Special values of automorphic L-functions for GL(r) × GL(r)

Loïc Grenié

February 9, 2001

Contents

1 Introduction 2

2 Critical values of motivic L-functions. 3

3 Rankin-Selberg convolutions. 4 3.1 Local convolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 Global convolutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Product decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4 Other expression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Finite dimension. 6 4.1 Notations, conventions, reminding. . . . . . . . . . . . . . . . . . . . . . . 6 4.2 One-dimensional representations in a tensor product. . . . . . . . . . . . . 7 4.3 Restriction to G_{r−1} × G_{1}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Infinite components. 8 5.1 Cohomology coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 Minimal K-types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 5.3 K-type supporting Eisenstein cohomology. . . . . . . . . . . . . . . . . . . 11

6 Cohomology. 13 6.1 Commutative diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.2 Tensor product of minimal K_{∞}-types. . . . . . . . . . . . . . . . . . . . . . 15

7 Rationality. 16 7.1 Eisenstein series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7.2 Rationality of cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 7.3 Rational vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.3.1 Essential vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7.3.2 Cuspidal representations . . . . . . . . . . . . . . . . . . . . . . . . 18 7.3.3 Eisenstein series representations . . . . . . . . . . . . . . . . . . . . 19

8 Special value. 20

References 22

1 Introduction

The aim of this article is to compute the special values of the automorphic L-function of a product of cuspidal representations of GL(r) up to a number in a fixed number field (depending on the representations). The method are very similar to the ones used by H. Hida in [Hid] and the results are of course generalization of these.

More precisely, let F be a totally imaginary number field and π and π′ two automorphic representations of GL(r, A_{F}). According to [Clo] the representation π is defined over a finite extension F (π) of F and we denote F (π, π′) = F (π)F (π′) the composite field. We can define a period C(π) for the representation π and another one C_{s}(η) for a certain Eisenstein series arising from Rankin-Selberg method. We then have, Theorem 8.1,

L_{f}(s, π × π′) = \frac{C(π)C(π′)C_{s}(η)}{Z_{∞}}c

with c ∈ F (π, π′) and Z_{∞} an archimedean integral, for s > 1 integer in a certain strip (computable strip). This result is valid under some rather strong condition of compatibility of the infinite components of π and π′.

To prove this, we use the (𝔤, K)-cohomology of the representations and the rationality of one-dimensional subspaces. On the infinite part, the dimension one comes from the classic theorem of [BW] on the cohomology of induced representation; on the finite part it comes from results of H. Jacquet and J. A. Shalika for the cuspidal representation and an appropriate choice for the Eisenstein series. Now the product of three elements of these onedimensional subspaces happens to be a top-degree differential form. This differential form is defined on the manifold G(F)\G(A_{F})/Z(F_{∞})K_{∞}K_{f}(π, π′) with K_{f}(π, π′) an appropriate open compact subgroup of G(A_{F,f}). This differential form can then be integrated over the manifold. Up to a finite number of places (including the archimedean), the L-function is represented by a Rankin–Selberg integral which is equal to the integration over the manifold. Up to the various periods C(π), C(π′) and C_{s}(η), the classes of cohomology are proportional to a rational class. The integration of the product of these rational classes leads to the rational number. We want to indicate that we where unable to prove the non vanishing of the integral Z_{∞}.

We would like to introduce the main notations used throughout the article. Let F be a totally imaginary number field with set of places Σ_{F}, infinite places S_{∞} and finite places Σ_{f}. We let h be its class number and d = \frac{[F:Q]}{2}. The algebraic group G = G_{r} will be the group GL(r) defined over O_{F}. For each v ∈ Σ_{f}, O_{v} will be the integer ring of F_{v} and K_{v} = G(O_{v}). For v ∈ S_{∞}, K_{v} = U_{r} and 𝔤_{v} will be the Lie algebra of the real Lie group G_{v} = G(F_{v}) (and accordingly 𝔨_{v} for K_{v}). The algebraic group Z will be the center of G, B = B_{r} will be the (standard) Borel subgroup of upper-triangular matrices and T the torus of diagonal matrices. For v an archimedean place, A_{v} will be the maximal split torus of B and for v finite, A_{v} will be the set of diagonal matrices whose elements are power of the uniformizer. When a Lie group is designed by a letter the same letter in lowercase gothic font will design its Lie algebra; a subscript C will indicate the complexification of a real Lie group or real Lie algebra. The set of weights for G_{r}(C) and U_{r} will be ordered by the order associated to B. The set of positive weights will be denoted Δ^{+}. Geometrically,

G(F)\G(A_{F})/Z(F_{∞})K_{∞}K_{f} = ∐_{j=1}^{h}Γ_{j}\Z

with Γ_{j} a congruence subgroup and Z = G^{1}(F_{∞})/K_{∞}^{1} is the usual symmetric domain. The L-functions will be complete L-functions, including the infinite L-factors. The finite part will be written L_{f} (as above).

For each representation τ of a group H, τ^{∨} will be its contragredient whenever this makes sense.

The vector spaces F^{r} and F_{v}^{r} will be made of row vectors and we put ϵ = (0, ..., 0, 1).

2 Critical values of motivic L-functions.

The point behind the calculus of special values of L-functions is to link these values to critical values of L-functions of motives à la Deligne. Suppose we have a motive whose L-function is equal to the automorphic L-function of π × π′. Suppose at archimedean place v the representation π_{v} is induced from the character χ(z)χ_{1}(a_{1})...χ_{r}(a_{r}) of the Borel

subgroup (with χ_{i}(a_{i}) = ( \frac{a_{i}}{|a_{i}|} )^{p_{i}+ \frac{r−1}{2}} , the p_{i} in decreasing order and z = a_{1}...a_{r}); similarly for π′ with p′_{i}’s. The corresponding archimedean L-factor is then equal to

L_{v}(s; π × π′) = ∏_{1⩽i,j⩽r} Γ(s + |p_{i} + p′_{j} + r − 1|).

The critical values of the global L-function are the integers s such that neither L_{v}(s, π ×π′) nor L_{v}(1 − s, π × π′) have poles. Given that Γ has poles for non positive integers, we must have s in [1 − m; m] with m = min_{i,j}(|p_{i} + p′_{j} + r − 1|). Suppose we must have the relations, consequences of proposition 6.2,

1 ⩽ min_{1⩽k⩽r}(p_{k} + p′_{r+1−k}) + r − 1  (1)

max_{1⩽k⩽r−1}(p_{k+1} + p′_{r+1−k}) + r − 1 ⩽ −1  (2)

\begin{matrix}p_{k} ⩾ p_{k+1} + 1 \\ p′_{k} ⩾ p′_{k+1} + 1 \end{matrix}  (3)

The p_{k} and p′_{k} are in decreasing order. So for i fixed, (1) and (2) means that the minimum of |p_{i} + p′_{j} + r − 1| will be for either j = r + 1 − i or j = r + 2 − i. So m will be the smallest of r − 1 + min(p_{k} + p′_{r+1−k}) and 1 − r − max(p_{k+1} + p′_{r+1−k}). These correspond exactly to the set of values for s given by proposition 6.2. We conclude that this article computes the critical values of the given motive, provided that it exists.

3 Rankin-Selberg convolutions.

We recall here what we will need from Rankin-Selberg integrals and their link to the automorphic L-function of GL(r) × GL(r). We fix two irreducible cuspidal representations of G_{r}, π and π′. They decompose as restricted tensor products π = ⊗′_{v}π_{v} and their central characters ω = ⊗′_{v}ω_{v} and ω′ = ⊗′_{v}ω′_{v} as well. We denote η = ωω′ and η_{v} = ω_{v}ω′_{v}.

