^{G}(ηγ)|_{v}^{1/2}Θ_{}_{π̃′}(ηγ).

According to character relation (6) the right-hand sides of these equations are equal. If γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}) is not F_{v}-elliptic in M then

I_{M}^{M}(γ, f̃) = 0 = I_{M}^{Σ}(γ, f̃),

by [Art87]. Applying Lemma 11.3 we obtain

I_{M}^{M}(1, f̃) = I_{M}^{Σ}(1, f̃).

Suppose u ∈ M(F_{v}) is unipotent and u ≠ 1. Then u can be represented by an induced unipotent conjugacy class u_{1}^{M}, where u_{1} ∈ (U_{M_{1}}(F_{v})), M_{1} ∈ L and M_{1} ⫋ M. Equation (20) and the descent formula of Corollary 8.2 [Art88a] (extended to metaplectic coverings) imply that

I_{M}^{M}(u, f̃) = Λ^{M}(u) ∑_{L∈L(M)} c_{M}^{L}(u^{n}, n)IL̃((u_{1}^{L})′, f̃)
= Λ^{M}(u) ∑_{L∈L(M)} c_{M}^{L}(u^{n}, n) ∑_{L_{1}∈L(M_{1})} d_{M_{1}}^{G}(L, L_{1})ÎL̃_{1} ; M̃_{1}(u′_{1}, f̃L̃_{1}).

Since f̃ is a supercusp form on G̃ we have f̃L̃_{1} = 0 for any L_{1} ∈ L such that L_{1} ⫋ G. It follows that the right-hand side of this equation vanishes. Using a similar, but simpler, argument we see that I_{M}^{Σ}(u, f̃) also vanishes. Collecting our results, we find that many of the terms in the respective germ expansions in Lemma 12.3 and Lemma 12.4 of I_{M}^{Σ}(γ, f̃) and I_{M}^{M}(γ, f̃) about the identity disappear. Assuming inductively that the lemma holds for G replaced by L ∈ L with L ⫋ G, we find that the difference of the remaining terms is

gG̃ ; M̃(s(γ), 1)I_{G}^{M}(1, f̃) − g_{M}^{G}(γ, 1)I_{G}^{Σ}(γ, f̃)

= (gG̃ ; M̃(s(γ), 1) − g_{M}^{G}(γ, 1)) |n|_{v}^{r^{2}/2}f̃(1)
∼^{(M,1)} 0.

The lemma now follows from the fact that f̃(1) ≠ 0.□

We now define a subspace H(G̃(F_{S}))^{0} of H(G̃(F_{S})) upon which our distributions shall be easier to compare. Set H(G̃(F_{S}))^{0} to be the subspace of H(G̃(F_{S})) generated by functions,

f̃ = ∏_{v∈S} f̃_{v}, f̃_{v} ∈ H(G(F_{v})),

such that for each nonarchimedean valuation v ∈ S the orbital integral of f̃_{v} vanishes at any element of the form

(δ_{v}u_{v}, ζ), δ ∈ A_{G}(F_{v}), u_{v} ∈ U_{G}(F_{v}), u_{v} ≠ 1, ζ ∈ μ_{n}.

Proposition 12.1 (7.3) Suppose σ is as above and f̃ ∈ H(G̃(F_{v}))^{0}. Then

I_{M}^{M}(γ, f̃) ∼^{(M,σ)} ∑_{L∈L(M)} ∑_{δ∈(U_{Lσ} (F_{v}))} g_{M}^{L}(γ, δ)I_{L}^{M}(δ, f̃), γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v}).

Proof. Suppose δ = σu, where u ∈ U_{L_{σ}}(F_{v}), and γ = σγ_{1}, where γ_{1} ∈ M_{σ}(F_{v}). Then according to Lemma 9.2 [Art88c] and the proof of Lemma 12.4,

g_{M}^{L}(γ, δ) = \{ \begin{matrix}g_{M_{σ}}^{L_{σ}}(γ_{1}, u), & if σ is F_{v}-elliptic in L \\ 0, & otherwise \end{matrix}

and

gL̃ ; M̃(s(γ), s(δ)) = \{ \begin{matrix}gL̃_{σ} ; M̃_{σ}(s(γ_{1}), s(u)), & if σ is F_{v}-elliptic in L \\ 0, & otherwise \end{matrix} .

By the germ expansion of Lemma 12.4, it suffices to show that

g_{M_{σ}}^{L_{σ}}(γ_{1}, u) ∼^{(M_{σ},1)} gL̃_{σ} ; M̃_{σ}(s(γ_{1}), s(u)),

if σ is F_{v}-elliptic in L, and L ∈ L(M). Suppose that σ is F_{v}-elliptic in L and that L_{σ} ≠ G. Then the induction assumption following Theorem A holds for L_{σ} and we may apply Lemma 12.5, with L_{σ} in place of G, to obtain

g_{M_{σ}}^{L_{σ}}(γ_{1}, u) ∼^{(M_{σ},1)} gL̃_{σ} ; M̃_{σ}(s(γ_{1}), s(u)).

By Lemma 2.1 [Art88c], it follows that (M_{σ}, 1)-equivalence of these germs as functions of γ_{1} is that same as (M, σ)-equivalence as functions of γ. That is,

g_{M}^{L}(γ, δ) = g_{M_{σ}}^{L_{σ}}(γ_{1}, u) = gL̃_{σ} ; M̃_{σ}(s(γ_{1}), s(u)) = gL̃ ; M̃(s(γ), s(δ)).

The remaining possibility is that σ ∈ A_{G}(F_{v}) and L = G. Suppose this is the case. Then, by our assumptions on σ and Lemma 4.5,

∑_{δ∈σ(U_{G}(F_{v}))} gG̃ ; M̃(s(γ), s(δ))I_{G}^{M}(δ, f̃) = ∑_{δ∈σ(U_{G}(F_{v}))} gG̃ ; M̃(s(γ), s(δ))Λ^{G}(δ)IG̃(δ, f̃).

Since f̃ ∈ H(G̃(F_{v}))^{0}, we see that IG̃(δ, f̃) vanishes in the above sum unless δ = σ. Since σ is central,

gG̃ ; M̃(s(γ), s(σ)) = gG̃ ; M̃(s(γ_{1}), 1) = g_{M}^{G}(γ_{1}, 1) = g_{M}^{G}(γ, σ)

by Lemma 9.2 [Art88c] and Lemma 12.6. Hence,

∑_{δ∈σ(U_{G}(F_{v}))} gG̃ ; M̃(s(γ), s(δ))I_{G}^{M}(δ, f̃) = gG̃ ; M̃(s(γ), s(σ))I_{G}^{M}(σ, f̃)

∼^{(M,σ)} g_{M}^{G}(γ, σ)I_{G}^{M}(σ, f̃)
= ∑_{δ∈σ(U_{G}(F_{v}))} g_{M}^{G}(γ, δ)I_{G}^{M}(δ, f̃)

and the lemma is complete.□

13 The distributions IM̃(π̃, X) and I_{M}^{Σ}(π̃, X)

Leaving the geometric side of the trace formula behind, we examine the spectral side. The spectral side contains invariant distributions,

IM̃(π̃, X, f̃), π̃ ∈ Π(M̃(F_{S})), X ∈ 𝔞_{M,S}, f̃ ∈ H(G̃(F_{S})),

which are introduced in §3 [Art88a] in the case of the trivial covering. We assume that the reader is to some degree familiar with this introduction and describe some of it below in the context of non-trivial coverings.

Given ρ̃ ∈ Σ(M̃(F_{S})), we define

I_{M}^{G,Σ}(ρ̃, X, f̃) = n^{− dim(A_{G})}Î_{M}^{G}(ρ̃′, X′, f̃′) = n^{−1}Î_{M}(ρ̃′, X′, f̃′),

for all X ∈ 𝔞_{M,S} and f̃ ∈ H(G̃(F_{S})). The coefficient n^{−1} appears in this definition in order absorb the same coefficient appearing in I^{Σ}(f̃) = n^{−1}I(f̃′). If L ∈ L(M) and λ ∈ 𝔞_{M,S}^{∗} is in general position, then the induced representation ρ̃L̃ ; λ belongs to Σ(L̃(F_{S})). When ρ̃L̃ ; λ appears as an argument of IL̃(·) or I_{L}^{Σ}(·), we will often suppress the superscript L̃. For π̃ ∈ Π(M̃(F_{S})), we define I_{M}^{Σ}(π̃, X, f̃) as

∑_{P} ω_{P} ∑_{L} ∑_{ρ̃} ∫_{ε_{P} +i𝔞_{M,S}^{∗}/i𝔞_{L,S}^{∗}} rL̃ ; M̃(π̃_{λ}, ρ̃_{λ})I_{L}^{Σ}(ρ̃_{λ}, h_{L}(X), f̃)e^{−λ(X)}dλ, (36)

where P, L and ρ̃ are summed over P(M), L(M) and Σ(M̃(F_{S})) respectively. For the definitions of h_{L}(X) and ω_{P} see §6 [Art89a] and §3 [Art88a]. The definition of rL̃ ; M̃(π̃_{λ}, ρ̃_{λ}) follows §6 [Art89a].

As on p. 127 [AC89], we identify a representation π̃ in Π(M̃(F_{S})^{1}) with the orbit {π̃_{λ} : λ ∈ 𝔞_{M,C}^{∗}} in Π(M̃(F_{S})), if π̃ is not unitary. If π̃ belongs to Π_{unit}(M̃(F_{S})^{1}), then we identify it with the orbit {π̃_{λ} : λ ∈ i𝔞_{M}^{∗}} in Π_{unit}(M̃(F_{S})). We make similar identifications for representations in Π(M̃(A)^{1}) and Π_{unit}(M̃(A)^{1}). If π̃ ∈ Π_{unit}(M̃(F_{S})^{1}), set

IM̃(π, f̃) = IM̃(π̃_{λ}, 0, f̃),

and

I_{M}^{Σ}(π̃, f̃) = I_{M}^{Σ}(π̃_{λ}, 0, f̃),

for any λ ∈ i𝔞_{M}^{∗}. It may be verified that these definitions are well-defined. Both of these definitions are independent of S, if S is large, and therefore may be extended to representations in Π_{unit}(M̃(A)^{1}). In complete analogy with the expressions of the geometric side of the trace formula, we shall compare IM̃(π̃, f̃) with the term I_{M}^{Σ}(π̃, f̃) occurring in the spectral sides of the trace formulas.

We may draw parallels between the local geometric and the local spectral terms of the trace formulas. In order to compare the local geometric terms of the trace formulas for G and G̃, we use the orbit map. One might surmise that the analogous transfer map for the local spectral terms might be given by the local metaplectic correspondence (11). Unfortunately, this map does not intrinsically relate the characters of the representations to each other. In what follows, we define certain constants which relate representations in Π(M̃(F_{S})) to representations in Π(M(F_{S})) in a fashion that is compatible with their characters.

By the Langlands quotient theorem and §5 [Art89a] there exist constants Δ(π̃, ρ̃) and Γ(ρ̃, π̃) for arbitrary ρ̃ ∈ Σ(M̃(F_{S})) and π̃ ∈ Π(M̃(F_{S})), such that

tr(ρ̃) = ∑_{π̃∈Π(M̃(F_{S}))} Γ(ρ̃, π̃)tr(π̃),

and

tr(π̃) = ∑_{ρ̃∈Σ(M̃(F_{S}))} Δ(π̃, ρ̃)tr(ρ̃).

For π ∈ Π(M(F_{S})) and ρ̃ ∈ Σ(M̃(F_{S})) define

Δ(π, ρ̃) = Δ(π, ρ̃′).

Given π̃ ∈ Π(M̃(F_{S})), we set

δ(π, π̃) = ∑_{ρ̃∈Σ(M̃(F_{S}))} Δ(π, ρ̃)Γ(ρ̃, π̃).

As can be seen from the next proposition, the map of virtual characters,

Θ_{π} ↦ ∑_{π̃∈Π(M̃(F_{S}))} δ(π, π̃)Θπ̃, (37)

is the transfer map which allows us to compare characters of representations.

Proposition 13.1 (8.2) For any f̃ ∈ H(M̃(F_{S})) and matching f ∈ H(M(F_{S})) we have

tr (π(f)) = ∑_{π̃∈Π(M̃(F_{S}))} δ(π, π̃)tr(π̃(f̃)), π ∈ Π(M(F_{S})).

Proof. According to the character relations satisfied by the local metaplectic correspondence

tr (π(f)) = ∑_{ρ∈Σ(M(F_{S}))} Δ(π, ρ)tr (ρ(f))
= ∑_{ρ̃∈Σ(M̃(F_{S}))} Δ(π, ρ̃′)tr(ρ̃(f̃))
= ∑_{ρ̃∈Σ(M̃(F_{S}))} ∑_{π̃∈Π(M̃(F_{S}))} Δ(π, ρ̃′)Γ(ρ̃, π̃)tr(π̃(f̃))
= ∑_{π̃∈Π(M̃(F_{S}))} δ(π, π̃)tr(π̃(f̃)).□

Corollary 13.1 (8.3) Suppose S consists of a single valuation v, for which |n|_{v} = 1, and that π̃ ∈ Π(M̃(F_{v})) is an unramified representation. Then for any π ∈ Π(M(F_{v}))

δ(π, π̃) = \{ \begin{matrix}1, & if π = π̃′ \\ 0, & otherwise \end{matrix} .

Proof. Take f̃ to be an arbitrary function in H(M(F_{v})) which is bi-invariant under s (K_{v} ∩ M(F_{v})). Theorem 16 [FK86] tells us that tr (π̃′(f)) = tr(π̃(f̃)) for any f ∈ H(M(F_{v})) matching f̃. The corollary now follows from Proposition 13.1 and the linear independence of characters. □

This corollary allows us to define an map

δ(π, π̃) = ∏_{v} δ(π_{v}, π̃_{v})

for adelic representations π̃ = ⊗_{v}π̃_{v} ∈ Π(M̃(A)) and π = ⊗_{v}π_{v} ∈ Π(M(A)). All of the above formulas remain valid in the adelic context. Given π̃ ∈ Π(M̃(A)^{1}) and π ∈ Π(M(A)^{1}), set

δ(π, π̃) = ∑_{λ∈𝔞_{M,C}^{∗}} δ(π_{λ_{1}}, π̃_{λ})

for arbitrary λ_{1} ∈ 𝔞_{M,C}^{∗}. This definition may be verified to be well-defined.

14 Statement of Theorem B

Now that we have a spectral transfer map (37), we can begin to compare the spectral sides of the trace formulas. The spectral side of the (conjectural) trace formula for G̃ is of the form

I(f̃) = ∑_{t≥0} I_{t}(f̃),

where

I_{t}(f̃) = ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∫_{Π(M̃,t)} aM̃(π̃)IM̃(π̃, f̃)dπ̃. (38)

A description of the terms occurring in this formula may be extrapolated from §4 [Art88b]. The definition of Π(M̃, t) is given here (cf. §4 [Art88b]), as we shall often consider it. Let M_{1} ∈ L and t be a positive real number. We are obliged to first define two other sets, Π(M̃(A)^{1}, t) and Π_{disc}(M̃_{1}, t), before we define Π(M̃, t). Given a representation π̃ ∈ Π(M̃(A)^{1}), let νπ̃ be the infinitesimal character of the archimedean factor of π̃. The set Π(M̃(A)^{1}, t) is defined to be the set of (equivalence classes) of representations π̃ ∈ Π(M̃(A)^{1}) such that ∥Im(νπ̃)∥ = t. We write Π_{disc}(M̃_{1}, t) for the subset of Π_{unit}(M̃(A)^{1}, t) consisting of irreducible constituents of induced representations

σ̃M̃_{1} ; λ, L ∈ L^{M_{1}}, σ̃ ∈ Π_{unit}(L̃(A)^{1}, t), λ ∈ i𝔞_{L}^{∗}/i𝔞_{M_{1}}^{∗},

in which σ̃_{λ} satisfies the following two conditions:

1. aL̃ ; disc(σ̃) ≠ 0 (see (4.4) [Art88b]).

2. There is an element s ∈ W^{M_{1}}(𝔞_{L})_{reg} such that sσ̃_{λ} = σ̃_{λ}.

Then Π(M̃, t) is defined as the disjoint union over M_{1} ∈ L^{M} of the sets

ΠM̃_{1}(M̃, t) = {π̃ = π̃_{1,λ} : π̃_{1} ∈ Π_{disc}(M̃_{1}, t), λ ∈ i𝔞_{M_{1}}^{∗}/i𝔞_{M}^{∗}}.

The measure in (38) is given by

dπ̃ = dπ̃_{1,λ} = |W_{0}^{M_{1}}||W_{0}^{M}|^{−1}dλ.

Let us again consider §10, where we examined the geometric sides of the trace formulas. In Proposition 10.1, the geometric side of the trace formula for G̃(A) was expressed in a manner that was compatible with the orbit map. We shall follow suit by expressing the spectral side of the trace formula for G̃(A) in a manner that is compatible with the spectral transfer map (37). This will be carried out in §17. For the the time being, we set up the appropriate grouping of representations for the global datum aM̃(π̃). In other words, we define the global datum a^{M,Σ}(π̃), which ought equal the global datum aM̃(π̃), occurring in the trace formula for G̃(A). This is similar in spirit to the grouping of the local geometric terms according to μ_{n}^{M} in §6. We first define a_{disc}^{M_{1},Σ}(π̃_{1}) for M_{1} ∈ L^{M}. Set

a_{disc}^{M_{1},Σ}(π̃_{1}) = ∑_{π∈Π(M_{1}(A)^{1})} a_{disc}^{M_{1}}(π)δ(π, π̃_{1}), π̃_{1} ∈ Π(M̃_{1}(A)^{1}).

