ALMOST UNRAMIFIED AUTOMORPHIC REPRESENTATIONS FOR SPLIT GROUPS OVER F_{q}(t)

AMRITANSHU PRASAD

Abstract. Let G be a split reductive group over a finite field F_{q}, F = F_{q}(t) and A the adèles of F. We describe the irreducible automorphic representations of G(A) which have non-zero vectors invariant under Iwahori subgroups at two places and maximal compact subgroups at all other places in terms of the irreducible representations of Iwahori-Hecke algebras and the Satake isomorphism.

Introduction. Let F_{q} denote the finite field with q elements and let G be a split reductive group over F_{q}. Fix a Borel subgroup B of G defined over F_{q} and a maximal F_{q}-split torus T contained in B. Consider the global field F = F_{q}(t) with ring of adèles A. For each valuation v of F let F_{v} denote the corresponding completion of F. Let O_{v} denote the ring of integers in F_{v}. Denote by 0 and ∞ the valuations uniformized by t and t^{−1} respectively. Set

K_{v} = \begin{cases}pre-image of B(F_{q}) under G(O_{v}) → G(F_{q}) & if v = ∞, 0 \\ G(O_{v}) otherwise. \end{cases}

Let K = ∏_{v} K_{v}. Let

M = L^{2}(G(F)\G(A)/K).

Denote by H_{v} the Hecke algebra of compactly supported measures on G(F_{v}) that are left and right invariant under K_{v}. An irreducible representation (π, V) of ⊗_{v}H_{v} occurring discretely in M has a decomposition

(π, V) = ⊗_{v}(π_{v}, V_{v}),

where π_{v} is an irreducible representation of H_{v}. The representation π_{v} is called the local constituent of π at v.

The main theorem of this article describes the local constituents of the irreducible representations of ⊗_{v}H_{v} that occur in the discrete part of M. For each (π, V) occurring discretely in M the representations (π_{v}, V_{v}) are explicitly described in terms of the irreducible square-integrable representations of an Iwahori-Hecke algebra (for v = ∞, 0) or the Satake isomorphism (for the remaining v).

Relation to Eisenstein series. Let X denote the space of all characters

T (F)\T (A)/T (O) → C^{×}.

There is a natural isomorphism of X with Spec(C[Pic(P^{1})⊗X_{∗}(T)]), where X_{∗}(T) is the lattice Hom(G_{m}, T) of rational cocharacters of T. The isomorphism deg : Pic(P^{1}) → Z induces an isomorphism Pic(P^{1}) ⊗ X_{∗}(T) → X_{∗}(T) and hence and isomorphism deg : Spec(C[Pic(P^{1}) ⊗ X_{∗}(T)]) → T̂, where T̂ = Spec(C[X_{∗}(T)]) is a maximal torus in the Langlands dual group of G. In particular X is isomorphic to a complex torus.

Date: 28 August, 2002.

For any function ϕ : B(O)\G(O)/K → C there exists a unique function ϕ_{χ} : T (F)T (O)N(A)\G(A)/K whose restriction to G(O) coincides with ϕ and such that

ϕ_{χ}(tg) = δ^{\frac{1}{2}} (t)χ(t)ϕ_{χ}(g)

where δ : B(A) → R^{+} is the modulus character of B(A). If |χ((t^{−1})α̌)| < 1 for each coroot α̌ of T in G which is positive with respect to B, then the sum

E_{ϕ}(χ, g) = ∑_{γ∈B(F)\G(F)} ϕ_{χ}(γg)

converges absolutely and normally in χ and defines a function on G(F)\G(A)/K. For each g, E_{ϕ}(χ, g) extends to a rational function on X.

Theorem 1. M_{discrete} is generated by functions of the form

m(g) = f(χ)E_{ϕ}(χ, g) (2)

where f is a rational function on X and χ ∈ X.

In view of Langlands’s theory of Eisenstein series ([4], [5]), in order to prove Theorem 1 it suffices to verify that

Lemma 3. Let m be a complex-valued function on G(F)\G(A)/K. If m_{N} ≡ 0 then m ≡ 0. Here m_{N} denotes the constant term of m along the unipotent radical N of B.