3.1 Local convolutions.

Let v be a place of F. We take W(π_{v}, ψ_{v}) to be the Whittaker model of π_{v} with respect to the character θ_{v}(n) = ψ_{v}(∑ n_{i,i+1}) of N. To unify notations, if v is finite, we take W_{0}(π_{v}, ψ_{v}) to be W(π_{v}, ψ_{v}) and if v is archimedean, W_{0}(π_{v}, ψ_{v}) is the subset of C^{∞}, K_{v}- finite vectors in W(π_{v}, ψ_{v}).

For W ∈ W_{0}(π_{v}, ψ_{v}), W′ ∈ W_{0}(π′_{v}, \overline{ψ_{v}}), Φ ∈ S(F^{r}) and s ∈ C, let

Z_{v}(s, W, W′; Φ) = ∫_{N_{v}\G^{v}} W (g)W′(g)Φ(ϵg)| det g|_{v}^{s} dg.

We take H_{s,v}(p) = | det p|_{v}^{s}|p_{rr}|_{v}^{−rs}η_{v}(p_{rr})^{−1} and we can define equivalently with f ∈ J_{s,v} = ind_{P_{v}}^{G_{v}} H_{s,v}

Z_{v}(s, W, W′; f) = ∫_{Z_{v}N_{v}\G^{v}} W (g)W′(g)f(g) dg,

the correspondence being

f(g, s; η, Φ) = | det g|_{v}^{s} ∫_{F_{v}^{×}} Φ(aϵg)|a|_{v}^{rs}η(a) d^{×}a.

These integrals converges for real part of s sufficiently large (actually greater than 1). Since this integral is trilinear on its last three arguments, we extend it to a function of C × (W_{0}(π_{v}, ψ_{v}) ⊗ W_{0}(π′_{v}, \overline{ψ_{v}}) ⊗ J_{s,v}). If v is a finite place, these integrals generate a fractional ideal of C(q^{−s}) (with q the number of elements of the residual field of F_{v}). This ideal is principal and generated by L_{v}(π × π′, s) = L(π_{v} × π′_{v}, s) = P (q^{−s})^{−1} with P (0) = 1. If, in addition, π_{v} and π′_{v} are unramified, the ideal on which ψ_{v} is trivial is O_{v}, W^{∘} and W′^{∘} are the essential vectors of the model (that is W^{∘} is fixed by K_{v} and W^{∘}(K_{v}) = 1) and Φ is the characteristic function of O_{v} ^{r}, then L(π_{v} × π′_{v}, s) = Z(s, W^{∘}, W′^{∘}; Φ).

3.2 Global convolutions.

The setup is nearly the same. We take W_{0}(π, ψ) to be the Whittaker model of π. For W ∈ W_{0}(π, ψ), W′ ∈ W_{0}(π′, \overline{ψ}), Φ ∈ S(A_{F}) and s ∈ C let

Z(s, W, W′; Φ) = ∫_{N(_{A_{F}})\G(A_{F})} W (g)W′(g)Φ(ϵg)| det g|_{A_{F}}^{s} dg.

We take H_{s} to be the character of P (A_{F}) defined by H_{s}(p) = | det p|_{A_{F}}^{s} |p_{rr}|_{A_{F}}^{−rs}η(p_{rr})^{−1} and we have equivalently with f ∈ J_{s} = ind_{P (A_{F})}^{G(A_{F})} H_{s}

Z(s, W, W′; f) = ∫_{Z(_{A_{F}})N(_{A_{F}})\G(A_{F})} W (g)W′(g)f(g) dg,

the correspondence being

f(g, s; η, Φ) = | det g|^{s} ∫_{_{A_{F}}_{×}} Φ(aϵg)|a|^{rs}η(a) d^{×}a.

Since the function Z is trilinear in its last three coefficients, we extend it as in the local case to a function of C × (W_{0}(π, ψ) ⊗ W_{0}(π′, \overline{ψ}) ⊗ J_{s}).

3.3 Product decomposition.

As representations of G(A_{F}) (by which we mean a (𝔤_{∞}, K_{∞}) × G(A_{F,f})-module), the Whittaker models decompose as restricted tensor products; explicitly:

W_{0}(π, ψ) ≃ ⨂′ W_{0}(π_{v}, ψ_{v}),

the product being taken all all places v of F and restricted to the essential element (at places where it is defined) at all but a finite number of places. The space of Schwartz functions decompose in the same way, the limiting element being the characteristic function of K_{v}.

Accordingly the zeta functions can be decomposed into products; take pure tensors in each space, that is W = ∏ W_{v}, W′ = ∏ W′_{v} and Φ = ∏ Φ_{v}, then we have

Z(s, W, W′; Φ) = ∏_{v} Z(s, W_{v}, W′_{v}; Φ_{v}).

The product converges for real part of s large enough (greater than 1). In the product, all but a finite number of the local components are the local L-function of the product. In that way, we see that

Z(s, W, W′; Φ) = L(π × π′, s) ∏_{v∈S} \frac{Z(s, W_{v}, W′_{v}; Φ_{v})}{L(π_{v} × π′_{v}, s)}

for a finite set of places S including the places at infinity. Note that a local L-factor can not be 0.

3.4 Other expression.

We come to another expression of the integral Z. Let Φ be a Schwartz function of A_{F}^{r}. We define the Eisenstein series E(g, s; η, Φ) to be equal to ∑_{γ} f(γg, s; η, Φ) where γ runs

through a set of representatives of P (F)\G(F). This is an automorphic function of g with central character η^{−1}. This series converges for real part of s large enough (namely greater than 1).

Let φ and φ′ be functions in the space of π and π′ respectively and let Φ ∈ S(A_{F}^{r}). We define

I(s, φ, φ′; Φ) = ∫_{Z(_{A_{F}})G(F)\G(A_{F})} φ(g)φ′(g)E(g, s; η, Φ) dg.

Of course, as before, we can change the definition of the function E and define it as an intertwining operator between J_{s} and the space of automorphic forms by the same token: E(f)(g) = ∑ f(γg) and we can extend I to be a function on C × (π ⊗ π′ ⊗ E(J_{s})).

To each element W in W_{0}(π, ψ) we can associate an element φ of the space of π by φ(g) = ∑ W (ξg) with ξ running through N_{P} (F)\P (F). We do the same for W′ and φ′ we then have ([JS1, p550])

I(s, φ, φ′; Φ) = Z(s, W, W′; Φ)  (4)

for any Schwartz function Φ; we have equivalently

I(s, φ ⊗ φ′ ⊗ E(f)) = Z(s, W ⊗ W′ ⊗ f)

for any f ∈ J_{s}.

4 Finite dimension.

We recall or state some results about finite dimensional representations of GL_{r}(C) (or of U_{r}).

4.1 Notations, conventions, reminding.

We will here treat only the case of finite dimensional holomorphic representations of G = GL_{r}(C) but that can be as well applied to finite dimensional representations of U_{r}. The irreducible ones are parameterized by their highest weight with respect to Δ^{+}, whose coordinates in the basis Φ are r integers in decreasing order. We will identify a highest weight with its coordinates in Φ. The contragredient of a representation of highest weight (λ_{i})_{1⩽i⩽r} is the representation of highest weight (−λ_{r+1−i}). The representation Sym^{a} (a ∈ N) is of highest weight (a, 0, ..., 0); if a ⩽ 0, Sym^{a} with be the contragredient of Sym^{−a} or equivalently the representation of highest weight (0, ..., 0, a). The one-dimensional representation det^{d}, d ∈ Z is of highest weight (d, ..., d); tensoring with det^{d} an irreducible representation of highest weight results in adding d to each coefficient. We remind for the irreducible smooth representation of G the analog of Pieri’s rule:

Proposition 4.1 ([FH, Proposition 15.25, p225]) The tensor product of the representation of highest weight (a_{1}, ..., a_{r}) with Sym^{k} (k ∈ N) decomposes as a direct sum of the representations of highest weight (b_{1}, ..., b_{r}) with a_{i} ⩽ b_{i} ⩽ a_{i−1} and ∑ b_{i} = k + ∑ a_{i}; each representation occurs with multiplicity one.