This sum may be shown to be finite using the arguments of Lemma 9.1 II [AC89]. Given π̃ = π̃_{1,λ}, where λ ∈ 𝔞_{M_{1},C}^{∗}/𝔞_{M,C}^{∗}, we set

a^{M,Σ}(π̃) = a_{disc}^{M_{1},Σ}(π̃_{1})rM̃ ; M̃_{1}(π̃_{1,λ}).

The number rM̃ ; M̃_{1}(π_{1,λ}) is not defined for arbitrary π̃_{1} ∈ Π(M̃_{1}(A)^{1}), and so the definition of a^{M,Σ}(π̃) is not valid as it now stands. The obstacle stems from the fact that rM̃ ; M̃_{1}(π̃_{1,λ}) is derived from the adelic version of the normalizing factors of intertwining operators (§5, §16). As such, it is defined as an infinite product over the valuations of F and might not converge. One expects such normalizing factors to converge and have analytic continuation for automorphic representations. This is borne out by the theory of Eisenstein series (§4 [Art89a]). In order to rectify the above definition, we make the following induction hypothesis. We assume that for any M_{1} ∈ L with M_{1} ≠ G, that

a_{disc}^{M_{1},Σ}(π̃_{1}) = a_{disc}^{M_{1}}(π̃_{1}), π̃_{1} ∈ Π(M̃_{1}(A)^{1}).

In this case a_{disc}^{M_{1},Σ}(π̃_{1}) vanishes unless π̃_{1} belongs to Π_{disc}(M̃_{1}, t). If M_{1} ∈ L^{M}, π̃_{1} ∈ Π_{disc}(M̃_{1}, t) and λ ∈ i𝔞_{M_{1},C}^{∗}/i𝔞_{M,C}^{∗} then rM̃ ; M̃_{1}(π̃_{1,λ}) is defined and the above definition of a^{M,Σ}(π̃_{1,λ}) makes sense.

The global datum a^{M,Σ}(π̃_{1}), suggests the definitions of sets of representations along the made along the same lines as the definitions of Π_{disc}(M̃_{1}, t), ΠM̃_{1}(M̃, t) and Π(M̃, t) above. We define the sets of (equivalence classes of) representations Π_{disc}^{Σ}(M̃_{1}, t), ΠΣ ; M̃_{1}(M̃, t) and Π^{Σ}(M̃, t) as above, except that aM̃ ; disc is replaced with a_{disc}^{M,Σ}.

We are now in the position to state the spectral analogue of Theorem A.

Theorem B (i) Suppose that S is a finite set of valuations with the closure property. Then

I_{M}^{Σ}(π̃, f̃) = IM̃(π̃, f̃), π̃ ∈ Π_{unit}(M̃(A)^{1}), f̃ ∈ H(G̃(F_{S})).

(ii) For any given

π̃ = π̃_{1,λ}, π̃_{1} ∈ Π(M̃_{1}(A)^{1}), λ ∈ 𝔞_{M_{1},C}^{∗}/𝔞_{M,C}^{∗},

we have

a^{M,Σ}(π̃) = aM̃(π̃).

The proof of this theorem will be completed at the end of §20.

15 Comparison of IM̃(π̃, X, f̃) and Î_{M}^{Σ}(π̃, X, f̃)

The purpose of this section is to show that Theorem A (i) implies Theorem B (i). We achieve this using the maps θL̃ ; M̃ and ^{c}θL̃ ; M̃ defined as in §4 [Art88a]. These maps are defined on H̃_{ac}(L̃(F_{S})) and take values in Ĩ_{ac}(M̃(F_{S})) for every pair of Levi subgroups M ⊂ L in L. The spaces H̃_{ac}(L̃(F_{S})) and Ĩ_{ac}(M̃(F_{S})) contain H(L̃(F_{S})) and I(M̃(F_{S})) respectively and are defined as in §11 [Art89a]. The map (13) extends in an obvious manner from a map on H(L̃(F_{S})) to a map on H̃_{ac}(L̃(F_{S})). The above maps satisfy the following properties:

∑_{L∈L(M)} θ̂L̃ ; M̃(^{c}θL̃(f̃)) = ∑_{L∈L(M)} ^{c}θ̂L̃ ; M̃(θL̃(f̃)) = 0,

IM̃(γ̃, f̃) = ∑_{L∈L(M)} ^{c}ÎL̃ ; M̃(γ̃, θL̃(f̃)), (39)

and

cIM̃(γ̃, f̃) = ∑_{L∈L(M)} ÎL̃ ; M̃(γ̃, ^{c}θL̃(f̃)), (40)

for γ̃ ∈ M̃(F_{S}) and f̃ ∈ H̃_{ac}(G̃(F_{S})). For the definition of ^{c}IM̃(γ̃) see §4 [Art88a]. Set

cI_{M}^{M}(γ, f̃) = Λ^{M}(γ)^{c}IM̃(γ′, f̃), γ ∈ M(F_{S}) ∩ G_{oreg}(F_{S}).

The analogue of property (40) for ^{c}I_{M}^{M}(γ) is then seen to be

cI_{M}^{M}(γ, f̃) = ∑_{L∈L(M)} Λ^{M}(γ)ÎL̃ ; M̃(γ′, ^{c}θL̃(f̃))
(41)  = ∑_{L∈L(M)} Î_{M}^{L,M}(γ, ^{c}θL̃(f̃)), γ ∈ M(F_{S}) ∩ G_{oreg}(F_{S}).

After a similar computation we may conclude that the analogue of property (39) is

I_{M}^{M}(γ, f̃) = ∑_{L∈L(M)} ^{c}Î_{M}^{L,M}(γ, θL̃(f̃)), γ ∈ M(F_{S}) ∩ G_{oreg}(F_{S}).

Properties (39) and (40) may also be adapted to the distributions of the form I_{M}^{Σ}(γ). We mimic the arguments of §6 to arrive at the equation

cÎ_{M}(ηγ, f̃′) = ^{c}Î_{M}(γ, f̃′), f̃ ∈ H̃_{ac}(G̃(F_{S})), γ ∈ M(F_{S}), η ∈ μ_{n}^{G}.

Thus, imitating the definition of I_{M}^{Σ}(γ), we set

cI_{M}^{Σ}(γ, f̃) = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{c}Î_{M}(ηγ, f̃′), γ ∈ M(F_{S}), f̃ ∈ H̃_{ac}(G̃(F_{S})).

The analogue of property (40) for ^{c}I_{M}^{Σ}(γ) is then seen to be

cI_{M}^{Σ}(γ, f̃) = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ∑_{L∈L(M)} Î_{M}^{L}(ηγ, ^{c}θ̂_{L}(f̃′))
= ∑_{L∈L(M)} ∑_{η∈μ_{n}^{M} /μ_{n}^{L}} ∑_{η_{1}∈μ_{n}^{L}/μ_{n}^{G}} Î_{M}^{L}(η_{1}ηγ, ^{c}θ̂_{L}(f̃′))
= ∑_{L∈L(M)} ∑_{η∈μ_{n}^{M} /μ_{n}^{L}} Î_{M}^{L}(ηγ, ∑_{η_{1}∈μ_{n}^{L}/μ_{n}^{G}} ^{η_{1}c}θ̂_{L}(f̃′)).

Here, ^{η_{1}c}θ̂_{M}(f̃′) is defined as (^{η_{1}}h)_{M} where h ∈ H̃_{ac}(M(F_{S})) such that h_{M} = cθ̂_{M}(f̃′) (Lemma 4.1 [Art88a]), and ^{η_{1}}h is the left-translate of h by η_{1} ∈ μ_{n}^{L}.

Theorem 15.1 (10.2) Fix an element M ∈ L and assume that

I_{L}^{M}(γ, f̃) = I_{L}^{Σ}(γ, f̃), f̃ ∈ H(G̃(F_{S})),

for each L ∈ L(M) and γ ∈ L_{comp}(F_{S}). Then for any f̃ ∈ H_{ac}(G̃(F_{S})) and X ∈ 𝔞_{M,S}, we have

cθM̃(f̃)′ = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′),

θM̃(f̃)′ = ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{η}θ̂_{M}(f̃′),

IM̃(ρ̃, X, f̃) = I_{M}^{Σ}(ρ̃, X, f̃), ρ̃ ∈ Σ(M̃(F_{S})),

and

IM̃(π̃, X, f̃) = I_{M}^{Σ}(π̃, X, f̃), π̃ ∈ Π(M̃(F_{S})).

Proof. We shall not prove this theorem in its entirety as its proof is almost identical to the proof of Theorem 10.2 II [AC89]. We shall however, provide a proof of the first and third assertions. We begin with the first assertion. We assume inductively that the theorem holds if G is replaced by L_{1} ∈ L such that L_{1} ⫋ G. We also make the inductive assumption that the theorem holds if M is replaced by L_{1} ∈ L(M) such that L_{1} ⫌ M. We may assume that f̃ ∈ H(G̃(F_{S})) as the restriction of any function in H̃_{ac}(G̃(F_{S})) to

G̃(F_{S}) = {γ̃ ∈ G̃(F_{S}) : H_{G}(p(γ̃)) = Z}, Z ∈ 𝔞_{G,S},

lies in H(G̃(F_{S})). Suppose γ ∈ M(F_{S})∩G_{oreg}(F_{S}) and consider the expression

cI_{M}^{M}(γ, f̃) − ^{c}I_{M}^{Σ}(γ, f̃)
= ∑_{L∈L(M)} Î_{M}^{L,M}(γ, ^{c}θL̃(f̃)) − ∑_{η∈μ_{n}^{M} /μ_{n}^{L}} Î_{M}^{L}(ηγ, ∑_{η_{1}∈μ_{n}^{L}/μ_{n}^{G}} ^{η_{1}c}θ̂_{L}(f̃′)).

Under the assumption of the theorem and the above induction hypotheses this difference reduces to

Î_{M}^{M,M}(γ, θM̃(f̃)) − Î_{M}^{M}(γ, ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′))

+ ∑_{{L∈L(M):L≠M}} Î_{M}^{L,M}(γ, ^{c}θL̃(f̃)) − Î_{M}^{Σ}(γ, ^{c}θL̃(f̃))
= Î_{M}^{M}(γ, θM̃(f̃)′ − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′)).

By Lemma 4.4 [Art88a],^{c}I_{M}^{M}(·, f̃)−^{c}I_{M}^{Σ}(·, f̃) is a function of compact support in the space M(F_{S})-conjugacy classes of M(F_{S}). Given X ∈ 𝔞_{M,S} and γ ∈ M(F_{S}) ∩ G_{oreg}(F_{S}) such that H_{M}(γ) = X, the expression Î_{M}^{M}(γ, θM̃(f̃)′ − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′)) is the orbital integral of a function defined on

M(F_{S})^{X} = {γ_{1} ∈ M(F_{S}) : H_{M}(γ_{1}) = X}.

The tempered characters of this function are

θM̃(f̃)′(π, X) − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′, π, X), π ∈ Π_{temp}(M(F_{S})).

Since Î_{M}^{M}(γ, θM̃(f̃)′) − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′)) has compact support in γ, and H_{M} maps M(F_{S}) ∩ G_{oreg}(F_{S}) onto 𝔞_{M,S}, this difference has compact support in X ∈ 𝔞_{M,S}. We shall combine this fact with the classical Paley-Wiener theorem in order to deform contours of integration later in the proof. We proceed by showing that ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′, π, X) vanishes if π ∈ Π_{temp}(M(F_{S})) is not metic, that is if π is not in the image of (10). It follows easily from the definitions that for π ∈ Π_{temp}(M(F_{S})),

∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′, π, X) = \{ \begin{matrix}n^{dim(A_{M} /A_{G})} ^{c}θ̂_{M}(f̃′, π, X), & if π is μ_{n}^{M}-invariant \\ 0, & otherwise \end{matrix} .

If π is metic then it is μ_{n}^{M}-invariant. Suppose therefore that π is not metic but is μ_{n}^{M}-invariant. Define

cÎ_{M}(ρ, X, f̃′) = ∑_{L∈L(M)} Î_{M}^{L}(ρ, X, ^{c}θ̂_{L}(f̃′)), ρ ∈ Σ(M(F_{S})), X ∈ 𝔞_{M,S}.

According to Lemma 5.2 [Art88a] and Proposition 5.4 [Art88a] there exist constants ω_{P} and a meromorphic function,

cθ̂_{M}(f̃′, ρ_{λ}) = ∫_{𝔞_{M,S}} ^{c}θ̂_{M}(f̃′, ρ_{λ}, X)e^{λ(X)}dX, λ ∈ 𝔞_{M,C}^{∗},

such that

cÎ_{M}(ρ, X, f̃′) = lim_{β} ∑_{P ∈P(M)} ω_{P} ∫_{ε_{P} +i𝔞_{M,S}^{∗}} β̂(λ)^{c}θ̂_{M}(f̃′, ρ_{λ})e^{−λ(X)}dλ. (42)

Here β is a test function in C_{c}^{∞}(𝔞_{M,S}) which approaches Dirac measure at the origin and X ∈ 𝔞_{M,S} is any point at which the left-hand side is smooth.

We take ρ = π_{ν}, where ν ∈ 𝔞_{M}^{∗} is in general position. We can then deform the contour of integration in the above integral so that

cÎ_{M}(π_{ν}, X, f̃′) = lim_{β} ∑_{P ∈P(M)} ω_{P} ∫_{i𝔞_{M,S}^{∗}} β̂(λ)^{c}θ̂_{M}(f̃′, π_{ν+λ})e^{−λ(X)}dλ

= e^{ν(X)} lim_{β} ∑_{P ∈P(M)} ω_{P} ∫_{ν+i𝔞_{M,S}^{∗}} β̂(λ − ν)^{c}θ̂_{M}(f̃′, π_{λ})e^{−λ(X)}dλ.

Notice that the function,

λ ↦ β̂(λ − ν),

is the Fourier transform of the function,

X ↦ e^{−ν(X)}β(X),

which approaches Dirac measure at the origin if β does. We may therefore replace β̂(λ − ν) with β̂(λ) in the previous equation to obtain

e^{−ν(X)} ^{c}Î_{M}(π_{ν}, X, f̃′) = lim_{β} ∑_{P ∈P(M)} ω_{P} ∫_{ν+i𝔞_{M,S}^{∗}} β̂(λ)^{c}θ̂_{M}(f̃′, π_{λ})e^{−λ(X)}dλ.

By definition λ ↦ ^{c}θ̂_{M}(f̃, π_{λ}) (43)

the Fourier transform of the function

X ↦ ^{c}θ̂_{M}(f̃′, π, X).

By our assumptions on π we have

X ↦ θM̃(f̃)′(π, X) − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′, π, X) = n^{dim(A_{M} /A_{G})} ^{c}θ̂_{M}(f̃′, π, X).

This function has compact support in 𝔞_{M,S}. It follows that (43) is the Fourier transform of a function of compact support and is therefore entire. We may consequently deform the contour of integration in the previous integral to i𝔞_{M,S}^{∗}. This implies that e^{−ν(X)} ^{c}Î(π_{ν}, X, f̃′) is independent of ν ∈ 𝔞_{M}^{∗}, for ν in general position, and by a comment on p. 143 [AC89] it follows that e^{−ν(X)} ^{c}Î(π_{ν}, X, f̃′) is independent of ν ∈ 𝔞_{M}^{∗} without restriction. According to Lemma 4.5 [Art88a] we have

∑_{P ∈P(M)} ω_{P} (X)e^{−ν_{P} (X)} ^{c}Î_{M}(π_{ν_{P}} , X, f̃′) = 0.

Here, ω_{P} (X) = vol(𝔞_{P} ∩ B_{X})vol(B_{X})^{−1}, where B_{X} is a small ball in 𝔞_{M} centered at X and ν_{P} is a point in the chamber (𝔞_{P}^{∗})^{+} which is far from the walls. It is obvious from this equation and the independence in ν just mentioned that ^{c}Î_{M}(π_{ν}, X, f̃′) = 0. Setting ν = 0 and applying Lemma 4.7 [Art88a] we obtain

cθ̂_{M}(f̃′, π, X) = ^{c}Î_{M}(π, X, f̃′) = 0.