Proof. Recall the following theorem [6]:

Theorem 4. The map W̃ → G(A) defined by

π^{μ}w ↦ (t^{−1})^{μ}ϕ_{∞}(w)

induces a bijection of sets

W̃ ˜→G(F)\G(A)/K′.

Therefore, to prove Lemma 3 it suffices to verify that

(5) for any w, w′ ∈ W̃ , (t_{w})_{N}(w′) = 0 if and only if w = w′.

Indeed,

(t_{w})_{N}(w′) = ∫_{N(F)\N(A)} t_{w}(nw′)dn
= measure (N(F)\(N(A) ∩ G(F)wKw^{′−1}))

The latter is positive if and only if w = w′.

Hecke algebras. For each valuation v of F let H_{v} denote the convolution algebra of compactly supported measures on G(F_{v}) which are left and right invariant under K_{v}. Let Z_{v} denote the center of H_{v} for each v and R_{v} denote the lattice T (F_{v})/T (O_{v}).

Proposition 6. For each v there exists an isomorphism

B_{v} : Z_{v} → R_{v}^{W}

such that Z_{v} acts on i_{B}^{G}χ_{v} by χ_{v} ∘ B_{v} for any character χ_{v} : T (F_{v})/T (O_{v}) → C^{×}.

automorphic representations over F_{q}(t) 3

Proof. For v ≠ ∞, 0, H_{v} is commutative so that Z_{v} = H_{v} and B_{v} is the Satake isomorphism. For v = ∞, or 0 B_{v} is the well known characterization of the center of an Iwahori-Hecke algebra attributed to Bernstein (see [8]).

The map X_{∗}(T) → R_{v} given by μ ↦ π_{v}^{μ}, where π_{v} is a uniformizing element for F_{v} is independent of π_{v} and is an isomorphism. This induces an isomorphism deg_{v} : X_{v} := Spec(C[R_{v}]) → T̂. We have bijections

\xymatrix{SpecZ_{v} \ar[r]^{B_{v}^{∗}} & SpecR_{v}^{W} \ar[r]^{^{deg^{v}}} & T̂/W}

(7)

Let χ_{s}^{v} : Z_{v} → C denote the character corresponding to an element s ∈ T̂/W under the above identifications.

Lemma 8. For each valuation v of F and each ω_{v} ∈ Z_{v}, we have

E_{ϕ}(χ) · ω_{v} = χ_{v} ∘ B_{v}(ω_{v})E_{ϕ}(χ),

where χ_{v} : T (F_{v})/T (O_{v}) → C^{×} is the restriction of χ to T (F_{v}).

Proof. We have

(E_{ϕ}(χ))_{N}(g) = ∑_{w∈W} meas(N_{w}(F)\N_{w}(A))I_{w}(χ)ϕ_{χ}(g),

where I_{w}(χ) : i_{B(A)}^{G(A)}χ → i_{B(A)}^{G(A)w}χ is the intertwining operator defined by

I_{w}(χ)ϕ_{χ} = ∫_{(N(A)∩wN(A)w^{−1})\N(A)} ϕ_{χ}(w^{−1}ng)dn (9)

for χ such that |χ(πα̌)| < 1 for each positive coroot α̌ and then extended to all χ as a rational function on X. The function ϕ_{χ} is of the form

ϕ_{χ} = ⊗_{v}ϕ_{χ,v}

where ϕ_{χ,v} is function in i_{B(F_{v})}^{G(F_{v})w}χ_{v}^{K_{v}}. The intertwining operator I_{w} is a product of local intertwining operators:

I_{w}ϕ_{χ} = ⊗_{v}I_{w}^{v}ϕ_{χ,v}

where I_{w}^{v} is defined as

I_{w}^{v}ϕ_{χ,v} = ∫_{(N(F_{v})∩wN(F_{v})w^{−1})\N(F_{v})} ϕ_{χ}(w^{−1}ng)dn

for χ such that |χ(πα̌_{i} ; v)| < 1 for each positive coroot α̌ and extended to X_{v} by a rational function.

Since I_{w}^{v}ϕ_{χ,v} ∈ i_{B(F_{v})}^{G(F_{v})w}χ and the image of B_{v} is W -invariant, Z_{v} also acts on I_{w}^{v}ϕ_{χ,v} by χ_{v} ∘ B_{v}. Since the intertwining operators and constant term operations commute with the action of the Hecke algebras and since the constant term map is injective in our setting, Lemma 8 follows.