4.2 One-dimensional representations in a tensor product.

We will be often confronted to tensor product of three representations. One of them, (usually the one associated to the Eisenstein series) will be (up to central character) a symmetric power. In these products we will have to find a trivial representation. This is the object of the following proposition.

Proposition 4.2 Let σ and τ two finite dimensional irreducible representation of highest weight respectively (λ_{i}) and (μ_{i}). Then det^{d} occurs in τ ⊗ σ ⊗ Sym^{a} if and only if max_{1⩽i⩽r}(λ_{i} + μ_{r+1−i}) ⩽ d ⩽ min_{1⩽i⩽r−1}(λ_{i} + μ_{r−i}) and a = rd − ∑(λ_{i} + μ_{i}) ⩾ 0.

Remark 4.1 We see that it is not always possible.

Proof: Let the data be as in the text of the proposition. We only have to compute for which values of a the tensor product σ ⊗ Sym^{a} contains τ^{∨} ⊗ det^{d}. To obtain the last one from σ, one needs to add d − μ_{r+1−i} − λ_{i} to the ith coefficient. To obtain that by tensoring with Sym^{a}, according to proposition 4.1, one has to have

λ_{i} ⩽ d − μ_{r+1−i} (1 ⩽ i ⩽ r) et d − μ_{r−i} ⩽ λ_{i} (1 ⩽ i ⩽ r − 1),

which leads to the conditions of the proposition. Then a must be the sum of all what have been added and that gives a = rd − ∑(λ_{i} + μ_{r+1−i}). □

Corollary 4.1 The det^{d} in the proposition above occurs with multiplicity one.

Proof: Suppose det^{d} occurs in σ ⊗ τ ⊗ Sym^{a}. According to proposition 4.1, τ occurs exactly once in det^{d} ⊗σ^{∨} ⊗ Sym^{a} so that det^{d} has multiplicity one in the given tensor product. □

Remark 4.2 In the particular case r = 2, the condition is

λ_{2} + μ_{2} ⩽ d ⩽ min(λ_{1} + μ_{2}, λ_{2} + μ_{1}).

As λ_{1} ⩾ λ_{2} and μ_{1} ⩾ μ_{2} this relation is satisfied for at least one value of d so that there exist at least one a such that σ ⊗τ ⊗Sym^{a} contains a one-dimensional representation. This is of course obvious because each representation if of the type det^{d} Sym^{a}.

4.3 Restriction to G_{r−1} × G_{1}.

We inject G_{r−1} × G_{1} in G_{r} by the “standard” application (g, x) ↦ ( \begin{matrix}g & 0 \\ 0 & x \end{matrix} ). We see, as

for instance in exercise 6.12, p80 of [FH]:

Proposition 4.3 The restriction of the irreducible representation of G_{r}(C) of highest weight (λ_{i}) to G_{r−1}(C) decomposes in direct sum of representations of highest weight (μ_{i})_{1⩽i⩽r−1} such that for each i, 1 ⩽ i ⩽ r − 1, λ_{i} ⩾ μ_{i} ⩾ λ_{i+1}; each representation occurs with multiplicity one.

We look for representations which have a fixed vector by G_{r−1} × G_{1}. Their highest weight (λ_{i}) has to be such that

∑_{i=1}^{n} λ_{i} = 0 (central character)
λ_{1} ⩾ 0 ⩾ λ_{2} ⩾ 0 ⩾ ... ⩾ λ_{r−1} ⩾ 0 ⩾ λ_{r}

because of last proposition. It follows that we must have λ_{1} = −λ_{r} = k ∈ N and all other λ_{i} are zero.

If we look for representations of G_{r}(C) with a one-dimensional sub-representation of G_{r−1}(C) × G_{1}(C), we have as above:

Proposition 4.4 Let (λ_{i}) be the highest weight of an irreducible representation of G_{r}(C). This representation has an invariant one-dimensional subspace under the action of the group G_{r−1}(C) × G_{1}(C) if and only if

\{ \begin{matrix}λ_{1} = & d + k & (d, k) ∈ Z × N \\ λ_{i} = & d & 2 ⩽ i ⩽ r − 1 \\ λ_{r} = & d − u & u ∈ N \end{matrix}

Then G_{r−1}(C) act on the invariant line by determinant to the power d and G_{1}(C) by the power d + k − u.

5 Infinite components.

In this section, v will be an infinite place, so that F_{v} ≃ C. We will usually drop the subscript v or even use P = P (F_{v}), G = G(F_{v}) and so on. This will not create confusion because everything is purely local in this section.

5.1 Cohomology coefficients.

We will use the fundamental result of Borel and Wallach in [BW, Theorem 3.3, p93]. We rephrased it to fit our notations and our particular case.

Theorem 5.1 (Borel–Wallach) Let λ ∈ 𝔱_{C}^{∗} be a dominant weight and F_{λ} a simple G_{C}- module with highest weight λ.

(i) If H^{•}(𝔤_{C}, K_{C}; J_{s} ⊗ F_{λ}) ≠ 0, then there exists s ∈ W^{P} such that

(1) s(ρ + λ) |_{A_{0}} + H_{s} |_{A_{0}} = 0;

(2) The infinitesimal character of H_{s} |_{A_{0}\B} is equal to the one parameterized by −s(ρ+ λ) (restricted to A_{0}\B).

Such an s is unique.

(ii) If s ∈ W^{P} satisfies conditions (1) and (2) then, for every q ∈ N, we have

H^{q+ℓ(s)}(𝔤_{C}, K_{C}; J_{s} ⊗ F_{λ}) = (H^{•}(^{0}𝔪, K_{P}; H_{σ} ⊗ E_{s(ρ+λ)−ρ)}) ⊗ ⋀^{•} 𝔞_{C}^{∗})^{q}

and all other cohomology groups are 0.

The algebra 𝔤_{C} is M_{r}(C ⊗_{R} C). We will fix some notations. We denote one of the elements in C whose square is −1 by J. Thus C ⊗_{R} C = C ⊕ JC with J not part of the C’s on the right hand side of the equality. Then 𝔤_{C} = M_{r}(C) ⊕ JM_{r}(C). For the diagonal torus, the weights are (x_{k} −x_{l})±i(y_{k} −y_{l}) where x_{k} +Jy_{k} is the k-th coordinate on the diagonal (and x_{k} and y_{k} are C-valued); the corresponding weight vectors are \frac{1∓J}{2} E_{kl}. The weights will be denoted by a pair of r-tuples of integers ((λ_{k}), (λ̸_{k})); the Weil group is the group 𝔖_{r} × 𝔖_{r} (each 𝔖_{r} acting on each r-tuple) so that the dominant weights will be chosen to be the ones with decreasing values in each r-tuple.

A little bit of combinatorics gives the following result.

Proposition 5.1 Let η be a character of C^{×} of the form η(z) = ( \frac{z}{\overline{z}} )^{k_{η}} with k_{η} half integer and for s ∈ C, let H_{s} the character of P (C) defined by H_{s}(p) = | det p|^{s}|p_{rr}|^{−rs}η(p_{rr})^{−1}. Let V be a finite dimensional irreducible representation of GL(r, C) tensor product of the representation of highest weight (λ_{j}) by the complex conjugate of the one of highest weight (λ̸_{j}). The cohomology H^{•}(𝔤_{C}, K_{C}; J_{s} ⊗ V) is non trivial if and only if k_{η} and s are integers and one of the following conditions are satisfied:

(0) s ⩾ 1 + \frac{|k_{η}|}{r},

λ_{1} = 1 − r + k_{η} + (r − 1)s, λ̸_{1}= 1 − r − k_{η} + (r − 1)s et λ_{j} =λ̸_{j}= 1 − s (2 ⩽ j ⩽ r). We then take ℓ(s) = 2(r − 1).