We have just proved that ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{ηc}θ̂_{M}(f̃′, π, X) vanishes if π is not metic. Now suppose that π̃ ∈ Π_{temp}(M̃(F_{S})). Define

cI_{M}^{Σ}(ρ̃, X, f̃) = n^{− dim(A_{G})} ^{c}Î_{M}(ρ̃′, X′, f̃′)
= ∑_{L∈L(M)} n^{− dim(A_{G})}Î_{M}^{L}(ρ̃′, X′, ^{c}θ̂_{L}(f̃′))
= ∑_{L∈L(M)} n^{− dim(A_{L})}Î_{M}^{L}(ρ̃′, X′, ∑_{η∈μ_{n}^{L}/μ_{n}^{G}} ^{ηc}θ̂_{L}(f̃′)),

for ρ̃ ∈ Σ(M̃(F_{S})) and X ∈ 𝔞_{M,S}. Equation (42) allows us to write^{c}Î_{M}^{Σ}(ρ̃, X, f̃) as

n^{− dim(A_{G})} lim_{β} ∑_{P ∈P(M)} ω_{P} ∫_{ε_{P} +i𝔞_{M,S}^{∗}} β̂(λ′)^{c}θ̂_{M}(f̃′, (ρ̃_{λ})′)e^{−λ′(X′)}dλ′

= lim_{β} ∑_{P ∈P(M)} ω_{P} ∫_{ε_{P} +i𝔞_{M,S}^{∗}} β̂(λ)n^{dim(A_{M} /A_{G})} ^{c}θ̂_{M}(f̃′, (ρ̃_{λ})′)e^{−λ(X)}dλ.

Defining

cÎM̃(ρ̃, X, f̃) = ∑_{L∈L(M)} ÎL̃ ; M̃(ρ̃, X, ^{c}θL̃(f̃)), ρ̃ ∈ Σ(M̃(F_{S})), X ∈ 𝔞_{M,S},

we follow the earlier argument to show that ^{c}IM̃(π̃_{ν}, X, f̃) − ^{c}I_{M}^{Σ}(π̃_{ν}, X, f̃) is equal to

lim_{β} ∑_{P ∈P(M)} ω_{P}

× ∫_{i𝔞_{M,S}^{∗}} β̂(λ) (^{c}θM̃(f̃)′((π̃_{ν+λ})′) − n^{dim(A_{M} /A_{G})} ^{c}θ̂_{M}(f̃′, (π̃_{ν+λ})′)) e^{−λ(X)}dλ,

and that this expression vanishes. It now follows as before that

cθM̃(f̃, π̃, X) − n^{− dim(A_{G})}θ̂_{M}(f̃′, π̃′, X′) = 0.

This equation may be rewritten as

n^{− dim(A_{M})} ^{c}θM̃(f̃)′(π̃′, X′) − n^{− dim(A_{G})} ^{c}θ̂_{M}(f̃′, π̃′, X′) = 0,

and the first assertion follows. The first assertion implies that

cθM̃(f̃)′((π̃_{λ})′) = n^{dim(A_{M} /A_{G})} ^{c}θ̂_{M}(f̃′, (π̃_{λ})′), π̃ ∈ Π_{temp}(M̃(F_{S})), λ ∈ 𝔞_{M,C}^{∗}.

By analytic continuation, this formula holds if π̃ is replaced by a standard representation. Recalling (42), this implies that

cIM̃(ρ̃, X, f̃) = ^{c}I_{M}^{Σ}(ρ̃, X, f̃), ρ̃ ∈ Σ(M̃(F_{S})).

By the induction assumption and the first assertion we have

0 = ^{c}IM̃(ρ̃, X, f̃) − ^{c}I_{M}^{Σ}(ρ̃, X, f̃)
= ∑_{L∈L(M)} ÎL̃ ; M̃(ρ̃, X, ^{c}θL̃(f̃)) − Î_{M}^{L,Σ}(ρ̃, X, ^{c}θL̃(f̃))
= IM̃(ρ̃, X, f̃) − I_{M}^{Σ}(ρ̃, X, f̃).

This is the third assertion.□

With Theorem 15.1 in place, the proof of of Theorem B (i) follows mutatis mutandis from the argument on p. 145 [AC89]. We include it here for the sake of continuity. We wish to show that

I_{M}^{Σ}(π̃, 0, f̃) = IM̃(π̃, 0, f̃), π̃ ∈ Π_{unit}(M̃(F_{S})).

The distribution on the left is defined by equation (36), while the distribution on the right is defined by

∑_{P} ω_{P} ∑_{L} ∑_{ρ̃} ∫_{ε_{P} +i𝔞_{M̃,S}^{∗}/i𝔞_{L̃,S}^{∗}} rL̃ ; M̃(π̃_{λ}, ρ̃_{λ})IL̃(ρ̃_{λ}, hL̃(X), f̃)e^{−λ(X)}dλ,

where P, L and ρ̃ are summed over P(M), L(M) and Σ(M̃(F_{S})) respectively. The number rL̃ ; M̃(π̃_{λ}, ρ̃_{λ}) vanishes if Δ(π̃, ρ̃) vanishes ((6.4) [Art89a]). Comparing the expansions of these two distributions, it is clear that it suffices to prove I_{L}^{Σ}(ρ̃_{λ}, h_{L}(X), f̃) = IL̃(ρ̃_{λ}, h_{L}(X), f̃), (44) for all L ∈ L(M), X ∈ 𝔞_{M}, λ ∈ 𝔞_{M,C}^{∗} with small real part, and ρ̃ ∈ Σ(M̃(F_{S})) such that Δ(π̃, ρ̃) ≠ 0. By using the splitting property for these distributions (Proposition 9.4 [Art88a]) it suffices to prove (44) for S = {v}. Suppose then that ρ̃ ∈ Σ(M̃(F_{v})) and Δ(π̃, ρ̃) ≠ 0. Then the central character of ρ̃ must be unitary. From the definition of standard representations, it follows that ρ̃ is either tempered or induced from a proper parabolic subgroup of M. Suppose first that ρ is tempered. Then by the proof of Lemma 3.1 [Art88a], we have

I_{L}^{Σ}(ρ̃_{λ}, h_{L}(X), f̃) = \{ \begin{matrix}0, & L ≠ G \\ ∫_{i𝔞_{G,v}^{∗}} tr (ρ̃_{λ+ν}(f̃)) e^{−ν(h_{G}(X))}dν, & L = G \end{matrix}
= IL̃(ρ̃_{λ}, h_{L}(X), f̃).

Now suppose that ρ̃ = ρ̃M̃ ; 1, where ρ̃_{1} ∈ Σ(M̃_{1}(F_{v})) and M_{1} is a proper Levi subgroup of M. We apply the descent property (Corollary 8.5 [Art88a]) to (44) and find that it suffices to show

Î_{M_{1}}^{L_{1},Σ}(ρ̃_{1,λ}, X_{1}, f̃L̃_{1}) = ÎL̃_{1} ; M̃_{1}(ρ̃_{1,λ}, X_{1}, f̃L̃_{1}), (45)

for X_{1} ∈ 𝔞_{M_{1}} and L_{1} ∈ L(M_{1}) with L_{1} ≠ G. The induction hypothesis of following Theorem A allows us to apply Theorem 15.1, with G replaced by L_{1}, in order to obtain (45). The proof is now complete. □

16 More on normalizing factors

This section is devoted to the construction of a few additional (G, M) families which we shall require for the comparison of the spectral sides of the trace formulas. We shall return to the actual comparison of the trace formulas in the following section.

Take π̃ = ⊗_{v}π̃_{v} and π = ⊗_{v}π_{v} to be representations in Π(M̃(A)) and Π(M(A)) respectively, and suppose that

δ(π, π̃) = ∏_{v} δ(π_{v}, π̃_{v})

does not vanish (§13). Define

r̃_{P̃_{1}|P̃_{2}}(π_{v}, π̃_{v}) = r_{P_{1}|P_{2}}(π_{v})^{−1}r_{P̃_{1}|P̃_{2}}(π̃_{v}), P_{1}, P_{2} ∈ P(M).

If |n|_{v} = 1 and both π̃_{v} and π_{v} are unramified representations then π̃′_{v} = π_{v} by Corollary 13.1. The normalization of §5 then implies that

r̃_{P̃_{1}|P̃_{2}}(π_{v}, π̃_{v}) = r_{P_{1}|P_{2}}(π_{v})^{−1}r_{P̃_{1}|P̃_{2}}(π̃_{v}) = r_{P_{1}|P_{2}}(π_{v})^{−1}r_{P_{1}|P_{2}}(π_{v}) = 1.

This fact allows us to define the infinite product

r̃_{P̃_{1}|P̃_{2}}(π_{λ′}, π̃_{λ}) = ∏_{v} r̃P̃_{1}|P_{2}(π_{v,λ′}, π̃_{v,λ}), P_{1}, P_{2} ∈ P(M), λ ∈ 𝔞_{M,C}^{∗}.

Given P_{1} ∈ P(M) we define a (G, M) family by

r̃P̃(ν, π_{λ′}, π̃_{λ}, P_{1}) = r̃_{P̃|P̃_{1}}(π_{λ′}, π̃_{λ})^{−1}r̃_{P̃|P̃_{1}}(π_{λ′+ν′}, π̃_{λ+ν}),

where P ∈ P(M) and ν ∈ i𝔞_{M}^{∗}.

Lemma 16.1 (11.3) (a) Take π̃ and π as above. Then for each L ∈ L(M), r̃L̃ ; M̃(π_{λ′}, π̃_{λ}) is independent of P_{1} and is also a rational function of the variables {λ(α^{∨}), q_{v}^{−λ(α^{∨})}}_{v∈S}, where α runs over the roots of (G, A_{M}), q_{v} is the order of the residue field of F_{v} and S is a finite set of valuations outside of which π̃ and π are unramified,.

(b) Suppose in addition that π̃ ∈ Π_{disc}(M̃, t) and π ∈ Π_{disc}(M, t). Then

r̃_{P̃_{1}|P̃_{2}}(π_{λ′}, π̃_{λ}) = r_{P_{1}|P_{2}}(π_{λ′})^{−1}r_{P̃_{1}|P̃_{2}}(π̃_{λ}), P_{1}, P_{2} ∈ P(M).

In particular, for each L ∈ L(M), the function r̃L̃ ; M̃(π_{λ′}, π̃_{λ}) is regular for λ ∈ i𝔞_{M}^{∗}. Moreover,

rL̃ ; M̃(π̃_{λ}) = ∑_{L_{1}∈L^{L}(M)} n^{dim(A_{M} /A_{L_{1}})}r_{M}^{L_{1}}(π_{λ′})r̃L̃ ; L̃_{1}(π_{λ′}, π̃_{λ}).

Proof. Part (a) of the lemma follows from the computations on p.149 [AC89]. Under the hypotheses of part (b), r_{P̃_{1}|P̃_{2}}(π̃_{λ}) and r_{P_{1}|P_{2}}(π_{λ′}) are regular functions in λ ∈ i𝔞_{M}^{∗} (§6 [Art82]). Thus, if one unravels the definition of r̃_{P̃_{1}|P̃_{2}}(π_{λ′}, π̃_{λ}), one obtains the first equality and the the regularity on i𝔞_{M}^{∗}. The last equality follows from an application of Lemma 6.5 [Art81] to

rP̃(ν, π̃_{λ}, P̃_{1}) = r̃P̃(ν, π_{λ′}, π̃_{λ}, P̃_{1})r_{P} (ν′, π_{λ′}, P_{1}).□

We may define further (G, M) families along the same lines as r̃M̃(π_{λ′}, π̃_{λ}). If we replace π̃ in the above discussion with some ρ̃ ∈ Σ(M̃(A)) such that Δ(π, ρ̃) ≠ 0, we obtain the (G, M) family

r̃P̃(ν, π_{λ}, ρ̃_{λ}, P̃_{1}) = r̃_{P̃|P̃_{1}}(π_{λ′}, ρ̃_{λ})^{−1}r̃_{P̃|P̃_{1}}(π_{λ′+ν′}, ρ̃_{λ+ν}).

We define yet another (G, M) family for representations π_{1}, π_{2} ∈ Π(M(A)) such that δ(π_{i}, π̃) ≠ 0 for i = 1, 2. Set

r̃P̃(ν, π_{1,λ′}, π_{2,λ′}, P̃_{1}) = r̃P̃(ν, π_{1,λ′}, π̃, P̃_{1})r̃P̃(ν, π_{2,λ′}, π̃_{λ}, P̃_{1})^{−1}.

This (G, M) family is independent of π̃. Lemma 6.5 [Art81] applied to this last (G, M) family yields

r̃L̃ ; M̃(π_{1,λ′}, π̃_{λ}) = ∑_{L_{1}∈L^{L}(M)} r̃L̃_{1} ; M̃(π_{1,λ′}, π_{2,λ′})r̃L̃ ; L̃_{1}(π_{2,λ′}, π̃_{λ}). (46)

For arbitrary π ∈ Π(M(A)), ρ̃ = ⊗_{v}ρ̃_{v} ∈ Σ(M̃(A)) and π̃ ∈ Π(M̃(A)). Set

rP̃(π̃_{λ}, ρ̃_{λ}) = Δ(π̃, ρ̃)r̃P̃(π̃_{λ}, ρ̃_{λ}),
rP̃(ν, π_{λ′}, π̃_{λ}) = δ(π, π̃)r̃P̃(ν, π_{λ′}, π̃_{λ}),

and

rP̃(ν, π_{λ′}, ρ̃_{λ}) = Δ(π, ρ̃)r̃P̃(ν, π_{λ′}, ρ̃_{λ}).

In the following lemma we compare the final (G, M) family in this list to another, keeping in mind the normalization in §5. Lemma 16.2 (11.4) For each L ∈ L(M) we have

rL̃ ; M̃(π_{λ′}, ρ̃_{λ}) = n^{dim(A_{M} /A_{L})}r_{M}^{L}(π_{λ′}, ρ̃′_{λ′}).

Proof. According to Lemma 6.2 [Art81], rL̃ ; M̃(π_{λ′}, ρ̃_{λ}) is equal to

Δ(π, ρ̃) lim_{ν→0} ∑_{P ∈P^{L}(M)} θ_{P}^{L}(ν)^{−1}r_{P |P_{1}}^{L}(π_{λ′})r_{P |P_{1}}^{L}(π_{λ′+ν′})^{−1}rL̃ ; P̃|P̃_{1}(ρ̃_{λ})^{−1}rL̃ ; P̃|P̃_{1}(ρ̃_{λ+ν}),

where

θ_{P}^{L}(ν) = vol (𝔞_{M}^{L}/Z(Δ_{P}^{L})^{∨})^{−1} ∏_{α∈Δ_{P}^{L}} ν(α^{∨}),

and Δ_{P}^{L} are the simple roots of (P, A_{M}). (We apologize to the reader for the similarity in notation of θ_{P}^{L} with the maps of §15.) Making the change of variables from ν to ν′ in the limit and noting that θ_{P}^{L}(n^{−1}ν) = n^{− dim(A_{M} /A_{L})}θ_{P}^{L}, this expression can be rewritten as the product of Δ(π, ρ̃′)n^{dim(A_{M} /A_{L})} with

lim_{ν→0} ∑_{P ∈P^{L}(M)} θ_{P}^{L}(ν)^{−1}r_{P |P_{1}}^{L}(π_{λ′})r_{P |P_{1}}^{L}(π_{λ′+ν})^{−1}r_{P |P_{1}}^{L}(ρ̃′_{λ′})^{−1}r_{P |P_{1}}^{L}(ρ̃′_{λ′+ν}).

This is by definition equal to n^{dim(A_{M} /A_{L})}r_{M}^{L}(π_{λ′}, ρ̃′_{λ′}).□

The other (G, M) families defined in this section satisfy versions of Lemma 16.2 as well. These versions are proven similarly.

17 A formula for I_{t}^{Σ}

The object of this section is to express the spectral side

∑_{t≥0} I_{t}^{Σ}(f̃) = ∑_{t≥0} n^{−1}I_{t}(f̃′)

in a manner that is compatible with the spectral expansion (37). This amounts to expressing the spectral side for G(A) in terms of the global datum a^{M,Σ} and the set of representations Π^{Σ}(M, t).

The integrals in I_{t}^{Σ}(f̃) are taken over spaces of representations which, on the face of it, are not metic. We need the following spectral vanishing property to ensure that these non-metic representations do not appear in I^{Σ}(f̃).

Proposition 17.1 Suppose ρ ∈ Σ(M(F_{S})). Then

Î_{M}(ρ, X, f̃′) = 0, f̃ ∈ H(G̃(F_{S})),

unless ρ is metic.

Proof. Recall that ρ is metic if there exists f̃_{1} ∈ H(M̃(F_{S})) such that f̃′_{1}(ρ) ≠ 0. Therefore, the proposition follows in the case M = G. We assume inductively that the proposition holds if G is replaced by L ∈ L with L ⫋ G. By the splitting property, Proposition 9.4 [Art88a] (and the remark immediately following it), it suffices to prove the proposition for S = {v}. Suppose that ρ is not metic. We may write ρ = π_{ν}^{M}, where M_{1} ∈ L^{M}, π ∈ Π_{temp}(M_{1}(F_{v})) and ν ∈ 𝔞_{M_{1}}^{∗}. Since the local metaplectic correspondence commutes with induction it is clear that π_{ν} is not metic. According to Corollary 8.5 [Art88a], the Fourier transform,

∫_{i𝔞_{M_{1},v}^{∗}/i𝔞_{M,v}^{∗}} Î_{M}(ρ_{λ}, X, f̃′)e^{−λ(X_{1})}dλ,

(where the projection h_{M}(X_{1}) of X_{1} ∈ 𝔞_{M_{1},v} onto 𝔞_{M,v} is X ∈ 𝔞_{M,v},) is equal to

∑_{L∈L(M_{1})} d_{M_{1}}^{G}(M, L)Î_{M_{1}}^{L}(π_{ν}, X, f̃′).