Local constituents at ∞ and 0. let F denote the field F_{q}((π)) of Laurent series in a variable π with ring of integers O. Let I be the Iwahori subgroup of G(F) which is the preimage of B(F_{q}) under the map G(O) → G(F_{q}) and let H denote the corresponding Iwahori-Hecke algebra. The assignment π ↦ t (resp. π ↦ t^{−1}) defines an isomorphism F → F_{0} (resp. F → F_{∞}) and hence an isomorphism θ_{0} : H → H_{0} (resp. θ_{∞} : H → H_{∞}). Let

N = L^{2}(ΔG(F)\(G(F)^{2})/I^{2}).

Then N is an H ⊗ H module whose discrete part is known by elementary and standard considerations (see e.g., [7]):

N_{discrete} = ⨁_{(π,V)∈Ĥ}(π ⊗ π̃, V ⊗ Ṽ).

Here Ĥ denotes the set of isomorphism-classes of irreducible square-integrable representations of H. Moreover, by [7], there is an isometry of Hilbert spaces

I : N → M

such that

I(n · (h ⊗ h′)) = I(n) · (θ_{∞} ∘ ι(h) ⊗ θ_{0} ∘ ι ∘ \overline{κ}(h′)) for h, h′ ∈ H, n ∈ N.

where ι is the Iwahori-Matsumoto involution (introduced at the very end of [3]) and \overline{κ} is an involution (defined in [7]) which when the derived group of G is adjoint takes a representation to its contragredient. It follows that as an H_{∞} ⊗ H_{0}-module

(10) M_{discrete} = ⨁_{(π,V)∈Ĥ}(π ∘ ι ∘ θ_{∞} ⊗ π̃ ∘ \overline{κ} ∘ ι ∘ θ_{0}, V ⊗ Ṽ).

The Iwahori-Matsumoto involution.

Theorem 11. Let (π, V) ∈ Ĥ. The central character of (π, V) equals the central character of (π ∘ ι, V).

Proof. As in (7), the characters of the center of H are parameterized by points in T̂/W. Let s ∈ T̂/W be the central character of H on (π, V). Let (r(π), U) denote the representation of R that is obtained by taking the A(O)-invariants of the normalized Jacquet module (see [9, §2.3] for the correct normalization) of the irreducible representation of G(F) whose I invariants form the irreducible representation (π, V) of H. By [2, Proposition 2.4] the canonical projection V → U is an isomorphism. Moreover, it follows easily from the definition of B_{v} (see [8]), [2, Proposition 2.5] and [1, Lemma 1.5.1] that the restrictions to R^{W} of the weights of r(π) is also s. By [9, Proposition 13], weights of r(π) and r(π∘ι) on U differ by conjugation by the longest element in W. Therefore, their restrictions to R^{W} are the same. This means that the central character of (π, V) equals that of (π ∘ ι, V).

The discrete spectrum. For (π, V) ∈ Ĥ, let s(π) ∈ T̂/W be the central character of π on V. Let χ_{s}^{v} be as defined in the line following (7).

Theorem 12. As an ⊗_{v}H_{v}-module,

(13) M_{discrete} = ⨁_{(π,V)∈Ĥ}(π ∘ ι ∘ θ_{∞} ⊗ π̃ ∘ \overline{κ} ∘ ι ∘ θ_{0} ⊗ (⊗_{v≠0,∞}χ_{s(π)^{deg(v)}}^{v}), V ⊗ Ṽ).

automorphic representations over F_{q}(t) 5

Proof. By Theorem 1 we may assume that m is of the form (2) for some χ ∈ X. Let s be the image of χ in T̂. By Lemma 8, the central character of H_{v} on M is χ_{v} ∘ B_{v}, where χ_{v} is the restriction of χ to T (F_{v}). Obviously deg_{v}(χ_{v}) = s^{deg(v)} (see (7)). Taking m in the direct summand corresponding to π in (10) together with the fact that s(π) = s(π ∘ ι) (Theorem 11) completes the proof of Theorem 12.

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