(1) 1 + \frac{k_{η}}{r} ⩽ s ⩽ −\frac{k_{η}}{r},

λ_{j} = −s (1 ⩽ j ⩽ r − 1),

λ_{r} = k_{η} + (r − 1)s,

λ̸_{1}= 1 − r − k_{η} + (r − 1)s et

λ̸_{j}= 1 − s (2 ⩽ j ⩽ r). We then take ℓ(s) = r − 1.

(2) 1 − \frac{k_{η}}{r} ⩽ s ⩽ \frac{k_{η}}{r},

λ_{1} = 1 − r + k_{η} + (r − 1)s, λ_{j} = 1 − s (2 ⩽ j ⩽ r), λ̸_{j}= −s (1 ⩽ j ⩽ r − 1) et λ̸_{r}= −k_{η} + (r − 1)s. We then take ℓ(s) = r − 1.

(3) s ⩽ −\frac{|k_{η}|}{r},

λ_{j} =λ̸_{j}= −s (1 ⩽ j ⩽ r − 1),

λ_{r} = k_{η} + (r − 1)s et

λ̸_{r}= −k_{η} + (r − 1)s. We then take ℓ(s) = 0.

The cohomology is then equal to C in degree ℓ(s), C^{2} in degree ℓ(s)+1, C in degree ℓ(s)+2 and 0 everywhere else.

Remark 5.1 We are interested in the cases (1) and (2); these are possible only if 1− \frac{|k_{η}|}{r} ⩽ \frac{|k_{η}|}{r}, which leads to |k_{η}| ⩾ \frac{r}{2}.

The situation for the cuspidal representations is already known. The π_{v} part of the cuspidal representation is unitarily induced from the Borel subgroup of the character χ(b) = χ(z) ∏ χ_{j}(b_{jj}) with χ_{j}(z) = ( \frac{z}{\overline{z}} )^{p_{j}+ \frac{r−1}{2}} and z = ∏ b_{jj}. The p_{j}’s are supposed to be in strictly decreasing order. The finite dimensional irreducible representation τ such that H^{•}(𝔤_{C}, K_{C}; π_{v} ⊗ τ) ≠ 0 is the tensor product of the representation of highest weight λ_{τ} = (−p_{r+1−j} + j − r), the conjugate of the one of highest weight (p_{j} + j − 1) and the inverse of χ ∘ det.

Definition 5.1 We define τ, τ′ and τ′′ be the finite dimensional representations in which π, π′ and J_{s} respectively have cohomology (provided that this is possible for J_{s}). We let k_{η} be ∑(p_{j} + p′_{j}) + r(r − 1).

5.2 Minimal K-types.

The representation π_{v} (or π′_{v}) is unitarizable thus the K-type supporting cohomology in degree \frac{r(r−1)}{2} is the minimal K-type. We want to compute it in terms of the parameters of the character of B from which the representation is induced. The representations π_{v} and τ are unitary representations so that the differentiation is trivial and the cohomology is equal to the complex ([BW, Proposition 3.1, p52]).

Proposition 5.2 The minimal K-type of π_{v} is of highest weight (2p_{j} + r − 1).

Proof: Let 𝔭 be the Lie algebra of matrices M ∈ 𝔤_{C} such that ^{t}\overline{M} = M (the conjugacy is taken with respect to the complex structure of 𝔤, not the one of 𝔤_{C}). We let 𝔭̃ be the sub-algebra of 𝔭 with zero trace. The complex computing the cohomology of π_{v} is:

C^{i}(𝔤̃_{C}, K̃_{C}; π_{v} ⊗ τ) = (⋀^{i} 𝔭̃^{∗} ⊗ π_{v} ⊗ τ)^{K_{C}}.

The first non zero group is the one for i = \frac{r(r−1)}{2} in which it is a one-dimensional C-vector space.

As a K_{C} representation, 𝔭̃ is of highest weight (1, 0, ..., 0, −1) (and thus is auto-dual). If V is the standard representation of 𝔨_{C}, we note that V ⊗ V^{∨} ≃ C ⊕ 𝔭̃. We then have two different expressions for exterior powers of V ⊗ V^{∨}: the easiest one is

⋀^{i}(V ⊗ V^{∨}) = ⋀^{i−1} 𝔭̃ ⊕ ⋀^{i} 𝔭̃

and the other one given by [FH, p80]:

⋀^{i}(V ⊗ V^{∨}) = ⨁_{λ} S_{λ}(V) ⊗ S_{λ′}(V^{∨})

with λ running through partitions of i with at most r lines and at most r columns, λ′ is the conjugate partition to λ and S_{λ} is the Schur functor associated to λ. In our case, S_{λ}(V) is the representation of highest weight λ and S_{λ′}(V^{∨}) is the contragredient of the representation of highest weight λ′.

We are interested in the case i = \frac{r(r−1)}{2}. We take for λ the partition (r − j)_{1⩽j⩽r}. Then λ′ = λ and S_{λ}(V)⊗S_{λ′}(V^{∨}) contains the representation τ′ of highest weight λ_{τ′} = (r+1−2j) because it is the sum of the highest weights of the given representations. As a representation of K_{C}, τ is the tensor square of the representation of highest weight λ_{τ}. We thus see that τ ⊗τ′ contains the irreducible representation of highest weight (−2p_{r+1−j} −r+1) as it is the sum of the highest weight of the representations of K_{C}. We see now that the representation of highest weight (2p_{j} +r−1) occurs is π_{v}, because of Frobenius reciprocity law and because ρ is trivial on B ∩ K. Thus there is a non trivial vector in (⋀^{i}(V ⊗ V^{∨}) ⊗ π_{v} ⊗ τ)^{K_{C}}. It can come either from C^{i−1}(𝔤̃_{C}, K̃_{C}; π_{v} ⊗ τ) or C^{i}(𝔤̃_{C}, K̃_{C}; π_{v} ⊗ τ). Since the H^{i−1} is zero and is equal to C^{i−1}, this means that this vector is part of C^{i} = H^{i} and the proposition is proved. □

5.3 K-type supporting Eisenstein cohomology.

We begin by the list of the K-types contained in an induced representation from P to G.

Proposition 5.3 Let σ be the (unitary) character of C such that σ(z) = ( \frac{z}{\overline{z}} )^{−k}, k ∈ Z and let H_{s} be the character of P defined by H_{s}(p) = | det p|_{C}^{s}|p_{rr}|^{−rs}σ(p_{rr}). All the K-types contained in J_{s} = ind_{P}^{G} H_{s} are all the representations of K with highest weight (λ − 2k, 0, ..., 0, −λ) for λ ⩾ max(0, 2k); each of these K-type has multiplicity one. A vector of highest weight for the K-type (λ − 2k, 0, ..., 0, −λ) is the function from G_{v} to C given by

f_{λ}(g) = \frac{| det g|_{v}^{s}g_{r1}^{λ−2k}\overline{g_{rr}}^{λ}}{(∑_{i} |g_{ri}|_{v})^{rs−k+λ}} .

Proof: As K representations, the values of s are irrelevant, so that we take s to be equal to 0 and put H = H_{0}. As a representation of K, ind_{P}^{G} H is naturally isomorphic to ind_{P ∩K}^{K}(H |_{P ∩K}). We then use Frobenius reciprocity law which states that for any K representation V

Hom_{K}(ind_{P ∩K}^{K}(H |_{P ∩K}), V) ≃ Hom_{P ∩K}(H |_{P ∩K} , V |_{P ∩K}).