If M_{1} ⫋ M then d_{M_{1}}^{G}(M, G) = 0 and the right-hand side vanishes by the induction assumption. Taking the inverse Fourier transform then implies that Î_{M}(ρ_{λ}, X, f̃′) vanishes as well. We may therefore suppose that M_{1} = M, in which case ρ = π_{ν}. If v is nonarchimedean the infinitesimal character of π_{ν} is given by a finite set of cuspidal pairs (ρ_{1}, M_{1}), …, (ρ_{s}, M_{s}) where M_{1}, …, M_{s} ∈ L^{M} and ρ_{1} ∈ Σ(M_{1}(F_{v})), …, ρ_{s} ∈ Σ(M_{s}(F_{v})). If v is an archimedean valuation, then the infinitesimal character of π_{ν} also determines a set of cuspidal pairs (ρ_{1}, M_{0}), …, (ρ_{s}, M_{0}) as above. It follows from the remark immediately preceding Theorem 27.3 [FK86] and Theorem 26 [FK86] in the nonarchimedean case; and §3 [Mezb] in the archimedean case, that these supercuspidal (and hence elliptic) representations are not metic. The discussion of §7 [Art89b] tells us that for ν ∈ 𝔞_{M}^{∗} in general position we may express Î_{M}(π_{ν}, X, f̃′)e^{−ν(X)} as

∑_{{L_{1},L∈L(M):L_{1}⊃L⫌M}}∫_{ν_{L}+i𝔞_{L,v}^{∗}/i𝔞_{M_{1},v}^{∗}} ∑_{i=1}^{t} ∑_{j=1}^{s} (Δ_{ij}(π)Φ) (λ_{i}, ρ_{j})dλ.

Here, Δ_{ij}(π) is a differential operator on 𝔞_{M,C}^{∗} × 𝔞_{M_{j},C}^{∗} and Φ(λ_{i}, ρ_{j}) is equal to the product of

lim_{ν_{1}→0} ∑_{Q∈P^{L_{1}}(L)} r̃_{Q|Q_{0}}(π_{λ_{i}+λ}^{L}, ρ_{j,λ})^{−1}r̃_{Q|Q_{0}}(π_{λ_{i}+λ+ν_{1}}^{L}, ρ_{j,λ+ν_{1}})θ_{Q}^{L_{1}}(ν_{1})^{−1}

with Î_{L_{1}}(ρ_{j,λ}^{L}, h_{L_{1}}(X), f̃′). It is obvious from this equation that Î_{M}(π_{ν}, X, f̃′) vanishes if

Î_{L_{1}}(ρ_{j,λ}^{L}, h_{L_{1}}(X), f̃′) = 0, L_{1} ⊃ L ⫌ M.

Since L_{1} ⫌ M and ρ_{j,λ} is not metic, the earlier descent argument shows that this is indeed true. We have thus succeeded in showing that Î_{M}(π_{ν}, X, f̃′) = 0 if ν ∈ 𝔞_{M}^{∗} is in general position. However, the mean value property (Lemma 3.2 [Art88a]) expresses the value of Î_{M}(π_{ν}, X, f̃′) at any ν ∈ 𝔞_{M}^{∗} in terms of nearby points. In consequence, Î_{M}(π_{ν}, X, f̃′) vanishes for all ν ∈ 𝔞_{M}^{∗}.□

The following lemma gives an expansion of the local distributions occurring in I_{t}^{Σ}(f̃) in terms of the distributions I_{M}^{Σ}(π̃).

Lemma 17.1 (12.1) Suppose that π ∈ Π_{unit}(M(A)^{1}) and f̃ ∈ H(G̃(F_{S})). Then n^{−1}Î_{M}(π, f̃′) is equal to

∑_{L∈L(M)} ∑_{π̃∈Π(M̃(A)^{1})} ∫_{ε_{M} +i𝔞_{M}^{∗} /i𝔞_{L}^{∗}} rL̃ ; M̃(π_{λ′}, π̃_{λ})I_{L}^{Σ}(π̃_{λ}, f̃)dλ. (47)

Proof. This proof is almost identical to the proof of Lemma 12.1 II [AC89]. It is included so that the reader may feel a sense of continuity. Any statements which seem unjustified may be compared to the analogous statements in Lemma 12.1, where the details are given. To begin, relabel the summation index L in (47) by L_{1}. We then replace I_{L_{1}}^{Σ}(π̃_{λ}, f̃) with expression (36),

∑_{Q∈P(L_{1})} ω_{Q} ∑_{L∈L(L_{1})} ∑_{ρ̃∈Π(M̃(A)^{1})} ∫_{ε_{Q}+i𝔞_{L_{1}}^{∗}/i𝔞_{L}^{∗}} rL̃ ; L̃_{1}(π̃_{λ+μ}, ρ̃_{λ+μ})I_{L}^{Σ}(ρ̃_{λ+μ}, f̃)dμ.

We deform the contour of integration in μ so that (47) is equal to the sum over L_{1}, L ∈ L(M), with L_{1} ⊂ L, of

∑_{π̃∈Π(M̃(A)^{1})} ∑_{ρ̃∈Σ(M̃(A)^{1})} ∫_{ε_{M} +i𝔞_{M}^{∗} /i𝔞_{L}^{∗}} rL̃ ; M̃(π_{λ′}, π̃_{λ})rL̃ ; L̃_{1}(π̃_{λ}, ρ̃_{λ})I_{L}^{Σ}(ρ̃_{λ}, f̃)dλ.

Taking the sums over L_{1} and π̃ inside the integral we find that

∑_{π̃∈Π(M̃(A)^{1})} ∑_{L_{1}∈L^{L}(M)} rL̃_{1} ; M̃(π_{λ′}, π̃_{λ})rL̃ ; L̃_{1}(π̃_{λ}, ρ̃_{λ})

= r̃L̃ ; M̃(π_{λ′}, ρ̃_{λ}) ∑_{π̃∈Π(M̃(A)^{1})} δ(π, π̃)Δ(π̃, ρ̃)
= r̃L̃ ; M̃(π_{λ′}, ρ̃_{λ}) ∑_{ρ̃_{1}∈Σ(M̃(A)^{1})} Δ(π, ρ̃_{1}) ∑_{π̃∈Π(M̃(A)^{1})} Γ(ρ̃_{1}, π̃)Δ(π̃, ρ̃)
= r̃L̃ ; M̃(π_{λ′}, ρ̃_{λ})Δ(π, ρ̃)
= rL̃ ; M̃(π_{λ′}, ρ̃_{λ}).

Expansion (47) is therefore equal to

∑_{L∈L(M)} ∑_{ρ̃∈Σ(M̃(A)^{1})} ∫_{ε_{M} +i𝔞_{M}^{∗} /i𝔞_{L}^{∗}} rL̃ ; M̃(π_{λ′}, ρ̃_{λ})I_{L}^{Σ}(ρ̃_{λ}, f̃)dλ. (48)

By Lemma 16.2 and the spectral vanishing property, Proposition 17.1, we have

∑_{ρ̃∈Σ(M̃(A)^{1})} rL̃ ; M̃(π_{λ′}, ρ̃_{λ})I_{L}^{Σ}(ρ̃_{λ}, f̃)

= n^{dim(A_{M} /A_{L})} ∑_{ρ̃∈Σ(M(A)^{1})} r_{M}^{L}(π_{λ′}, ρ̃′_{λ′}) (n^{−1}Î_{L}(ρ̃′_{λ′}, f̃′))
= n^{dim(A_{M} /A_{L})} ∑_{ρ∈Σ(M(A)^{1})} r_{M}^{L}(π_{λ′}, ρ_{λ′}) (n^{−1}Î_{L}(ρ_{λ′}, f̃′)) .

Substituting back into (48) and noting that dλ′ = n^{dim(A_{M} /A_{L})}dλ, we obtain

∑_{L∈L(M)} ∑_{ρ∈Σ(M(A)^{1})} ∫_{ε_{M} +i𝔞_{M}^{∗} /i𝔞_{L}^{∗}} r_{M}^{L}(π_{λ}, ρ_{λ}) (n^{−1}Î_{L}(ρ_{λ}, f̃′)) dλ.

Since π is unitary, this expression is equal to n^{−1}Î_{M}(π, f̃′).□

Proposition 17.2 (12.2) Suppose that t ≥ 0 and f̃ ∈ H(G(A)). Then

I_{t}^{Σ}(f̃) = ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∫_{Π^{Σ}(M,t)} a^{M,Σ}(π̃)I_{M}^{Σ}(π̃, f̃)dπ. (49)

Proof. From §13 (cf. (38) we know that I_{t}^{Σ}(f̃) equals

∑_{{M_{1},M∈L:M⊃M_{1}}} |W_{0}^{M_{1}}||W_{0}^{G}|^{−1}

× ∑_{π∈Π_{disc}(M_{1},t)} ∫_{i𝔞_{M_{1}}^{∗}/i𝔞_{M}^{∗}} a_{disc}^{M}(π)r_{M_{1}}^{M}(π_{λ′})n^{−1}Î_{M}(π_{λ′}, f̃′)dλ′.

Substituting expression (47), deforming the contour of integration appropriately, and noting that dλ′ = n^{dim(A_{M_{1}}/A_{M})}dλ, we find that I_{t}^{Σ}(f̃) equals

∑_{{M_{1},M,L∈L:M_{1}⊂M⊂L}} ∑_{π̃_{1}∈Π(M̃_{1}(A)^{1},t)} |W_{0}^{M_{1}}||W_{0}^{G}|^{−1}n^{dim(A_{M_{1}}/A_{M})}

(50) × ∫_{ε_{M} +i𝔞_{M_{1}}^{∗}/i𝔞_{L}^{∗}} ∑_{π∈Π_{disc}(M_{1},t)} a_{disc}^{M_{1}}(π)r_{M_{1}}^{M}(π_{λ′})rL̃ ; M̃(π_{λ′}, π̃_{1,λ})I_{L}^{Σ}(π̃_{1,λ}, f)dλ.

The term rL̃ ; M̃(π_{λ′}, π̃_{1,λ}) in the above sum vanishes unless δ(π, π̃_{1}) ≠ 0. Fix some π_{1} ∈ Π_{disc}(M_{1}, t) such that δ(π_{1}, π̃_{1}) ≠ 0. Then for any other π ∈ Π_{disc}(M_{1}, t) with δ(π, π̃_{1}) ≠ 0 we may write

n^{dim(A_{M_{1}}/A_{M})}r_{M_{1}}^{M}(π_{λ′})
= ∑_{{L_{1}:M_{1}⊂L_{1}⊂M}} n^{dim(A_{M_{1}}/A_{L_{1}})}r_{M_{1}}^{L_{1}}(π_{1,λ′})n^{dim(A_{L_{1}}/A_{M})}r̃_{L_{1}}^{M}(π_{1,λ′}, π_{λ′})
= ∑_{{L_{1}:M_{1}⊂L_{1}⊂M}} n^{dim(A_{M_{1}}/A_{L_{1}})}r_{M_{1}}^{L_{1}}(π_{1,λ′})r̃M̃ ; L̃_{1}(π_{1,λ′}, π_{λ′}).

We substitute this expression into (50) and deform the contour of integration from ε_{M} +i𝔞_{M_{1}}^{∗}/i𝔞_{L}^{∗} to ε_{L_{1}} +i𝔞_{M_{1}}^{∗}/i𝔞_{L}^{∗} for some small regular point ε_{L_{1}} in 𝔞_{L_{1}}^{∗}. We then bring the sum over M inside the integral. Notice that

∑_{{M:L_{1}⊂M⊂L}} r̃M̃ ; L̃_{1}(π_{1,λ′}, π_{λ′})rL̃ ; M̃(π_{λ′}, π̃_{1,λ}) = δ(π, π̃_{1})r̃L̃ ; L̃_{1}(π_{1,λ′}, π̃_{1,λ})

by equation (46). Observe also that

∑_{π∈Π_{disc}(M_{1},t)} a_{disc}^{M_{1}}(π)δ(π, π̃_{1}) = a_{disc}^{M_{1},Σ}(π̃_{1}).

Now (50) may be written as

∑_{{M_{1},L∈L:M_{1}⊂L}} |W_{0}^{M_{1}}||W_{0}^{G}|^{−1} ∑_{π̃_{1}∈Π(M̃_{1}(A)^{1},t)} ∑_{{L_{1}:M_{1}⊂L_{1}⊂L}} n^{dim(A_{M_{1}}/A_{L_{1}})}

(51) × ∫_{ε_{L_{1}}+i𝔞_{M_{1}}^{∗}/i𝔞_{L}^{∗}} a_{disc}^{M_{1},Σ}(π̃_{1})r_{M_{1}}^{L_{1}}(π_{1,λ′})r̃L̃ ; L̃_{1}(π_{1,λ′}, π̃_{1,λ})I_{L}^{Σ}(π̃_{1,λ}, f̃)dλ.

The summand indexed by M_{1} = G in (51) reduces to

(52)  ∑_{π̃_{1}∈Π_{disc}^{Σ}(G̃,t)} a_{disc}^{Σ}(π̃_{1})I_{G}^{Σ}(π̃_{1}, f̃).

If M_{1} ≠ G then the induction hypothesis stated after the definition of a_{disc}^{M_{1},Σ} implies that

a_{disc}^{M_{1},Σ}(π̃_{1}) = aM̃_{1} ; disc(π̃_{1}), π̃_{1} ∈ Π(M̃_{1}(A)^{1}, t).

It is immediate from this equation that a_{disc}^{M_{1},Σ}(π̃_{1}) vanishes unless π̃_{1} ∈ Π_{disc}(M̃_{1}, t). Thus, by a variant of Lemma 16.1, the integrand of (51) is analytic for λ near i𝔞_{M_{1}}^{∗}, and we may deform the contour of integration from ε_{L_{1}} + i𝔞_{M_{1}}^{∗}/i𝔞_{L}^{∗} to i𝔞_{M_{1}}^{∗}/i𝔞_{L}^{∗}. This allows us to take the sum over L_{1} inside the integral. It is a simple exercise in (G, M) families (cf. proof of Lemma 16.1 (b)) to show that

∑_{{L_{1}:M_{1}⊂L_{1}⊂L}} n^{dim(A_{M_{1}}/A_{L_{1}})}r_{M_{1}}^{L_{1}}(π_{1,λ′})r̃L̃ ; L̃_{1}(π_{1,λ′}, π̃_{1,λ}) = rL̃ ; M̃_{1}(π̃_{1,λ}).

Therefore the summand indexed by M_{1} ≠ G in (51) is equal to

|W_{0}^{M_{1}}||W_{0}^{G}|^{−1} ∑_{L∈L(M_{1})} ∑_{π̃_{1}∈Π_{disc}(M̃_{1},t)} ∫_{i𝔞_{M_{1}}^{∗}/i𝔞_{L}^{∗}} a_{disc}^{M_{1},Σ}(π̃_{1})rL̃ ; M̃_{1}(π̃_{1,λ})I_{L}^{Σ}(π̃_{1,λ}, f̃)dλ.

Combining this expression with (52) we obtain

I_{t}^{Σ}(f̃) = ∑_{L∈L} |W_{0}^{L}||W_{0}^{G}|^{−1} ∫_{Π^{Σ}(L̃,t)} a^{L,Σ}(π̃)I_{L}^{Σ}(π̃, f̃)dπ̃.□

We can now apply Proposition 17.2 to obtain a striking comparison between the spectral sides of the trace formulas.

Lemma 17.2 (12.3) Suppose that t ≥ 0 and f̃ ∈ H(G̃(A)). Then

I_{t}^{Σ}(f̃) − I_{t}(f̃) = ∑_{π̃∈Π(G̃(A)^{1},t)} (a_{disc}^{Σ}(π̃) − a_{disc}(π̃)) tr(π̃(f̃^{1})),

where f̃^{1} is the restriction of f̃ to G̃(A)^{1}.

Proof. This proof is almost identical to Lemma 12.3 II [AC89] and is included solely for the sake of continuity. Consider the difference of (49) and

I_{t}(f̃) = ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∫_{Π(M̃,t)} a^{M}(π̃)IM̃(π̃, f̃)dπ̃.

If M_{1} ∈ L and M_{1} ⫋ M ⊂ G then the induction hypothesis of §14 implies that

a^{M,Σ}(π̃) = aM̃(π̃), π̃ ∈ ΠM̃_{1}(M̃, t),

and ΠΣ ; M̃_{1}(M̃, t) = ΠM̃_{1}(M̃, t). If π̃ is not unitary both a^{M,Σ}(π̃) and a^{M}(π̃) vanish. When π̃ is unitary we know from the discussion at the end of §15 that

I_{M}^{Σ}(π̃, f̃) = IM̃(π̃, f̃).

Therefore, the only terms which remain in the difference, I_{t}^{Σ}(f̃) − I_{t}(f̃), are those parameterized by M_{1} = M = G. The lemma now follows from

I_{G}^{Σ}(π̃, f̃) = tr(π̃(f̃^{1})) = IG̃(π̃, f̃), π̃ ∈ Π_{disc}(G̃, t).□

18 The map ε_{M}

Having simplified the comparison of the spectral sides, we do the same for the geometric sides. We lighten the burden of this task by adding yet another induction hypothesis. From this point on, fix M ∈ L such that M ≠ G. The additional induction hypothesis is that

I_{L}^{M}(γ, f̃) = I_{L}^{Σ}(γ, f̃), γ ∈ L_{comp}(F_{S}), f̃ ∈ H(G̃(F_{S})),

where L ∈ L(M) with L ≠ M.

The following proposition brings us closer to the proof of Theorem A (i) in that it expresses the difference of our invariant distributions in terms of orbital integrals, which are easier to manage.

Proposition 18.1 (13.2) There exists a unique map

ε_{M} : H(G̃(F_{S}))^{0} → I_{ac}(M(F_{S})),

such that

I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃) = Î_{M}^{M}(γ, ε_{M}(f̃)), γ ∈ M_{comp}(F_{S}), f̃ ∈ H(G̃(F_{S}))^{0}.

The map satisfies the splitting property

ε_{M}(f̃) = ε_{M}(f̃_{1})f̃′_{2,M} + f̃′_{1,M}ε_{M}(f̃_{2}),

where S is the disjoint union of S_{1} and S_{2} (each satisfying the closure property) and f̃ = f̃_{1}f̃_{2} is a corresponding decomposition.

Proof. If ε_{M} satisfies the first property of the proposition then the earlier splitting properties (Proposition 6.2, (23)) imply that the splitting property for ε_{M}. Therefore, we only need to show that the map ε_{M} exists. In other words, must show that

ε_{M}(γ, f̃) = I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃), γ ∈ M_{comp}(F_{S}),

is an orbital integral at γ of a function in H_{ac}(M(F_{S})). The splitting properties (Proposition 6.2, (23)) allow us to restrict our proof to the case that S is comprised of a single valuation v. Also, Lemma 11.3 allows us to restrict to the case that γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}). Suppose first that v is archimedean. By our assumptions on F, this means that F_{v} = C. We may therefore apply Jordan canonical form to conclude that every element of M(C) ∩ G_{oreg}(C) is G(C)-conjugate to an element in M_{0}(C). If M ⫌ M_{0} then Lemma 10.2 implies that ε_{M}(γ, f̃) vanishes. In other words, if M ⫌ M_{0} then

ε_{M}(f̃) = 0, f̃ ∈ H(G̃(C)). (53)

We therefore need only consider the case M = M_{0}. We proceed by showing that, for fixed f̃ ∈ H(G̃(C)), the function,

γ ↦ ε_{M_{0}}(γ, f̃), γ ∈ M_{0}(C) ∩ G_{oreg}(C),

extends to a smooth function of M_{0}(C). Suppose γ_{0} is an arbitrary element of M_{0}(C). Then, by the definition of I_{M_{0}}^{M}(γ_{0}, f̃) − I_{M_{0}}^{Σ}(γ_{0}, f̃) (§§6-7) and Lemma 11.2, the limit,

lim_{{a→1:a∈A_{M_{0},reg}(C)}} ∑_{L∈L} r_{M_{0}}^{L}(γ_{0}^{n}, a) (I_{L}^{M}(aγ_{0}, f̃) − I_{L}^{Σ}(aγ_{0}, f̃))

= lim_{{a→1:a∈A_{M_{0},reg}(C)}} ε_{M_{0}}(aγ_{0}, f̃),

exists. In consequence, ε_{M_{0}}(·, f̃) extends to a continuous function on M_{0}(C). To see that this function is smooth, suppose that z is a differential operator in Z_{C}, the center of the universal enveloping algebra of the complexified Lie algebra of G(C). This operator passes to a differential operator on G̃(C) ≅ G(C) × μ_{n} by virtue of its action on G(C). By Lemma 7.2 [Mezb], we have

ε_{M_{0}}(γ_{0}, zf̃) = lim_{{a→1:a∈A_{M_{0},reg}(C)}} ε_{M_{0}}(aγ_{0}, zf̃)
= lim_{{a→1:a∈A_{M_{0},reg}(C)}} z′_{M_{0}}ε_{M_{0}}(aγ_{0}, f̃).

Here, z_{M_{0}} is the image of z under the Harish-Chandra isomorphism and

z_{M_{0}} ↦ z′_{M_{0}}

is an algebra automorphism (§6 [Mezb]). We can choose a set of generators, z_{1}, …, z_{2r}, of Z_{C} and apply a well-known argument involving the fundamental theorem of calculus to show that z′_{i,M_{0}}ε_{M_{0}}(γ_{0}, f̃), 1 ≤ i ≤ 2r, exists and is equal to

lim_{{a→1:a∈A_{M_{0},reg}(C)}} z′_{i,M_{0}}ε_{M_{0}}(aγ_{0}, f̃) = ε_{M_{0}}(γ_{0}, z_{i}f̃).

A simple induction argument on the number of generators occurring in z then implies that z′_{M_{0}}ε_{M_{0}}(γ_{0}, f̃) exists and

z′_{M_{0}}ε_{M_{0}}(γ_{0}, f̃) = ε_{M_{0}}(γ_{0}, zf̃). (54)

This proves that ε_{M_{0}}(·, f̃) extends to a smooth function on M_{0}(C). Combining this fact with estimate (5.7) [Art94] and the inductive definitions of our invariant distributions we find that ε_{M_{0}}(·, f̃) is actually in the Schwartz space of M_{0}(C). Let ε̂_{M_{0}}(f̃) be its Fourier transform. If ε̂_{M_{0}}(f̃) belongs to I_{ac}(M_{0}(C)) then we may define ε_{M_{0}}(f̃) by

(ε_{M_{0}}(f̃))_{M_{0}} = ε̂_{M_{0}}(f̃)

and the proposition is proven at the archimedean valuations. The only requirement that ε̂_{M_{0}}(f̃) does not obviously satisfy for it to be in I_{ac}(M̃_{0}(C)) is the K_{v} ∩ M_{0}(C)-finiteness requirement. More precisely, we must show that there exists a finite set Γ_{M_{0}} of (equivalence classes of) irreducible admissible representations of K_{v} ∩ M_{0}(C) such that ε̂_{M_{0}}(f̃)(π) vanishes unless the restriction of π ∈ Π_{temp}(M_{0}(C)) to K_{v} ∩ M_{0}(C) contains a representation in Γ_{M_{0}}. Since f̃ is K̃_{v}-finite, there exists a finite set Γ̃ of (equivalence classes of) irreducible admissible representations of K̃_{v} such that f̃M̃_{0}(π̃), π̃ ∈ Π_{temp}(M̃_{0}(C)), vanishes unless the restriction of π̃G̃ to K̃_{v} contains a representation in Γ̃. Clearly, Γ̃ determines a finite set of (equivalence classes of) irreducible admissible representations Γ of K_{v} such that f̃′_{M_{0}}(π), π ∈ Π_{temp}(M_{0}(C)), vanishes unless the restriction of π^{G} to K_{v} contains a representation in Γ. Let Γ_{M_{0}} be the finite set of irreducible constituents of restrictions of representations in Γ to K_{v} ∩ M_{0}(C). It is a straightforward consequence of equation (54) that

f̃ ↦ ε̂_{M_{0}}(f̃)(π̃′), π̃ ∈ Π_{temp}(M̃_{0}(C)),

is an invariant eigendistribution of Z_{C}. It then follows from Harish-Chandra’s work ([HC65]) that there exists a smooth function c on Π_{temp}(M̃_{0}(C)) such that

ε̃_{M_{0}}(f̃)(π̃′) = c(π̃)f̃M̃_{0}(π̃) = c(π̃)f̃′_{M_{0}}(π̃′), π̃ ∈ Π_{temp}(M̃_{0}(C)). (55)

Moreover, if π ∈ Π_{temp}(M_{0}(C)) is not metic then it is not μ_{n}^{M_{0}}-invariant (Lemma 3.1 [Mezb]). Since ε_{M_{0}}(·, f̃) is μ_{n}^{M_{0}}-invariant, we must have ε̂_{M_{0}}(f̃)(π) = 0. This vanishing property together with equation (55) imply that Γ_{M_{0}} satisfies the K_{v} ∩ M_{0}(C)-finiteness requirement for ε̂_{M_{0}}(f̃).

Now suppose v is nonarchimedean. For γ ∈ M(F_{v}) ∩ G_{oreg}(F_{v}) define

cε_{M}(γ, f̃) = ^{c}I_{M}^{M}(γ, f̃) − ^{c}I_{M}^{Σ}(γ, f̃), f̃ ∈ H(G̃(F_{v}))^{0},

(cf. §15). By Lemma 4.4 [Art88a], this function has compact support as a function of M(F_{v})-conjugacy classes of M(F_{v}). It is related to ε_{M}(γ, f̃) by

ε_{M}(γ, f̃) − ^{c}ε_{M}(γ, f̃) = Î_{M}^{M}(γ, θM̃(f̃)′ − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{η}θ̂_{M}(f̃′)).

To see this, recall that by the induction assumption after Theorem A, Theorem 15.1, equations (39)-(40) and their analogues for I_{M}^{Σ}(γ), we have

ε_{M}(γ, f̃) − ^{c}ε_{M}(γ, f̃)

= ∑_{{L∈L(M):L⫋G}} ^{c}Î_{M}^{L,M}(γ, θL̃(f̃)) − ∑_{η_{1}∈μ_{n}^{M} /μ_{n}^{L}} Î_{M}^{L}(η_{1}γ, ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{η}θ̂_{L}(f̃′))
= ∑_{{L∈L(M):M⫋L⫋G}} ^{c}Î_{M}^{L,M}(γ, θL̃(f̃)) − ^{c}Î_{M}^{L,Σ}(γ, θL̃(f̃))
+ Î_{M}^{M}(γ, θM̃(f̃)′ − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{η}θ̂_{M}(f̃′))
= Î_{M}^{M}(γ, θM̃(f̃)′ − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{η}θ̂_{M}(f̃′)).

Since θM̃(f̃)′ − ∑_{η∈μ_{n}^{M} /μ_{n}^{G}} ^{η}θ̂_{M}(f̃′) belongs to I_{ac}(M(F_{v})) (§4 [Art88a]), this equation implies that^{c}ε_{M}(γ, f̃) is an orbital integral of a function in H_{ac}(M(F_{v})) if and only if ε_{M}(γ, f̃) is as well. More generally, given a semisimple element σ ∈ M(F_{v}), it implies that

cε_{M}(γ, f̃) ∼^{(M,σ)} 0, γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v})

if and only if

ε_{M}(γ, f̃) ∼^{(M,σ)} 0, γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v}).

We shall complete the proposition by showing that

ε_{M}(γ, f̃) ∼^{(M,σ)} 0, γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v})

and then showing that this implies that ^{c}ε_{M}(γ, f̃) is an orbital integral of a function in H_{ac}(M(F_{v})). If σ is not F_{v}-elliptic in M, that is, if 𝔞_{M_{σ}} ≠ 𝔞_{M}, then there is a proper Levi subgroup M_{1} ∈ L^{M} of M such that M_{σ}(F_{v}) ⊂ M_{1}(F_{v}). Lemma 10.2 then implies that

ε_{M}(γ, f̃) ∼^{(M,σ)} 0, γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v}).

Suppose that σ is F_{v}-elliptic in M. We may assume that σ is F_{v}-elliptic in L ∈ L(M) if and only if σ^{n} is F_{v}-elliptic in L. Indeed, we know by Lemma 12.1 that a translate of σ by an element in μ_{n}^{M} satisfies this property and we also know that

ε_{M}(ηγ, f̃) = ε_{M}(γ, f̃), η ∈ μ_{n}^{M}.

We may therefore apply Lemma 12.3 and Proposition 12.1 to see that ε_{M}(γ, f̃) is (M, σ)-equivalent to

∑_{L∈L(M)} ∑_{δ∈σ(U_{Lσ} (F_{v}))} g_{M}^{L}(γ, δ) (I_{L}^{M}(δ, f̃) − I_{L}^{Σ}(δ, f̃)) , γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v}).

Since g_{M}^{M}(γ, δ) = 0, we may apply the induction hypothesis of this section to conclude that the right-hand side vanishes. From our earlier remark we now have

cε_{M}(γ, f̃) ∼^{(M,σ)} ε_{M}(γ, f̃) ∼^{(M,σ)} 0, γ ∈ M_{σ}(F_{v}) ∩ G_{oreg}(F_{v}),

for any semisimple element σ ∈ M(F_{v}). We use a partition of unity argument on the compact support of ^{c}ε_{M}(γ, f̃) in the space of M(F_{v})-conjugacy classes to conclude that ^{c}ε_{M}(γ, f̃) is in fact an orbital integral.□

Suppose v is a valuation for which |n|_{v} = 1. Then it is an immediate consequence of Lemma 11.4 that ε_{M}(f̃_{v}^{0}) = 0. It therefore follows from the splitting property of Proposition 18.1 that the map ε_{M} extends to a map on H(G̃(A))^{0}.

19 Comparison for f̃ ∈ H(G̃(A), M)

We now give a sketch of the proof that I(f̃) = I^{Σ}(f̃) for f̃ in a certain subspace H(G̃(A), M) of H(G̃(A)). The train of reasoning in this section is based entirely on §§15-16 II [AC89]. We shall outline the arguments found there and leave the confirmation of the details to the reader.

Recall that M ∈ L was fixed in the previous section. Let H(G̃(A), M) be the subspace of H(G̃(A)) spanned by functions

f̃ = ∏_{v} f̃_{v}, f̃_{v} ∈ H(G̃(F_{v})),

which satisfy the following property. For two nonarchimedean places v_{1} and v_{2}, which are not in {v : |n|_{v} ≠ 1},

f̃_{v_{i},L̃} = 0, L ∈ L, i = 1, 2,

unless L contains a conjugate of M. If S contains {v : |n|_{v} ≠ 1} and at least two other nonarchimedean places, we define H(G̃(F_{S}), M) in the same way.

The proof of Lemma 13.1 II [AC89] may be imitated to obtain the following lemma.

Lemma 19.1 (13.1) For f̃ ∈ H(G̃(A), M), the distribution

I(f̃) − I^{Σ}(f̃)

equals the sum of

|W (𝔞_{M})|^{−1} ∑_{γ∈(M(F))_{M,S}/μ_{n}^{M}} a^{M}(S, γ) (I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃))

and

∑_{δ∈A_{G}(F)\μ_{n}^{G}} ∑_{u∈(U_{G}(F))_{G,S}} (aM̃(S, u′) − a(S, u)) I_{G}^{M}(δu, f̃).

Proof. The lemma follows from the splitting properties and the properties of f̃. See Lemma 13.1 II [AC89] for details.□

Let H(G̃(A), M)^{0} be the space of functions f̃ in

H(G(A), M) ∩ H(G(A))^{0}

which satisfy one additional condition, namely, that f̃ vanishes at any element in G̃(A) whose component at each nonarchimedean place v belongs to A_{G}(F_{v}). This ensures that f̃ ∈ H(G̃(A), M)^{0} vanishes at

(δu, ζ), δ ∈ A_{G}(F), u ∈ U_{G}(F), ζ ∈ μ_{n}.

Lemma 19.2 (15.1) Suppose that f̃ ∈ H(G̃(A), M)^{0}. Then

I(f̃) − I^{Σ}(f̃) = n^{dim(A_{M})}|W (𝔞_{M})|^{−1}Î^{M}(ε_{M}(f̃)),

where I^{M} is the analogue for M of I = I^{G}.

Proof. By the properties of f̃ Lemma 19.1 and Proposition 18.1, we see that I(f̃) − I^{Σ}(f̃) is equal to

n^{dim(A_{M})}|W (𝔞_{M})|^{−1} ∑_{γ∈(M(F))_{M,S}} a^{M}(S, γ)Î_{M}^{M}(γ, ε_{M}(f̃)),

for a large set of valuations S. By the descent property (Corollary 8.3 [Art88a]) and Lemma 10.2 it follows that

Î_{M_{1}}^{M_{1}}(γ, ε_{M}(f̃)_{M_{1}}) = Î_{M}^{M}(γ, ε_{M}(f̃)) = 0, γ ∈ M_{1}(F_{S}) ∩ G_{oreg}(F_{S}),

for any M_{1} ∈ L such that M_{1} ⫋ M. Therefore, for such M_{1}, ε_{M}(f̃)_{M_{1}} = 0. Combining this with the splitting property (Proposition 9.1 [Art88a]) applied to I_{M_{1}}^{M}(γ), we find that

Î_{M_{1}}^{M}(γ, ε_{M}(f)) = 0, γ ∈ M_{1}(F) ∩ G_{oreg}(F),

for any M_{1} ∈ L^{M} such that M_{1} ⫋ M. Thus by (25),

n^{dim(A_{M})}|W (𝔞_{M})|^{−1} ∑_{γ∈(M(F))_{M,S}} a^{M}(S, γ)Î_{M}^{M}(γ, ε_{M}(f̃))

= n^{dim(A_{M})}|W (𝔞_{M})|^{−1} ∑_{M_{1}∈L^{M}} |W_{0}^{M_{1}}||W_{0}^{M}|^{−1} ∑_{γ∈(M_{1}(F))_{M_{1},S}} a^{M_{1}}(S, γ)Î_{M_{1}}^{M}(γ, ε_{M}(f̃))
= n^{dim(A_{M})}|W (𝔞_{M})|^{−1}Î^{M}(ε_{M}(f̃)).□

We now give an outline of the method of separation by infinitesimal characters, which is to be found in §15 II [AC89]. Let S_{∞} denote the set of archimedean (in our case complex) valuations of F. Then G(F_{S_{∞}}) may be regarded as a real Lie group. Let 𝔥_{C} denote the standard Cartan subalgebra of its complexified Lie algebra. Let 𝔥 be the real form of 𝔥_{C} associated to the split real form of G(F_{S_{∞}}). Then 𝔥 contains 𝔞_{M_{0}} as a vector space and in turn contains all vector spaces of the form 𝔞_{L}, L ∈ L. Let 𝔥^{1} be the orthogonal complement of 𝔞_{G} in 𝔥. We recall the theory of multipliers ([Art83]). Let α belong to E(𝔥^{1})^{W}, the convolution algebra of compactly supported, distributions on 𝔥^{1}, which are invariant under the complex Weyl group W of G(F_{S_{∞}}). Then there is an action, f̃ ↦ f̃_{α}, on H(G̃(A)) such that

f̃_{α,M̃}(π̃) = α̂(νπ̃)f̃M̃(π̃), π̃ ∈ Π(M̃(A)).