The group P ∩ K is isomorphic to U_{r−1} × U_{1} and through that isomorphism the restriction of H to P ∩ K is the character ((0), −2k) where (0) is the null weight of U_{r−1}. If V is the representation of highest weight (λ_{i})_{1⩽i⩽r} its restriction to P ∩ K decomposes in direct sum of representations of highest weight ((μ_{i})_{1⩽i⩽r−1}, μ_{r}) with:

• ∀i, 1 ⩽ i ⩽ r − 1, λ_{i} ⩾ μ_{i} ⩾ λ_{i+1}.

• ∑_{i=1}^{r} λ_{i} = ∑_{i=1}^{r} μ_{i} (equality of central character).

This is the content of proposition 4.3. In that case all μ_{i}’s are zero except possibly μ_{1} and μ_{r} which gives the first result of the proposition.

We suppose k ⩾ 0, the situation is exactly the same when k < 0. Under the action of K, the vector f_{λ} generates a vector space contained in the space W of functions

g ⟼ \frac{P ((g_{rj})_{j})\overline{Q}((g_{rj})_{j})}{(∑ |g_{ri}|_{C})^{λ−k}}

with (P, Q) ∈ Sym^{λ−2k} × Sym^{λ}. The space W is finite dimensional and f_{λ} is a highest weight vector of W. □

Proposition 5.4 The K-type of J_{s} supporting cohomology in degree r − 1 is the minimal K-type of J_{s}, that is the one of highest weight (max(0, −2k), 0, ..., 0, min(0, −2k)).

Proof: The induced representation J_{s} is not unitarizable which makes things slightly harder. We will use Zuckerman-Jantzen translation functor. For more information see [Vog] and of course [Zuc]. We will use the translation functor which brings J_{s} to an induced representation with coefficientless cohomology, and we will get in the process an unitarizable representation.

We will proceed the other way round, it’s actually easier. We suppose that the characters χ and χ′ used in the parameterization of π_{v} and π′_{v} are trivial. As a representation of 𝔤_{C}, the representation τ′′ is irreducible and has highest weight

((r − 1)(s − 1) + k_{η}, 1 − s, ..., 1 − s), (−s, ..., −s, (r − 1)s − k_{η}))

so that there is up to a scalar a unique vector of weight

((1 − s, ..., 1 − s, (r − 1)(s − 1) + k_{η}), (−s, ..., −s, (r − 1)s − k_{η}))

because this weight is extremal. Call one of these vectors v_{s}. Under the action of P (F_{v}) the line Cv_{s} is a one-dimensional representation. We call H′ the character of P (F_{v}) defined by H′(p) = p_{rr}^{r} and J′ the induced representation J′ = ind_{P (F_{v})}^{G(F_{v})} H′. As characters of P (F_{v}), H_{s} ⊗ Cv_{s} and H′ are isomorphic. Moreover, ind_{P}^{G}(H_{s} ⊗ Cv_{s}) ⊂ ind_{P}^{G}(H_{s} ⊗ τ′′) and as stated by [Vog, Lemma 4.5.2(a), p206], ind_{P}^{G}(H_{s} ⊗ τ′′) ≃ ind_{P}^{G}(H_{s}) ⊗ τ′′ = J_{s} ⊗ τ′′. This last isomorphism is reciprocal to the one defined by:

\begin{aligned}T′_{s} : & J_{s} ⊗ τ′′ & ⟶ & ind_{P}^{G}(H_{s} ⊗ τ′′) \\ & f ⊗ v & ⟼ & (g ↦ f(g) ⊗ (gv)) \end{aligned}

By composing these various morphisms, we obtain an injective morphism from J′ to J_{s}⊗τ′′:

J′ = ind_{P}^{G} H′ ≃ ind_{P}^{G}(H_{s} ⊗ Cv_{s}) ⟶ ind_{P}^{G}(H_{s} ⊗ τ′′) ≃ J_{s} ⊗ τ′′.

Using theorem 5.1, we see that J′ is indeed a cohomological representation with trivial coefficients and that the only non trivial cohomology groups are those of degree r − 1, r and r + 1 (of dimension respectively 1, 2 and 1). Let’s compare the cohomologies of J_{s} and J′. The complexes are respectively

C^{•} (J_{s}) = (⋀^{•}(𝔤/𝔨)^{∗} ⊗ J_{s} ⊗ τ′′)^{K}

and

C^{•} (J′) = (⋀^{•}(𝔤/𝔨)^{∗} ⊗ J′)^{K} .

The action of T_{s} induces and injection T_{s}^{•} from C^{•} (J′) to C^{•} (J_{s}). But now J′ is unitarizable because it is unitarily induced from the character H′_{∘} = |δ_{P} |^{−1/_{2}}H′ which is given by H′_{∘}(p) =

( \frac{det p}{\overline{det p}} )^{\frac{1}{2}} ( \frac{p_{rr}}{\overline{p_{rr}}} )^{− \frac{r}{2}} so that it is unitary. That way, the differentiation of the complex C^{•} (J′) is trivial and the cohomology is isomorphic to the complex. Thus C^{r−2} (J′) is zero and C^{r−1} (J′) is one-dimensional. As in proposition 5.3 the K-types of J′ are of the form (1 + λ, 1, ..., 1, 1 − r − λ) for λ ∈ N and each K-type has multiplicity one. As J′ is unitary, the cohomology is supported in the (r − 1)th group by the minimal K-type, that is (1, ..., 1, 1 − r).

The part of J_{s} ⊗ τ′′ which has same infinitesimal character as J′ is exactly the image of T_{s} because of [Vog, Proposition 7.4.1, p488]. Now the differentiation operator of the complex C^{•} (J_{s}) is defined by the action of the Lie algebra 𝔤 so that it commutes with the action of center of the universal enveloping algebra. Thus if a given K type is in the image of the differentiation, its preimage has the same infinitesimal character. The image of C^{r−1} (J′) in C^{r−1} (J_{s}) is a one-dimensional vector space L. Its image by the differentiation d has same infinitesimal character hence it is the image of T_{s}^{r}. So now (T_{s}^{r})^{−1} (dL) is in the image of the differentiation of C^{r−1} (J′) which is trivial. As T_{s}^{•} is injective, that means that dL = 0. Now this L cannot be in the image of d because its preimage would be in the image of T_{s}^{r−1} ∘ d which is zero. Thus T_{s}^{r−1} (C^{r−1} (J′)) = H^{r−1}(𝔤_{C}, K_{C}; J_{s} ⊗ τ′′).

On the level of K-types, T_{s} is simply translation by the character of Cv_{s} because it is simply tensoring with this vector. Thus the K-type of J_{s} supporting cohomology is the minimal K-type because it is the image of the minimal K-type of J′ by T_{s}.

If then χ and χ′ are non trivial, we tensorize J_{s} by χχ′ ∘ det and τ′′ by the inverse, but the result remains the same. □

6 Cohomology.

6.1 Commutative diagram.

We fix the following notations. The sub-algebra 𝔭 ⊂ 𝔤_{C} is the sub-algebra of Lie(B)_{C} such that diagonal elements are “real”. Whenever there is a tilde over a letter, this means that the center is mod-ed out. We denote p_{c} = \frac{r(r−1)}{2}d and p_{E} = (r − 1)d (recall that [F : Q] = 2d). We use the natural basis of weight vectors on 𝔭̃. We define a morphism

from ⋀^{p_{c}} 𝔭̃^{∗} ⊗ ⋀^{p_{c}} 𝔭̃^{∗} ⊗ ⋀^{p_{E}} 𝔭̃^{∗} to C by using the natural contraction to (⋀^{r^{2}−1} 𝔭̃^{∗})^{⊗d} and dividing by the wedge product of the elements of the basis of 𝔭̃^{∗} we chose.

We consider the following diagram:

⋀^{p_{c}} 𝔭̃^{∗} ⊗ ⋀^{p_{c}} 𝔭̃^{∗} ⊗ ⋀^{p_{E}} 𝔭̃^{∗} ⨂ π ⊗ π′ ⊗ E(J_{s}) ⨂ τ ⊗ τ′ ⊗ τ′′  ⟶^{m⊗Z(s)⊗id} C ⨂ C ⨂ τ ⊗ τ′ ⊗ τ′′
↑  ↓^{×⊗id}
H^{p_{c}}(𝔤̃_{C}, K̃_{C}; π ⊗ τ) ⊗ H^{p_{c}}(𝔤̃_{C}, K̃_{C}; π′ ⊗ τ′) ⊗ H^{p_{E}}(𝔤̃_{C}, K̃_{C}; E(J_{s}) ⊗ τ′′) ⟶_{∨}  τ ⊗ τ′ ⊗ τ′′

(5)

with the following arrows:

• The ascending arrow is the inclusion of the K_{∞}-invariants in the total space of vectors. Note that differentiations are trivial so that the cohomologies are equal to the complexes. Note also that the tensor elements have been reorganized.

• The descending arrow is only the product of the scalars with the vector.

• The bottom arrow is the cup product of the differential forms on the manifold.

Proposition 6.1 Up to a multiplication of m by a constant, the diagram is commutative.

Proof: this is clear, this is only the contents of the equality of the Z and the I function as seen in (4). What remains to prove is the following.

Lemma 6.1 The operator m is non trivial on the product of K_{∞}-types of the ⋀^{p_{▽}} 𝔭̃^{∗} occurring in the cohomologies.

Proof: In the case of the infinite part of the cuspidal representations, the K_{∞}- type of ⋀^{p_{c}} 𝔭̃^{∗} occurring is the one with highest weight k_{c} = (r + 1 − 2j) while for the Eisenstein part it is the one of highest weight k_{E} = (r − 1, −1, ..., −1). The representation of highest weight k_{c} is its own contragredient. When it is tensored with the one of highest weight k_{E} (which is Sym^{r} ⊗ det^{−1}) we see by using Pieri’s rule (Proposition 4.1) that one can obtain only once the representation of highest weight k_{c} (graphically, one has to add each square of the Young diagram of Sym^{r} to one line of the Young diagram for k_{c}). This means that the trivial representations occurs with multiplicity one in the tensor product of the 3 K_{∞}-types. Now, if m is non vanishing on one vector of the tensor product it will be the intertwining operator. between the tensor product and the trivial representation.

In the k_{c} representations, the vectors v_{h} = ⋀_{1⩽j<k⩽r}e_{jk} and v_{l} = ⋀_{1⩽k<j⩽r}e_{jk} are re-

spectively highest and lowest weight vectors. We define the sequence of vectors v_{k} by

\{ \begin{matrix}v_{0} = & ⋀_{2⩽l⩽r} e_{1l} \\ v_{k+1} = & (E_{1,r−k} − E_{r−k,1}) · v_{k} \end{matrix}

We can then check inductively that v_{k} has a non zero (integer) coefficient against ⋀_{2⩽l⩽r−k}e_{1l} ∧ ⋀_{r−k⩽l<r}(e_{ll} − e_{rr}) and all other components are of degree smaller than k

in the e_{ll} − e_{rr}. This proves that there is a vector v_{m} with a non zero coefficient in ⋀_{1⩽l<r}(e_{ll} − e_{rr}) in the representation of highest weight k_{E} and m(v_{h} ⊗ v_{b} ⊗ v_{m}) is a

non zero integer. □

We even have proved that the constant is rational. □

6.2 Tensor product of minimal K_{∞}-types.

We have to ponder when the diagram (5) gives a non zero multiple of the Z function in the bottom-right corner. The first point is that there must be a trivial representation in the tensor product τ ⊗ τ′ ⊗ τ′′. This is easily accomplished using proposition 4.2:

Proposition 6.2 The tensor product τ ⊗ τ′ ⊗ τ′′ contains the trivial representation if and only if s is in the following bounds:

• if k_{η} ⩾ \frac{r}{2}

r + max_{1⩽j⩽r−1}(p_{j+1} + p′_{r+1−j}) ⩽ s ⩽ r − 1 + min_{1⩽i⩽r}(p_{j} + p′_{r+1−j})
2 − r − min_{1⩽i⩽r}(p_{j} + p′_{r+1−j}) ⩽ s ⩽ 1 − r − max_{1⩽i⩽r−1}(p_{j+1} + p′_{r+1−j})

• if k_{η} ⩽ −\frac{r}{2}

−r + max_{1⩽j⩽r}(p_{j} + p′_{r+1−j}) ⩽ s ⩽ −r − 1 + min_{1⩽j⩽r−1}(p_{j} + p′_{r−j})
2 + r − min_{1⩽j⩽r−1}(p_{j} + p′_{r−j}) ⩽ s ⩽ 1 + r − max_{1⩽j⩽r}(p_{j} + p′_{r+1−j})

In that case, the trivial representation has multiplicity one in the tensor product.

This is obviously not always possible. In the case k_{η} ⩾ \frac{r}{2}, the parameters p_{j} and p′_{j} must satisfy the conditions max(p_{j+1}+p′_{r+1−j}) ⩽ −1+min(p_{j} +p′_{r+1−j}), max(p_{j+1}+p′_{r+1−j})+r ⩽ 1−r−max(p_{j+1} +p′_{r+1−j}) and − min(p_{j} +p′_{r+1−j})+2−r ⩽ −1+r−min(p_{j} +p′_{r+1−j}). They lead respectively to p_{j} ⩾ 1+p_{j+1}, max(p_{j+1}+p′_{r+1−j})+r ⩽ \frac{1}{2} and min(p_{j}+p′_{r+1−j})+r−1 ⩾ \frac{1}{2}. Since k_{η} = ∑(p_{j} + p′_{j}) + r(r − 1) we see that the last inequality implies k_{η} ⩾ r. Then we see that k_{η} ⩾ r − 1 + min(p_{j} + p′_{r+1−j}) so that the s verifying the proposition 6.2 falls in case (2) of 5.1.

We define the computable strip of π × π′.

Definition 6.1 The set of values for s defined by proposition 6.2 is called the computable strip of the pair of representations π × π′.

The last point is that the Z integral must be non-vanishing on the minimal K_{∞}-types of the representations.

Proposition 6.3 Whenever s is in the computable strip of π × π′, the tensor product of the K_{∞}-types of π_{∞}, π′_{∞} and E(J_{s})_{∞} contains the trivial representation with multiplicity one.

Proof: this is only proposition 4.2. □ Unfortunately we did not manage to prove that Z is an intertwining operator with the trivial representation so that we have to suppose it:

Conjecture 6.1 The integral Z is non trivial on the tensor product of the minimal K_{∞}- types of π, π′ and E(J_{s}).

7 Rationality.

We want to prove that the cohomology spaces we are interested in are rational subspaces of the de Rham cohomology. In the case of the cuspidal cohomology, this is a result of L. Clozel ([Clo, Théorème 3.19, p129]).

7.1 Eisenstein series.

This section recalls only what is written in [Har1, p102ff].

Let φ be a character the Borel subgroup B(A_{F}) of G(A_{F}). We will denote I_{φ} the unitarily induced representation of φ from B to G:

I_{φ} = Ind_{B(A_{F})}^{G(A_{F})} φ.