As usual, νπ̃ is taken to be the infinitesimal character of the archimedean factor of π̃. This action of E(𝔥^{1})^{W} on H(G̃(A)) affects only the archimedean factor of f̃, and is supported on characters. The map,

α ↦ α′, α ∈ E(𝔥^{1})^{W} ,

given by α′(ν) = n^{− dim(𝔥^{1})}α(n^{−1}ν), ν ∈ 𝔥^{1}, (56)

is compatible with map (13). More precisely, given α ∈ E(𝔥^{1})^{W} and π̃ ∈ Π_{temp}(G̃(F_{S})), we have νπ̃ = ν_{n^{−1}π̃′} (§3 [Mezb]) and in turn

(f̃_{α})′(π̃′) = α̂(n^{−1}ν_{}_{π̃′})f̃G̃(π̃) = α̂′(ν_{}_{π̃′})f̃′(π̃′) = f̃′_{α′}(π̃′).

This map is also compatible with the map ε_{M} of Proposition 18.1.

Lemma 19.3 (14.4) Suppose f̃ ∈ H(G̃(A)), and α ∈ E(𝔥^{1})^{W}. Then ε_{M}(f̃_{α}) = ε_{M}(f)_{α′}.

Proof. Suppose f̃ = ∏_{v} f̃_{v} ∈ H(G̃(A)) and suppose S contains all valuations at which f̃_{v} ≠ f̃_{v}^{0}. If v is nonarchimedean then α acts trivially on f̃_{v}, that is f̃_{v,α} = f̃_{v}. If v is complex, it follows from equations (53) and (55) that ε_{M}(f̃_{v,α}) = ε_{M}(f̃)_{α′}. Repeated applications of the splitting property of Proposition 18.1 yield

ε_{M}(f̃_{α}) = ∑_{v_{1}∈S} ε_{M}(f_{v_{1},α}) ∏_{w∈S−v_{1}}(f̃_{w,α})′_{M} ∏_{v∉S}(f̃_{v}^{0})′_{M}
= ∑_{v_{1}∈S} ε_{M}(f_{v_{1}})_{α′} ∏_{w∈S−v_{1}}((f̃_{w})′_{α′})_{M} ∏_{v∉S}(f̃_{v}^{0})′_{M}
= ε_{M}(f̃)_{α′}.□

Let 𝔥_{u}^{∗} be the set of points ν in 𝔥_{C}^{∗}/i𝔞_{G}^{∗} such that the complex conjugate of ν with respect to 𝔥^{∗} is equal to −sν for some element s ∈ W of order two. The archimedean infinitesimal character of νπ̃ associated to any π̃ ∈ Π_{unit}(G̃(A)^{1}) belongs to 𝔥_{u}^{∗}. Given ν_{1} ∈ 𝔥_{u}^{∗} define

Π_{ν_{1}}(G̃(A)^{1}) = {π̃ ∈ Π(G̃(A)^{1}) : νπ̃ = sν_{1} for some s ∈ W }.

We can use multipliers to prove the following lemma.

Lemma 19.4 (15.4) For each f̃ ∈ H(G̃(A), M)^{0} and ν_{1} ∈ 𝔥_{u}^{∗}, we have

∑_{π̃∈Π_{ν_{1}}(G̃(A)^{1})} (a_{disc}(π̃) − a_{disc}^{Σ}(π̃)) tr (π̃(f̃^{1})) = 0.

As the proof of Lemma 19.4 is almost identical to that of Lemma 15.4 II [AC89], we shall only sketch the proof. Given f̃ ∈ H(G̃(A), M)^{0}, T ≥ 0 and ν_{1} ∈ 𝔥_{u}^{∗} we choose α_{1} ∈ E(𝔥^{1})^{W} as in Lemma 15.2 II [AC89]. The multiplier α_{1} has the property that 0 ≤ α̂_{1}(νπ̃) ≤ 1 for any π̃ ∈ Π_{unit}(G̃(A)) whose K-type is the same as that of f̃. Furthermore the inverse image of 1 under α̂_{1} is {sν_{1} : s ∈ W }. Following Corollaries 14.2-14.3 II [AC89], we find that ε_{M}(f̃) is a moderate function in I_{ac}(M(A)) in the sense of §6 [Art88b]. We may then apply Corollary 6.5 [Art88b] (cf. (15.5) II [AC89]) to obtain

|∑_{t≤T} I_{t}(f̃_{α_{1}^{m}}) − I_{t}^{Σ}(f̃_{α_{1}^{m}}) − n^{dim(A_{M})}|W (𝔞_{M})|^{−1}Î_{t}^{M}(ε_{M}(f̃)_{(α′_{1})^{m}})| ≤ Ce^{−kNm}

for some positive constants C, k and N. Thus

∑_{t≤T} I_{t}(f̃_{α_{1}^{m}}) − I_{t}^{Σ}(f̃_{α_{1}^{m}}) (57)

approaches

∑_{t≤T} n^{dim(A_{M})}|W (𝔞_{M})|^{−1}Î_{t}^{M}(ε_{M}(f̃)_{(α′_{1})^{m}}) (58)

as m approaches infinity. According to Lemma 15.3 II [AC89] we may write (58) as

∑_{t≤T} n^{dim(A_{M})}|W (𝔞_{M})|^{−1} ∑_{π∈Π_{disc}(M,t)} a_{disc}^{M}(π) ∫_{i𝔞_{M}^{∗} /i𝔞_{G}^{∗}} ε_{M}(f̃^{1}, π, λ)α̂′_{1}(ν_{π} + λ)^{m}dλ,

for some Schwartz function

λ ↦ ε_{M}(f̃^{1}, π, λ), λ ∈ i𝔞_{M}^{∗}/i𝔞_{G}^{∗}.

The multiplier α_{1} was chosen so that

0 ≤ α̂′_{1}(ν_{π} + λ) < 1,

for all but finitely many λ ∈ i𝔞_{M}^{∗}/i𝔞_{G}^{∗} in the above integral. Thus, by the dominated convergence theorem, the integral approaches zero as m approaches infinity. Expression (57) therefore also approaches zero as m approaches infinity. Moreover, by Lemma 17.2 it is equal to

∑_{t≤T} ∑_{π̃∈Π(G̃(A)^{1},t)}(a_{disc}(π̃) − a_{disc}^{Σ}(π̃))tr (π̃(f̃^{1})) α̂_{1}(νπ̃)^{m}.

By our choice of α_{1}, the limit of this expression as m approaches infinity is the expression in Lemma 19.4.

In §16 II [AC89] it is shown, using the Plancherel formula, how to extend Lemma 19.4 to functions in H(G̃(A), M). We shall not repeat the arguments here. We merely translate the final result of that section into the following proposition.

Proposition 19.1 (16.2) For any f̃ ∈ H(G̃(A), M), we have

I(f̃) = I^{Σ}(f̃).

Proof. By Lemma 17.2 and the extension of Lemma 19.4 to H(G̃(A), M), the difference I_{t}(f̃) − I_{t}^{Σ}(f̃), t ≥ 0, is equal to

∑_{{ν_{1}:∥Im(ν_{1})∥=t}} ∑_{π̃∈Π_{ν_{1}}(G̃(A)^{1})} (a_{disc}(π̃) − a_{disc}^{Σ}(π̃)) tr (π̃(f̃^{1})) = 0.

It is then obvious that

I(f̃) = ∑_{t} I_{t}(f̃) = ∑_{t} I_{t}^{Σ}(f̃) = I^{Σ}(f̃).□

20 Completion of the proofs of Theorems A and B

We first prove Theorem A (i). The structure of the proof is as follows. We begin by showing that for any valuation v of F and each γ_{v} ∈ M(F_{v}) ∩ G_{oreg}(F_{v}) there exists a constant ε_{M}(γ_{v}) such that

I_{M}^{M}(γ_{v}, f̃) − I_{M}^{Σ}(γ_{v}, f̃) = ε_{M}(γ_{v})I_{G}^{M}(γ_{v}, f̃), f̃ ∈ H(G̃(F_{v})).

This part of the proof follows §17 II [AC89] almost exactly. It then follows from the splitting properties that for any S_{0} containing {v : |n|_{v} ≠ 1} and

γ = ∏_{v∈S_{0}} γ_{v} ∈ M(F_{S_{0}}) ∩ G_{oreg}(F_{S_{0}}),

we have

I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃) = (∑_{v∈S_{0}} ε_{M}(γ_{v})) I_{G}^{M}(γ, f̃), f̃ ∈ H(G̃(F_{S_{0}})). (59)

Using the strong approximation theorem, we show that ∑_{v∈S_{0}} ε_{M}(γ_{v}) vanishes unless M = M_{0}.^{1} The induction hypothesis of §18 and Corollary 11.1 then imply that ε_{M}(γ_{v}) = 0 unless M = M_{0}. We then take care of the case M = M_{0} by using the local trace formula as in the proof of Corollary 11.1.

Suppose f̃ ∈ H(G̃(A), M). According to Lemma 19.1 and Proposition 19.1 the sum of

|W (𝔞_{M})|^{−1} ∑_{γ∈(M(F))_{M,S}/μ_{n}^{M}} a^{M}(S, γ) (I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃)) (60)

and

∑_{δ∈A_{G}(F)\μ_{n}^{G}} ∑_{u∈(U_{G}(F))_{G,S}} (a^{M}(S, u) − a(S, u)) I_{G}^{M}(δu, f̃) (61)

vanishes. Recall that S is a large finite set of valuations which depends on f̃ and contains {v : |n|_{v} ≠ 1}. We may assume that S = {v_{1}, …, v_{k}}, where v_{1} and v_{2} are any two distinct valuations satisfying

|n|_{v_{1}} = |n|_{v_{2}} = 1.

Take γ to be an element in M(F) ∩ G_{oreg}(F) such that γ is F_{v_{i}}-elliptic in M(F_{v_{i}}), i = 1, 2. Choose f̃_{i} ∈ H(G̃(F_{v_{i}})) such that it is supported on a very small neighborhood of γ′i(μ_{n}) in G̃(F_{v_{i}}) and

Î_{M}^{M,M}(γ, f̃_{i,M̃}) = I_{G}^{M}(γ, f̃_{i}) = 1, i = 1, 2.

For the remaining 3 ≤ i ≤ k, let f̃_{i} ∈ H(G̃(F_{v_{i}})) be arbitrary and set f̃ = f̃_{1} ⋯ f̃_{k}. This function lies in H(G̃(F_{S}), M) by the construction of f̃_{1} and f̃_{2}. If we shrink the support of f̃_{1} and f̃_{2} around γ′i(μ_{n}) the set S is not affected. Bearing in mind that M̃(F) is a discrete subgroup of M̃(A), we may then assume that the support of f̃_{1} and f̃_{2} is so small that (61) vanishes and the only contribution to the sum in (60) comes from G̃(F_{S})-conjugates of γ′. The distributions in (60) are constant on G̃(F_{S})-conjugacy classes. By Theorem 8.2 [Art86] we have

1I am indebted to J. Arthur for providing the underlying ideas for this portion of the proof.

a^{M}(S, γ) = vol(M_{γ}(F)\M_{γ}(A)^{1}),

for S large enough (in a sense depending only on γ). Consequently,

I_{M}^{M}(γ, f̃) − I_{M}^{Σ}(γ, f̃) = 0. (62)

We apply the splitting properties (Proposition 6.2, equation (23)) and the induction hypothesis of §10 to this equation repeatedly in order to obtain

∑_{i=1}^{k} (I_{M}^{M}(γ, f̃_{i}) − I_{M}^{Σ}(γ, f̃_{i})) ∏_{j≠i} Î_{M}^{M,M}(γ, f̃_{j,M̃}) = 0. (63)

Suppose 3 ≤ i ≤ k. Choose γ ∈ M(F) as in (63), f̃_{i} ∈ H(G̃(F_{v_{i}})) such that

Î_{M}^{M,M}(γ, f̃_{i,M̃}) = I_{G}^{M}(γ, f̃_{i}) = 0,

and the remaining f̃_{j} ∈ H(G̃(F_{v_{j}})), j ≠ i, such that

Î_{M}^{M,M}(γ, f̃_{j,M̃}) ≠ 0.

In this case, the left-hand side of (63) is a nonzero multiple of

I_{M}^{M}(γ, f̃_{i}) − I_{M}^{Σ}(γ, f̃_{i}).

Consequently, this distribution vanishes for any f̃_{i} ∈ H(G̃(F_{v_{i}})) such that I_{G}^{M}(γ, f̃_{i}) vanishes. This implies that there exists a constant ε_{M}(γ) such that

I_{M}^{M}(γ, f̃_{i}) − I_{M}^{Σ}(γ, f̃_{i}) = ε_{M}(γ)I_{G}^{M}(γ, f̃_{i}), (64)

for any f̃_{i} ∈ H(G̃(F_{v_{i}})). Observe that if M ⫌ M_{0} and F_{v_{i}} = C then the lefthand side of this equation vanishes by Jordan canonical form and Lemma 10.2. This implies that ε_{M}(γ) = 0 at the archimedean places whenever M ⫌ M_{0}.

Let V_{i} = {v_{i}, v_{1}, v_{2}}. Suppose that v_{i} is nonarchimedean. The set

{∏_{v∈V_{i}} γ_{v} ∈ M(F_{V_{i}}) ∩ G_{oreg}(F_{V_{i}}) : γ_{v_{j}} is F_{v_{j}}-elliptic in M(F_{v_{j}}), j = 1, 2}

is open in M(F_{V_{i}}). Since M(F) is dense in M(F_{V_{i}}), it is also dense in the above open subset. It follows that we may approximate any γ_{i} ∈ M(F_{v_{i}}) by an element γ occurring in (64). Since all of the distributions in (64) are smooth on M(F_{v_{i}}) ∩ G_{oreg}(F_{v_{i}}), the function ε_{M} also extends to a smooth function on this space and we have

I_{M}^{M}(γ_{i}, f̃_{i}) − I_{M}^{Σ}(γ_{i}, f̃_{i}) = ε_{M}(γ_{i})I_{G}^{M}(γ_{i}, f̃_{i}), γ_{i} ∈ M(F_{v_{i}}) ∩ G_{oreg}(F_{v_{i}}). (65)

Now suppose that F_{v_{i}} = C. If M ⫌ M_{0} then, as we have already remarked, equation (65) holds with ε_{M}(γ_{i}) = 0. Otherwise, it is obvious that one can approximate γ_{i} ∈ M_{0}(C) ∩ G_{oreg}(C) by an element γ ∈ M(F) ∩ G_{oreg}(F) which is F_{v_{j}}-elliptic in M_{0}(F_{v_{j}}), for j = 1, 2. After all, the latter condition is satisfied for any element in M_{0}(F_{v_{j}}), j = 1, 2. As a result, equation (65) holds for the archimedean places as well. Finally, notice that since our choice of v_{1} and v_{2} was arbitrary as long as |n|_{v_{1}} = |n|_{v_{2}} = 1, equation (65) holds for any 1 ≤ i ≤ k.

Suppose M ⫌ M_{0} and S_{0} ⊃ {v : |n|_{v} ≠ 1}. We may assume that S ⊃ S_{0}. By approximating elements of M(F_{S}) with elements of M(F) as above, repeated use of the splitting properties (Proposition 6.2, equation (23)) and the induction hypothesis of §10, we obtain equation (59) from (62). We would like to show that ∑_{v∈S_{0}} ε_{M}(γ_{v}) vanishes and then apply Corollary 11.1.

Suppose that v ∉ S_{0} and γ_{v} lies in K_{v} ∩ M(F_{v}) ∩ G_{oreg}(F_{v}). It is obvious that I_{G}^{M}(γ_{v}, f̃_{v}^{0}) ≠ 0 ((27)). However, according to Lemma 11.4 the left-hand side of equation (65) vanishes. Therefore,

ε_{M}(γ_{v}) = 0, γ_{v} ∈ K_{v} ∩ M(F_{v}) ∩ G_{oreg}(F_{v}), v ∉ S_{0}. (66)

Recall decomposition (4),

M = ∏_{i=1}^{ℓ} M(i) ≅ ∏_{i=1}^{ℓ} GL(r_{i}).