If α is a root, we define N_{α} to be the one parameter subgroup corresponding to α. Explicitly, if α = e_{j} − e_{k}, j ≠ k, N_{α}(A) = I_{r} + AE_{jk} where A is any O_{F} -algebra. If w is an element of W, we define

N^{(w)} = ∏_{{a>0 \atop wα<0}} N_{α}.

Definition 7.1 Let T (w, φ) be the non-normalized intertwining operator from I_{φ} to I_{w^{−1}φ} defined by:

∀f ∈ I_{φ}, ∀g ∈ G(A_{F}), T (w, φ)(f)(g) = ∫_{N(A_{F})} f(wng) dn.

The operators T (w, φ) are multiplicative in the following sense: for all w, w′ in W, T (ww′, φ) = T (w′, w^{−1}φ) ∘ T (w, φ). This operator is meromorphic in φ but not holomorphic: it may have (and has) poles. We decompose T (w, φ) into the product of a scalar function c(w, φ) and a holomorphic operator T^{loc}(w, φ). To define the scalar function (and thus determine the operator), let us define for a root α = e_{j} − e_{k}, χ_{α} to be the group morphism:

\begin{aligned}χ_{α} : & A_{F}^{×} & → & B(A_{F}) \\ & t & ↦ & I_{r} + (t − 1)E_{jj} + (t − 1)E_{kk} \end{aligned}

If w is in W, and k is the number of positive roots that become negative under the action of w^{−1} let

c(w, φ) = |d_{F} |^{− \frac{k}{2}} ∏_{{a>0 \atop w^{−1}α<0}} \frac{L(φ ∘ χ_{α})}{L(φ ∘ χ_{α} · | |)}

The corresponding operator T^{loc}(w, φ) is holomorphic and as a consequence of what is written on the bottom of page 120 of [Har1], it is non zero and even surjective on Ind_{P}^{G}.

With these definitions, we can express the constant term of the Eisenstein series. If

f ∈ I_{φ} and Eis(g, f) = ∑_{γ∈B\G(F)} f(γg), we have

∫_{N(F)\N(A_{F})} Eis(ng, f) dn = ∑_{w∈W} T (w, φ)(f)(g).

7.2 Rationality of cohomology.

Proposition 7.1 Let V be a finite dimensional representation of G(F_{∞}). The space H^{(r−1)d}(𝔤_{C}, K_{C}; E(J_{s}) ⊗ V) is a rational subspace of H^{(r−1)d}(𝔤_{C}, K_{C}; A ⊗ V).

Remark 7.1 The cohomology space is zero most of the time, except when V = τ′′ is the right coefficient space. We suppose it that way.

Proof: We will use the general methods of Günter Harder as in [Har1] and [Har2] and in particular [Har1, 4.3, p85]. The principle is the following: let r_{B} be the restriction to the Borel stratum of the Borel-Serre compactification; if V is a representation such that Hom(V_{f}, ker r_{B}) = 0 then for any intertwining operator T : V → A, T (V) maps to a rational subspace of the cohomology if r_{B} ∘ T^{•} is non trivial (T^{•} is the mapping induced on cohomology by T).

Here we take V = J_{s} and T = E. We already know that Hom(J_{s,f}, ker r_{B}) = 0 ([Har2, 2.2.4, p125]). All what remains to prove is that r_{B} ∘ E^{•} is non zero. Harder has already proved that r_{B} ∘ Eis_{max}^{•} is non zero, so that it is enough to prove that Eis_{max} is isomorphic to E(J_{s}). Now this is clear because we just have to compare the constant terms; they are both linear combinations of the operators T^{loc}, they coincide (up to a factor) and these T^{loc} are onto J_{s}. □

7.3 Rational vectors.

7.3.1 Essential vector

Let v be a finite place of F. Suppose the maximal ideal on which ψ_{v} is trivial is O_{v}. In theorem (4.1), page 208 of [JPSS1], it is shown that there exists an unique function W^{∘} called essential vector of the representation such that

• ∀g ∈ G_{v}, ∀k ∈ G_{r−1}(O_{v}), W^{∘} (g ( \begin{matrix}k & 0 \\ 0 & 1 \end{matrix} )) = W^{∘}(g)

• a certain integral is covariant.

If the representation is non ramified, this is the only vector fixed by K_{v} such that its value at I_{r} is 1.

Now let ψ_{1} be a character of F_{v}^{×}, and let a_{1} be an element of F_{v} such that for all x ∈ F_{v}, ψ_{1}(x) = ψ_{v}(a_{1}x). We define an isomorphism between W_{0}(π_{v}, ψ_{v}) and W_{0}(π_{v}, ψ_{1}) sending a vector W of the first space to the function

g ⟼ W ( diag_{1⩽j⩽r}(a_{1}^{r−j})g).

This isomorphism sends the essential vector to the essential vector if it is already defined. If not, we define the essential vector W^{∘} of W_{0}(π_{v}, ψ_{1}) by requiring that it is mapped to an essential vector by this isomorphism.

7.3.2 Cuspidal representations

Let’s see first the cuspidal representation. We decompose it as π = ( ⨁_{v∈Σ_{f}}π_{v}) ⊗ π_{f}.

According to [Clo, Théorème 3.19, p129], the cohomology H^{•}(𝔤_{C}, K_{C}; π_{∞} ⊗ τ) is defined over some finite extension F (π) of F. We take E to be equal to F (π, π′) = F (π)F (π′).

As stated in [JPSS1, Theorem (5.1), p211], there exist an ideal n(π) of O_{F} such that the compact open subgroup of G(A_{F,f}) defined by

K_{f}(π) = \{A ∈ G ( ∏_{v∈Σ_{f}} O_{v}) /A ≡ ( \begin{matrix}k_{1} & ∗ \\ 0 & 1 \end{matrix} ) (mod n(π)), k_{1} ∈ G_{r−1} (∏ O_{v})\}

has a one-dimensional space of fixed vector in π_{f}. The essential vector W_{v}^{∘}, [JPSS1, definition (4.4), p211], of W_{0}(π_{v}, ψ_{v}) has the property that W_{f}^{∘} = ⨂ W_{v}^{∘} is in π_{f}^{K_{f} (π)}. We denote K(π) = Z_{G}(F_{∞})K_{∞}K_{f}(π). The space

H^{\frac{r(r−1)}{2} d}(𝔤_{C}, K_{C}; π ⊗ τ)^{K_{f} (π)} = (⋀^{\frac{r(r−1)}{2} d}(𝔤/𝔨)^{∗} ⊗ π ⊗ τ)^{K(π)}

is thus one-dimensional.

On each irreducible finite dimensional representation of G_{v}, there is a natural Q-rational structure associated to each highest weight vector. It comes from the natural Q-rational structure of the exterior powers of the standard representation and tensors thereof. By picking a highest weight vector in any finite-dimensional G_{v}, we fix an embedding into a tensor product of exterior powers of G_{v}, giving it a Q-rational structure. This of course applies as well to (irreducible finite-dimensional) representations of K_{v}.

We choose a highest weight vector in τ and give to τ the corresponding Q-rational structure. On the minimal K_{v} type of W_{0}(π_{v}, ψ_{v}) we fix a Q-rational structure by picking a non-zero highest weight vector. For 𝔭^{∗} we take the usual basis as a Q-rational basis. We have now fixed a Q-rational structure on the abovementioned one-dimensional space. But it still has the F (π)-rational structure coming from cohomology. We denote C(π) the quotient of the two rational structures (that is: a rational vector coming from the cohomology, is C(π) times a rational vector coming from the Whittaker models). This number C(π) is well defined up to an element of F (π). This number does not depend on the rational structure given to the coefficient system because it is a ratio.