We consider the subgroup ∏_{i=1}^{ℓ} SL(r_{i}) of ∏_{i=1}^{ℓ} GL(r_{i}) and identify it with its image in M using the above isomorphism. We shall show that

∑_{v∈S_{0}} ε_{M}(γ_{v}) = 0, γ_{0} = ∏_{v∈S_{0}} γ_{v} ∈ ∏_{i=1}^{ℓ} SL(r_{i}, F_{S_{0}}) ∩ G_{oreg}(F_{S_{0}}). (67)

One should keep in mind that the summands parameterized by archimedean valuations all vanish, as we are assuming that M ⫌ M_{0}. We may suppose that v_{1}, v_{2} are as earlier and v_{1}, v_{2} ∉ S_{0}. For i = 1, 2 we choose γ_{v_{i}} to be F_{v_{i}}-elliptic elements in M(F_{v_{i}}) which also lie in ∏_{j=1}^{ℓ} SL(r_{j}, F_{v_{i}}), K_{v_{i}} and G_{oreg}(F_{v_{i}}). To see that this is possible, consider a finite Galois extension E of F_{v_{i}}. Let x_{1} be an element of E such that F_{v_{i}}(x_{1}^{n}) = E. Choose a non-trivial element σ in Gal(E/F_{v_{i}}) and set x = x_{1}/σ(x_{1}). Then |x|_{E} = 1, N_{E/F_{v_{i}}}(x) = 1 and F (x^{n}) = E. These properties of x can be translated respectively into the above context as γ_{v_{i}} ∈ K_{v_{i}}, γ_{v_{i}} ∈ ∏_{i=1}^{ℓ} SL(r_{i}, F_{v_{i}}) and γ_{v_{i}} ∈ G_{oreg}(F_{v_{i}}) being F_{v_{i}}-elliptic in M(F_{v_{i}}). Now suppose γ_{0} ∈ ∏_{i=1}^{ℓ} SL(r_{i}, F_{S_{0}})∩G_{oreg}(F_{S_{0}}). The the strong approximation theorem ([Kne65]) allows us to choose γ ∈ ∏_{i=1}^{ℓ} SL(r_{i}, F) ∩ G_{oreg}(F) such that γ is close to γ_{0} at the nonarchimedean valuations in S_{0}, γ is close to γ_{v_{i}} in M(F_{v_{i}}) for i = 1, 2, and γ ∈ K_{v} for any v ∉ S_{0} ∪ {v_{1}, v_{2}}. Choose f̃_{i} ∈ H(G̃(F_{v_{i}})), 1 ≤ i ≤ k, such that f̃_{1}, f̃_{2} are supported on very small neighborhoods of γ′i(μ_{n}) and f̃_{i} = f̃_{v_{i}}^{0} if v_{i} ∉ S_{0}. It then follows from equation (63), equation (66) and our earlier observations that equation (67) holds.

Suppose δ = ∏_{v∈S_{0}} δ_{v} ∈ M(F_{S_{0}}) ∩ G_{oreg}(F_{S_{0}}) and

γ_{0} = ∏_{v∈S_{0}} γ_{v} ∈ ∏_{i=1}^{ℓ} SL(r_{i}, F_{S_{0}}) ∩ G_{oreg}(F_{S_{0}})

such that δγ_{0} ∈ M(F_{S_{0}}) ∩ G_{oreg}(F_{S_{0}}). Our next step is to show that

∑_{v∈S_{0}} ε_{M}(δ_{v}γ_{v}) = ∑_{v∈S_{0}} ε_{M}(δ_{v}). (68)

It is easily seen that we may choose γ_{v_{i}}, i = 1, 2, as in the previous paragraph, but with the additional requirement that γ_{v_{i}}^{2} satisfies the same conditions as γ_{v_{i}}. Weak approximation allows us to choose γ_{1} ∈ M(F) ∩ G_{oreg}(F) such that γ_{1} is close to γ_{v_{i}} in M(F_{v_{i}}) for i = 1, 2, and γ_{1} is close to δ at the nonarchimedean valuations of S_{0}. Let V be the finite set of valuations {v ∉ S_{0} : γ_{1} ∉ K_{v}}. Choosing S large enough and f̃_{1}, …, f̃_{k} appropriately in equation (63), we may conclude that

0 = ∑_{v∈S_{0}∪V} ε_{M}(γ_{1}) = ∑_{v∈S_{0}} ε_{M}(δ_{v}) + ∑_{v∈V} ε_{M}(γ_{1}). (69)

(In this equation we are abusing notation as ε_{M}(γ_{1}) depends on v ∈ S_{0} ∪ V.) As earlier, by strong approximation, we may choose γ_{2} ∈ ∏_{i=1}^{ℓ} SL(r_{i}, F) ∩ G_{oreg}(F) such that it is close to γ_{v_{i}} in M(F_{v_{i}}) for i = 1, 2; it is close to γ_{0} at the nonarchimedean valuations in S_{0}; it is close to the identity in M(F_{V}); and it lies in K_{v} for v ∉ S_{0} ∪V. We may assume that the product γ_{1}γ_{2} ∈ M(F) is also in G_{oreg}(F). By construction, γ_{1}γ_{2} is close to γ_{v_{i}}^{2} in M(F_{v_{i}}) for i = 1, 2; close to δγ_{0} at the nonarchimedean valuations of S_{0}; and close to γ_{1} in M(F_{V}). Furthermore γ_{1}γ_{2} lies in K_{v} for valuations v ∉ S_{0} ∪ V. Again, a judicious choice of S and f̃_{1}, …, f̃_{k} in equation (63) implies that

0 = ∑_{v∈S_{0}∪V} ε_{M}(γ_{1}γ_{2}) = ∑_{v∈S_{0}} ε_{M}(δ_{v}γ_{v}) + ∑_{v∈V} ε_{M}(γ_{1}).

Comparing this equation with equation (69), we find that (68) is true.

It is a simple exercise to show that any δ_{0} = ∏_{v∈S_{0}} δ_{0,v} ∈ M(F_{S_{0}}) ∩ G_{oreg}(F_{S_{0}}) can be written as a product δγ_{0}, where δ ∈ M_{0}(F_{S_{0}}) ∩ G_{oreg}(F_{S_{0}}) and γ_{0} is as above. As M ⫌ M_{0}, equation (68) and Lemma 10.2 imply that

∑_{v∈S_{0}} ε_{M}(δ_{0,v}) = ∑_{v∈S_{0}} ε_{M}(δ_{v}) = 0.

By equation (59) and Corollary 11.1 this constitutes a proof of Theorem A (i) for M ⫌ M_{0}.

To take care of the case M = M_{0}, we must show that

ε_{M_{0}}(γ) = 0, γ ∈ M_{0}(F_{v}) ∩ G_{oreg}(F_{v}).

Bearing in mind the induction hypothesis of §18, we may argue as in the proof of Corollary 11.1 (cf. equation (33)) to conclude that

∫_{M_{0}(F_{v})/μ_{n}^{M}} ε_{M_{0}}(γ)IG̃(γ′, f̃_{1})IG̃(γ′, f̃_{2})dγ = 0,

for any genuine Hecke function f̃_{1} on G̃(F_{v}) and f̃_{2} ∈ H(G̃(F_{v})). Fix γ_{1} ∈ M_{0}(F_{v}) ∩ G_{oreg}(F_{v}). Since H(G̃(F_{v})) is dense in C_{c}^{∞}(G̃(F_{v})) we may let f̃_{2} approach the antigenuine analogue of Dirac measure on G̃(F_{v}) at γ′_{1}. The Weyl integration formula then implies that the function,

γ̃ ↦ |D^{G}(p(γ̃))|^{1/2}IG̃(γ̃, f̃_{2}), γ̃ ∈ M̃_{0}(F_{v}),

approaches the antigenuine analogue of Dirac measure on M̃_{0}(F_{v}) at γ′_{1}. It then follows from our equation that

ε_{M_{0}}(γ_{1})|D^{G}(γ_{1}^{n})|^{−1/2}IG̃(γ′_{1}, f̃_{1}) = 0,

and in turn that ε_{M_{0}}(γ_{1}) = 0. This concludes the proof of Theorem A (i).

All that remains to be done now is to prove Theorem A (ii) and Theorem B. Proof of Theorem A (ii). We wish to show that

aM̃(S, γ′) = a^{M}(S, γ), γ ∈ M(F).

Suppose first that γ ∈ M(F) has Jordan decomposition γ = σu, where the semisimple element σ is not in A_{G}(F) if M = G. Then

dim(M_{σ}) < dim(G),

so we may apply the induction hypothesis of §10 to decompositions (28) and (29) and the lemma follows. On the other hand, if M = G and σ ∈ A_{G}(F) then

aG̃(S, γ′) = aG̃(S, u′)

and

a^{G}(S, γ) = a^{G}(S, u),

by (28) and (29) respectively. It follows from Theorem A (i) and Lemma 19.1, where we may now take M = M_{0}, that

∑_{σ∈A_{G}(F)} ∑_{u∈(U_{G}(F))} (aG̃(S, u′) − a^{G}(S, u)) I_{G}^{M}(σu, f̃) = 0,

for any f̃ ∈ H(G̃(A)). We may choose f̃ ∈ H(G̃(A)) above so that for a fixed element u_{1} ∈ U_{G}(F), we have

I_{G}^{M}(σu, f̃) = \{ \begin{matrix}1, & if σ = 1 and u = u_{1} \\ 0, & otherwise \end{matrix}

(§3.3 [Vig81]). This clearly implies that aG̃(S, u′_{1}) = a^{G}(S, u_{1}).□

It has already been shown in §15 that Theorem B (i) follows from Theorem A (i). This leaves us with a single proof to be completed. Proof of Theorem B (ii). By the induction hypothesis of §14, we need only show that

a_{disc}^{G,Σ}(π̃) = aG̃ ; disc(π̃), π̃ ∈ Π(G̃(A)^{1}).

Let ν_{1} be the infinitesimal character of the archimedean factor of some fixed π̃ ∈ Π(G̃(A)^{1}), and let K_{1} be a compact open subgroup of ∏_{v∉S_{∞}} K_{v} such that π̃ is bi-s(K_{1})-invariant. Let Π_{ν_{1},K_{1}}(G̃(A)^{1}) be the set of bi-s(K_{1})- invariant representations in Π(G̃(A)^{1}) with infinitesimal character ν_{1}. In the process of proving Proposition 19.1 (cf. (16.6) II [AC89]), one obtains

∑_{π̃∈Π_{ν_{1},K_{1}}(G̃(A)^{1})} (a_{disc}^{G,Σ}(π̃) − aG̃ ; disc(π̃)) tr (π̃(f̃^{1})) = 0,

for any f̃ ∈ H(G̃(A)^{1}) which is bi-s(K_{1})-invariant. This sum is finite by Lemma 4.2 [Art88b], and the linear forms,

f̃ ↦ tr (π̃(f̃^{1})) , π̃ ∈ Π_{ν_{1},K_{1}}(G̃(A)^{1}),

on the space of bi-s(K_{1})-invariant functions in H(G̃(A)) are linearly independent. The result follows.□

21 A global correspondence

We derive a global correspondence from a special case of Theorem B (ii). The global datum aG̃ ; disc(π̃) is defined by the equation (cf. (4.3) [Art88b], (9.2) II [AC89]),

∑_{π̃∈Π(G̃(A)^{1},t)} aG̃ ; disc(π̃)IG̃(π̃, f̃)

= ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{s∈W (𝔞_{M})_{reg}} | det(s − 1)_{𝔞_{M}^{G}}|^{−1}tr (M̃(s, 0)ρ_{Q̃,t}(0, f̃)) . (70)

Here, f̃ ∈ H(G̃(A)), Q is any element of P(M) and ρ_{Q̃,t}(0) is the induced representation of G̃(A)^{1} obtained from the genuine subrepresentation of M̃(A)^{1} on L^{2}(s_{0}(M(F))\M̃(A)^{1}) which decomposes into a discrete sum of elements in Π(M̃(A)^{1}, t). The term M̃(s, 0) is the global intertwining operator associated to an element in

W (𝔞_{M})_{reg} = {s ∈ W (𝔞_{M}) : det(s − 1)_{𝔞_{M}^{G}} ≠ 0}.

By Theorem B (ii) and the definitions of §§13-15, the right-hand side of (70) is equal to

∑_{π̃∈Π(G̃(A)^{1},t)} a_{disc}^{G,Σ}(π̃)I_{G}^{Σ}(π̃, f̃)

= ∑_{π∈Π(G(A)^{1},nt)} a_{disc}^{G}(π)n^{−1}Î_{G}(π, f̃′)
= n^{−1} ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{s∈W (𝔞_{M})_{reg}} | det(s − 1)_{𝔞_{M}^{G}}|^{−1}tr (M(s, 0)ρ_{Q,nt}(0, f)) ,

where f ∈ H(G(A)) is a function such that f_{G} = f̃′. We would like to convert this identity of representations on G̃(A)^{1} and G(A)^{1} to an identity of representations on G̃(A) and G(A). We may embed the connected component A_{G}(R)^{0} of A_{G}(R) diagonally into ∏_{v∈S_{∞}} A_{G}(F_{v}). The map,

H_{M} : A_{G}(R)^{0} → 𝔞_{G},

is an isomorphism which allows us to pull back the Haar measure on 𝔞_{G} to A_{G}(R)^{0}. Given λ ∈ i𝔞_{G}^{∗}, we define

I_{disc,t,λ}(f̃) = ∫_{A_{G}(R)^{0}} ∑_{π̃∈Π(G̃(A)^{1},t)} a_{disc}^{G}(π̃)IG̃(π̃, f̃_{a′})e^{λ(H_{G}(p(a′)))}da′,

where

f̃_{a′}(γ̃) = f̃(a′γ̃), γ̃ ∈ G̃(A), a ∈ A_{G}(R)^{0}.

Since f̃′_{a′} = f̃′_{a}, λ(H_{G}(p(a′))) = λ′(H_{G}(a)) and da′ ∘ s = nda our earlier identity becomes

(71) I_{disc,t,λ}(f̃) = I_{disc,nt,λ′}(f).

This identity may be rewritten as

∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{s∈W (𝔞_{M})_{reg}} | det(s − 1)_{𝔞_{M}^{G}}|^{−1}tr (M̃(s, 0)ρ_{Q̃,t,λ}(0, f̃))

= ∑_{M∈L} |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{s∈W (𝔞_{M})_{reg}} | det(s − 1)_{𝔞_{M}^{G}}|^{−1}tr (M(s, 0)ρ_{Q,nt,λ′}(0, f)) .

Here

ρ_{Q̃,t,λ}(0) = ⨁_{π̃∈Π_{disc}(M̃(A),t,λ)} IndG̃ ; Q̃π̃,

where Π_{disc}(M̃(A), t, λ) is the set of irreducible, λ-equivariant and genuine subrepresentations of M̃(A) on L^{2}(s_{0}(M(F))\M̃(A)) whose infinitesimal characters’ imaginary parts have norm equal to t ≥ 0. Since M̃(s, 0) intertwines ρ_{Q̃,t,λ}(0) with itself, Schur’s lemma implies that there exist complex numbers c_{s,π̃} such that

tr (M̃(s, 0)ρ_{Q̃,t,λ}(0, f̃)) = ∑_{π̃∈Π_{disc}(M̃(A),t,λ)} c_{s,π̃}tr ((IndG̃ ; Q̃π̃)(f̃)) .

Set

cπ̃ = |W_{0}^{M}||W_{0}^{G}|^{−1} ∑_{s∈W (𝔞_{M})_{reg}} | det(s − 1)_{𝔞_{M}^{G}}|^{−1}c_{s,π̃}, π̃ ∈ Π_{disc}(M̃(A), t, λ).

Identity (71) now has the form

∑_{M∈L} ∑_{π̃∈Π_{disc}(M̃(A),t,λ)} cπ̃tr (π̃G̃(f̃)) − ∑_{π∈Π_{disc}(M(A),nt,λ′)} c_{π}tr (π^{G}(f)) = 0. (72)

Suppose π̃ = ⊗_{v}π̃_{v} ∈ Π(G̃(A)) and π = ⊗_{v}π_{v} ∈ Π(G(A)). We say that π̃ corresponds (or lifts) weakly to π if

tr (π̃_{v}(f̃_{v})) = tr (π_{v}(f_{v})) , (73)

for any matching functions f̃_{v} ∈ H(G̃(F_{v})), f_{v} ∈ H(G(F_{v})) and almost every valuation v of F (§3). This relation is referred to as a quasi-lift in §28 [FK86]. If (73) holds for all valuations v then we say that π̃ corresponds (or lifts) to π. We say that π is metic if π_{v} is metic (§2) for all valuations v and that f = ∏_{v} f_{v} ∈ H(G(A)) matches f̃ = ∏_{v} f̃_{v} ∈ H(G̃(A)) if f_{v} matches f̃_{v} for all valuations v.

Theorem 21.1 (i) Suppose t ≥ 0, λ ∈ i𝔞_{G}^{∗} and π̃_{0} = ⊗_{v}π̃_{0,v} belongs to Π_{disc}(G̃(A), t, λ). Then there exist a unique Levi subgroup L ∈ L and a unique representation π_{0} ∈ Π_{disc}(L(A), nt, λ′) such that π_{0} is metic and π̃_{0} corresponds weakly to π_{0}^{G}.

(ii) Suppose t ≥ 0, λ ∈ i𝔞_{G}^{∗} and π_{0} = π_{0,v} ∈ Π_{disc}(G(A), nt, λ′) is metic. Then there exists π̃_{0} ∈ Π_{disc}(G̃(A), t, λ) such that π̃_{0} corresponds weakly to π_{0}. Moreover, if π̃ ∈ Π_{disc}(M̃(A), t, λ) and π̃G̃ corresponds weakly to π_{0} then M = G.

(iii) Suppose the representation π_{0} of (ii) is cuspidal. Then there exists a unique representation π̃_{0} ∈ Π_{disc}(G̃(A), t, λ) which corresponds to π_{0}. Moreover, if π̃_{1} ∈ Π_{disc}(M̃(A), t, λ) and π̃G̃ ; 1 corresponds weakly to π_{0} then π̃_{1} = π̃_{0}.

Proof of (i). We assume inductively that (i) holds if G is replaced by a proper Levi subgroup in L. Let V be the finite set of valuations at which π̃_{0} is ramified. Suppose w ∉ V. By §§11, 16, 17 [FK86], we can assign to each unramified representation in Π(G̃(F_{w})) a Satake parameter. Recall that a Satake parameter corresponds to an element of (C^{r})^{S_{r}}, the set of equivalence classes of C^{r} under permutation. Given a valuation v such that |n|_{v} = 1, let H(G̃(F_{v}), K_{v}) be the subset of H(G̃(F_{v})) of s(K_{v}) bi-invariant functions. Suppose f̃ = ∏_{v} f̃_{v} ∈ H(G̃(A)) such that f̃_{w} ∈ H(G̃(F_{w}), K_{w}), f = ∏_{v} f_{v} ∈ H(G_{v}) matches f̃, and t_{w} ∈ (C^{r})^{S_{r}}. Set

a(t_{w}) = ∑_{M∈L} (∑_{π̃} cπ̃ ∏_{v≠w} tr (π̃G̃ ; v(f̃_{v})) − ∑_{π} c_{π} ∏_{v≠w} tr (π_{v}^{G}(f_{v}))) ,

where the first sum is parameterized by the representations π̃ = ⊗_{v}π̃_{v} belonging to Π_{disc}(M̃(A), t, λ) such that π̃_{w} has Satake parameter t_{w} and the second sum is parameterized by the representations π = ⊗_{v}π̃_{v} belonging to Π_{disc}(M(A), nt, λ′) such that π_{w} has Satake parameter nt_{w}. Equation (72) implies that

∑_{t_{w}∈(C^{r})^{Sr} −{0}} a(t_{w})f̃_{w}^{∨}(t_{w}) = 0,

where f̃_{w}^{∨} is the Satake transform (§11 [FK86]) of f̃_{w}. It follows from the Satake isomorphism that

a(t_{w}) = 0, t_{w} ∈ (C^{r})^{S_{r}} − {0}.

Suppose t_{π̃_{0},w} is the Satake parameter of π_{0,w}. We can combine the earlier argument with an induction argument on the number of valuations outside of V to conclude that

∑_{M∈L} ∑_{π̃∈Π(M̃,π̃_{0},V)} cπ̃ ∏_{v∈V} tr (π̃G̃ ; v(f̃_{v})) = ∑_{M∈L} ∑_{π∈Π(M,π̃_{0},V)} c_{π} ∏_{v∈V} tr (π_{v}^{G}(f_{v})) . (74)

Here, Π(M̃, π̃_{0}, V) is the set of representations

π̃ = ⊗_{v}π̃_{v} ∈ Π_{disc}(M̃(A), t, λ)

which satisfy

tr (π̃G̃ ; w(h̃)) = h̃^{∨}(t_{π̃_{0},w}), h̃ ∈ H(G̃(F_{w}), K_{w}), w ∉ V,

and Π(M, π̃_{0}, V) is the set of representations

π = ⊗_{v}π_{v} ∈ Π_{disc}(M(A), nt, λ′)

which satisfy

tr (π_{w}^{G}(h)) = h^{∨}(nt_{π̃_{0},w}), h ∈ H(G(F_{w}), K_{w}), w ∉ V.

Of course, π̃_{0} is an element of set Π(G̃, π̃_{0}, V) and cπ̃_{0} = 1. If Π(M̃, π̃_{0}, V) is empty for every M ≠ G then Lemma 16.1.1 [JL70], applied to equation (74), implies that Π(M, π̃_{0}, V) is not empty for some M ∈ L. If Π(M̃, π̃_{0}, V) is not empty for some M ≠ G then our induction assumption implies that Π(M, π̃, V_{1}) is not empty for some finite set V_{1} ⊃ V. In any case, we have shown the existence of π_{0} as in the theorem. The uniqueness of this representation follows from the strong multiplicity one property for the cuspidal representations of general linear groups (Theorem 4.4 [JS81]) and the construction of the discrete spectrum of general linear groups in terms of the cuspidal representations ([MW89]). Proof of (ii). Suppose V is the finite set of valuations at which π_{0} is ramified. Given w ∉ V, set t_{π_{0},w} to be the Satake parameter of π_{0,w}. Given a finite set V_{1} ⊃ V, define Π(M̃, π_{0}, V_{1}) to be the set of representations

π̃ = ⊗_{v}π̃_{v} ∈ Π_{disc}(M̃(A), t, λ)

which satisfy

tr (π̃G̃ ; w(h̃)) = h̃^{∨}(nt_{π_{0},w}), h̃ ∈ H(G̃(F_{w}), K_{w}), w ∉ V.

Arguing as in (i) and applying the strong multiplicity property for general linear groups, we obtain

∏_{v∈V_{1}} tr (π_{0,v}^{G}(f_{v})) = ∑_{M∈L} ∑_{π̃∈Π(M̃,π_{0},V_{1})} cπ̃ ∏_{v∈V_{1}} tr (π̃G̃ ; v(f̃_{v})) . (75)

Since the left-hand side is not zero there exists a representation π̃_{0} belonging to Π_{disc}(M̃(A), t, λ) such that π̃G̃ corresponds weakly to π_{0}. Furthermore, any π̃ ∈ Π_{disc}(M̃(A), t, λ) such that π̃G̃ corresponds weakly to π_{0} occurs as a summand on the right-hand side for some V_{1} ⊃ V. By (i) (with G replaced by M) there exist M_{1} ∈ L^{M} and π_{1} ∈ Π_{disc}(M_{1}, nt, λ′) such that π̃_{0} corresponds weakly to π_{1}^{M}. In particular, π_{1,v} and π_{0,v} have the same Satake parameters at almost every valuation v. By the strong multiplicity one property mentioned earlier, it follows that π_{1} = π_{0} and in turn that M_{1} = M = G. Proof of (iii). According to Theorem 27.3 [FK86], for every valuation v of F, there exists a unitary representation π̃_{0,v} ∈ Π(G̃(F_{v})) such that

tr (π̃_{0,v}(f̃_{v})) = tr (π_{0,v}(f_{v})) ,

for any matching functions f̃_{v} ∈ H(G̃(F_{v})) and f_{v} ∈ H(G(F_{v})). Together with the second assertion of (ii) this allows us to rewrite equation (75) as

∏_{v∈V_{1}} tr (π̃_{0,v}(f̃)) = ∑_{π̃∈Π(G̃,π_{0},V_{1})} ∏_{v∈V_{1}} tr (π̃_{v}(f̃)) .

It follows from Lemma 16.1.1, applied to this equation, that there exists a unique representation π̃_{0} ∈ Π(G̃, π_{0}, V_{1}) which corresponds to π_{0}. The second assertion of (iii) follows from the second assertion of (ii) and the fact that the finite set V_{1} ⊃ V is arbitrary.□

Theorem 21.1 (iii) yields the multiplicity one and strong multiplicity one property for the representations of G̃(A) which lift to cuspidal representations of G(A). We expect the same properties to hold for representations of G̃(A) which lift to any representation in the discrete spectrum of G(A). One could attempt to prove this by showing that the local metaplectic correspondence preserves character identities for the local factors of representations in the residual spectrum of G(A) and then argue as in Theorem 21.1 (iii). Let us describe the residual spectrum of G(A) in more detail so that we can state these character identities as a precise conjecture.

Recall decomposition (4),

M = M(1) × ⋯ × M(ℓ) ≅ GL(r_{1}) × ⋯ GL(r_{ℓ}).

Suppose that r_{1} = ⋯ = r_{ℓ}, so that r = ℓr_{1}. Suppose further that π = ⊗_{v}π_{v} is a unitary, cuspidal and metic representation of GL(r_{1}, A). Let ⊗_{i=1}^{ℓ}π| det(·)|^{(ℓ(2i−1))/2} be the representation of M(A) given by

( \begin{matrix}γ_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & γ_{ℓ} \end{matrix} ) ↦ ∏_{i=1}^{ℓ} π(γ_{i})| det(γ_{i})|^{\frac{ℓ−(2i−1)}{2}} , γ_{1}, …, γ_{ℓ} ∈ GL(r_{1}, A),

where |·| = ⊗_{v}|·|_{v}, and let P ∈ P(M) be the unique parabolic subgroup containing the group of upper-triangular matrices. The induced representation Ind_{P}^{G} (⊗_{i=1}^{ℓ}π| det(·)|^{(ℓ−(2i−1))/2}) has a unique irreducible quotient which belongs to the discrete spectrum of G(A) (§2 [Jac84]). Conversely, any discrete representation of G(A) is of the above form ([MW89]).

Conjecture 1 Suppose that v is a nonarchimedean valuation of F and π_{v} ∈ Π(GL(r_{1}, F_{v})) and P ∈ P(M) are as above. Suppose further that π_{v} is metic and that π_{1} is the unique irreducible quotient of Ind_{P}^{G} (π_{v}| det(·)|_{v}^{(ℓ−(2i−1))/2}).

Then there exists π̃_{v} ∈ Π(G̃L(r_{1}, F_{v})) such that

IndG̃ ; P̃ (π̃_{v}| det ∘p(·)|v^{\frac{ℓ−(2i−1)}{2}} )

has a unique irreducible quotient π̃_{1} which satisfies

tr (π̃_{1}(f̃)) = tr (π_{1}(f))

for any matching functions f̃ ∈ H(G̃(F_{v})) and f ∈ H(G(F_{v})).

This conjecture can be shown to hold in the case that π_{v} is supercuspidal by using an argument in the proof of Theorem 29.1 [FK86]. Conjecture 1 would follow from a character identity of unitary representations if one could generalize the work of Bernstein, Tadić and Zelevinsky ([BZ77], [Zel80], [Tad95]) to G̃(F_{v}) (see also [Hua95]).

Taking the case r = 2 into consideration ([Fli80]), we make a second conjecture.

Conjecture 2 Suppose t ≥ 0, λ ∈ i𝔞_{G}^{∗} and π̃ belongs to Π_{disc}(G̃(A), t, λ). Then there exists a unique metic representation π ∈ Π_{disc}(G(A), nt, λ′) such that π̃ corresponds to π. Moreover if π̃_{1} is a representation in Π_{disc}(M̃(A), t, λ) such that π̃G̃ ; 1 corresponds weakly to π then π̃_{1} = π̃. This correspondence maps cuspidal representations to cuspidal representations and residual representations to residual representations.

This conjecture holds true given the general strong multiplicity one property, which follows from Conjecture 1, and given a characterization of the residual spectrum of G̃(A) as in [MW89]. We refer the reader to [Sun] for a discussion of the residual spectrum of G̃(A). The final assertion of Conjecture 2 is false if (†) is not assumed. This can be seen in [Fli80] and §29 [FK86].

22 Appendix. Tensor products of metaplectic representations

Suppose v is a nonarchimedean valuation of F. In §26.2 [FK86] a method of induction from parabolic subgroups of G̃(F_{v}) is delineated. Recall decomposition (4),

M = M(i) × ⋯ × M(ℓ).

This method of induction relates tensor products of representations of M̃(i)(F_{v}) to representations of M̃(F_{v}). We describe this relationship and prove all of the claims made in §26.2 [FK86] concerning it under assumption (†). As mentioned earlier, it seems that the claims are not true in general ([Sun97]). For fixed M assumption (†) is stronger than needed.

Let (·, ·)_{F_{v}} : F_{v}^{×} ×F_{v}^{×} → μ_{n} be the nth Hilbert symbol on F_{v} and let B be a maximal subgroup of F_{v}^{×} with respect to the property that (x_{1}, x_{2})_{F_{v}} = 1 for all x_{1}, x_{2} ∈ B. For 1 ≤ i ≤ ℓ, set

M̃^{B}(i)(F_{v}) = {γ̃ ∈ M̃(i)(F_{v}) : det(p(γ̃)) ∈ B}.

It is a simple matter to check that M̃^{B}(i)(F_{v}) is a normal subgroup of finite index in M̃(i)(F_{v}). Let π̃_{i} be a genuine irreducible admissible representation of M̃(i)(F_{v}). The restriction of π̃_{i} to M̃^{B}(i)(F_{v}) is the sum of conjugates of some irreducible representation ρ̃_{i} of M̃^{B}(i)(F_{v}). More precisely,

π̃_{i}|_{M̃^{B}(i)} = ∑_{γ} ρ̃_{i}^{γ},

where the sum runs over representatives γ of cosets in M̃(i)(F_{v})/M̃^{B}(i)(F_{v}) and

ρ̃_{i}^{γ}(γ_{1}) = ρ̃_{i}(γγ_{1}γ^{−1}), γ_{1} ∈ M̃^{B}(i)(F_{v}).

Lemma 22.1 If γ is as above then ρ̃_{i} is not equivalent to ρ̃_{i}^{γ} unless M(i) ≅ GL(1) or γ ∈ M̃^{B}(i)(F_{v}).

Proof. If M(i) = GL(1) then M̃(i)(F_{v}) ≅ F_{v}^{×} × μ_{n}. In particular M̃(i)(F_{v}) is abelian and ρ̃_{i} = ρ̃_{i}^{γ}. Suppose that M(i) is not isomorphic to GL(1). By using the Iwasawa decomposition, it is easy to see that representatives of M̃(i)(F_{v})/M̃^{B}(i)(F_{v}) may be taken to be diagonal matrices. Let γ be such a representative corresponding to the diagonal element

( \begin{matrix}γ_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & γ_{r_{i}} \end{matrix} ) ∈ GL(r_{i}, F_{v}),

and suppose that ρ̃_{i}^{γ} is equivalent to ρ̃_{i}. In other words, suppose that there exists a linear isomorphism T such that

T ∘ ρ̃_{i}^{γ}(γ̃) = ρ̃′_{i}(γ̃) ∘ T, γ̃ ∈ M̃^{B}(i)(F_{v}).

Suppose x ∈ B and choose γ̃ ∈ M̃^{B}(i)(F_{v}) such that p(γ̃) corresponds to the scalar matrix

( \begin{matrix}x & & & & 0 \\ & . & . & . \\ 0 & & & & x \end{matrix} ) ∈ GL(r_{i}, F_{v}).

By Proposition 0.1.5 [KP84] and the bilinearity of the Hilbert symbol, we have

ρ̃_{i}(γ̃) = T ∘ ρ̃_{i}^{γ}(γ̃) ∘ T^{−1}
= T ∘ ρ̃_{i}(γγ̃γ^{−1}) ∘ T^{−1}
= ((det(γ), det(p(γ̃)))_{F_{v}}^{1+2m}/ ∏_{j=1}^{r_{i}}(γ_{i}, x)_{F_{v}}) T ∘ ρ̃_{i}(γ̃) ∘ T^{−1}.

It may be verified by following 0.1.1 [KP84] that γ̃ is in the center of M̃^{B}(i)(F_{v}) and so, by Schur’s lemma, ρ̃_{i}(γ̃) is a nonzero scalar operator. Consequently the above identity reduces to

(det(γ), x)_{F_{v}}^{r_{i}(1+2m)−1} = 1.

Since gcd(n, r_{i}(1+2m)−1) = 1 under assumption (†), we have (det(γ), x)_{F_{v}} = 1. The element x ∈ B was chosen arbitrarily so this means that γ ∈ M̃^{B}(i)(F_{v}).□

Continuing with the discussion on tensor products, we set ρ̃ = ⊗_{i=1}^{ℓ}ρ̃_{i}. This representation passes to an irreducible representation of the subgroup

M̃^{B}(F_{v}) = {γ̃ ∈ M̃(F_{v}) : det(p(γ̃)) ∈ B}.

If M ≠ M_{0} then Lemma 22.1 implies that ρ̃ is inequivalent to any of its conjugates by elements in M̃(F_{v})−M̃^{B}(F_{v}). If M = M_{0}, we obtain the same result by Proposition 3 [FK86]. Applying Mackey’s criterion, we find that the representation of M̃(F_{v}) induced from ρ̃ is irreducible. This process may be reversed without difficulty. Hence, every genuine irreducible admissible representation of M̃(F_{v}) corresponds to a unique set of genuine irreducible admissible representations of M̃(i)(F_{v}), for 1 ≤ i ≤ ℓ.

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