7.3.3 Eisenstein series representations

For the Eisenstein series, the idea is the same. On the infinite part, the only thing to do is to choose the “good” multiple. We need to choose a finite part. We decompose the ideal n(π) as n(π) = ∏_{v∈Σ_{f}} n_{v}(π) with n_{v}(π) a power of the prime ideal corresponding to v. With that, we take

K_{v}(π) = \{A ∈ G(O_{v}) /A ≡ ( \begin{matrix}k_{1} & ∗ \\ 0 & 1 \end{matrix} ) (mod n_{v}(π)), k_{1} ∈ G_{r−1}(O_{v})\}

for each finite place v and we do the same for π′. We choose for Φ_{v} the characteristic function of ϵK_{v}(π, π′) with K_{v}(π, π′) = K_{v}(π) ∩ K_{v}(π′). Gluing together all these local components, we obtain a Schwartz function for A_{F}^{r} and the associated Eisenstein series. We use it, as for the cuspidal representations as a rational vector and we define C_{s}(η) to be the corresponding period. We can compute this C_{s}(η). Let Δ^{1} be the positive real number

such that \frac{dx}{Δ} is an F -rational differential form on C. Let’s start from ω = ∑_{i=1}^{dim τ′′}E(f_{i})v_{i} dg_{i}.

The v_{i}’s are the elements of the rational basis of τ′′ we chose. We take v_{1} to be a rational vector of highest weight,

f_{1}(g) = \frac{| det g|_{A_{F}}^{s} |g_{rr}|_{A_{F}}^{k_{η}} η(g_{rr})^{−1}}{(∑_{j=1}^{r} |g_{jr}|_{A_{F}} )^{k_{η}+rs}}

and dg_{1} = dg_{1,r} ∧ ... ∧ dg_{r−1,r}. To evaluate the rationality of this differential form, we first restrict it to the Borel stratum of the Borel-Serre compactification (operation rational since E(J_{s}) is embedded in Eis_{max}), then take the cup-product with the rational form \frac{1}{Δ}^{r−1} d\overline{g_{1,r}}∧...∧ d\overline{g_{r−1,r}} and integrate over the domain whose representatives are the matrices

1The definition is a bit long and can be found p493 of [Hid]

( \begin{matrix}1 \\ 0 & & I_{r−2} \\ x_{1} & ... & x_{r−1} & 1 \end{matrix} )

The result is the constant term of E(f_{1}) along the corresponding parabolic (times \frac{v_{1}}{Δ}^{r−1}) evaluated at I_{r}; this is just f_{1}(I_{r}) = 1. ¿From that we see that Δ^{r−1}.ω is a rational form, so that C_{s}(η) = Δ^{r−1}. We will need an additional notation: we call D(s, η) the E-vector space generated by the rational vectors of E(J_{s}).

8 Special value.

Theorem 8.1 Let π and π′ be two irreducible automorphic cuspidal cohomological representations of G_{r}. Let η be the product of their central characters. Let s_{0} > 1 be an integer greater than 1 in the computable strip of π×π′. Let ∑ φ⊗φ′⊗Φ be an element of the trivial K_{∞}-type of W_{0}(π_{∞}, ψ_{∞}) ⊗ W_{0}(π′_{∞}, \overline{ψ_{∞}}) ⊗ S(A_{F}^{r}) such that each φ, φ′ and f(s, g; η, Φ) are rational elements of each minimal K_{∞}-type. Then:

L_{f}(s_{0}, π × π′) = \frac{Δ^{r−1}C(π)C(π′)}{∑ Z_{∞}(s_{0}, φ ⊗ φ′ ⊗ Φ)}c

where c ∈ F (π, π′).

Proof: it is now straightforward. We take two irreducible cuspidal cohomological representations π and π′ and s_{0} > 1 an integer in the computable strip of π × π′. We begin with pure tensors. Let φ and φ′ be non zero rational elements of respectively W_{0}(π_{∞}, ψ_{∞}) and W_{0}(π′_{∞}, \overline{ψ_{∞}}). We take then W = φ ⊗ ⨂_{v∈Σ_{f}} W^{∘}(π_{v}, ψ_{v}) and W′ accordingly (changing ψ to

\overline{ψ}). The isomorphism between π and W(π, ψ) is canonical because it is fixed by Fourier decomposition thus to W and W′ correspond φ_{t} ∈ π and φ′_{t} ∈ π′.

Now take Φ to be a non zero element of D(s_{0}, η). We need to find local components for finite places. Let v ∈ Σ_{f}. We will show that for a good choice of Φ_{v} ∈ S(F_{v}^{r}), Z_{v} = Z(s_{0}, W_{v}^{∘}, W′_{v}^{∘}; Φ_{v}) is a non zero element of F (π, π′)(q_{v}^{−s_{0}}) (where q_{v} = N v is the number of element of the residue field of F_{v}). Suppose first that the ideal on which ψ_{v} is trivial is O_{v}. We then have W_{v}^{∘}(I_{r}) = W′_{v}^{∘}(I_{r}) = 1. We put K_{v}(π, π′) = K_{v}(π) ∩ K_{v}(π′) and take Φ_{v} to be the characteristic function of ϵK_{v}(π, π′) in F_{v}^{r}. As K_{v}(π) and K_{v}(π′) are open compact subgroup of G_{r}(F_{v}), K_{v}(π, π′) is an open compact subgroup of G_{r}(F_{v}) and Φ_{v} ∈ S(F_{v}^{r}). If G_{v} is decomposed as G_{v} = P_{v}K_{v}(π, π′).

Z_{v} = ∫_{K_{v}(π,π′)} ∫_{N\P/P ∩K_{v}(π,π′)} W_{v}^{∘}(pk)W′_{v}^{∘}(pk)Φ_{v}(ϵpk)| det p|^{−s_{0}} dpdk.

The integral on N_{v}\P_{v} is the integral on (N_{r−1}(F_{v})\G_{r−1}(F_{v})) × G_{1}(F_{v}) injected in G_{v} as in paragraph 4.3. We look for the (p, k) such that the integrand in non vanishing. We have Φ_{v}(ϵpk) = Φ_{v}(ϵp_{rr}k). As Φ_{v} is the characteristic function of ϵK_{v}(π, π′), we must have p_{rr} ≡ 1 (mod n_{v}(π, π′)) with n_{v}(π, π′) = n_{v}(π) ∩ n_{v}(π′). It follows that k ∈ (P ∩ K)K_{v}(π, π′). But all functions are invariant under the action of K_{r−1,v} = GL_{r−1}(O_{v}), thus it comes:

\begin{matrix}Z_{v} = & c ∫_{A^{v}/_{Av∩Kv}} W_{v}^{∘}(a)W′_{v}^{∘}(a)| det a|^{−s_{0}} da \\ = & c ∑_{{n_{1},...,n_{r} \atop n_{r}⩾0}} W_{v}^{∘}(d)W′_{v}^{∘}(d)q^{−(n_{1}+...+n_{r})s_{0}} \\ = & c ∑_{n_{1}⩾...⩾n_{r}⩾0} W_{v}^{∘}(d)W′_{v}^{∘}(d)q^{−(n_{1}+...+n_{r})s_{0}} \end{matrix}

where, with ϖ an uniformizer of F_{v}, d = diag(ϖ^{n_{1}}, ..., ϖ^{n_{r}}) and c > 0 is the measure of (P ∩K)K_{v}(π, π′). The last equality comes from the explicit calculus of W_{v}^{∘} in [JPSS1, p212] and [Shi]. Measures are normalized by requiring that all GL_{α}(O_{v}), 1 ⩽ α ⩽ r have measure 1. Then Z_{v} is non zero because the constant coefficient is equal to c, that is the inverse of the number of elements of K_{v}/(P ∩ K)K_{v}(π, π′). Now Z_{v} is an entire series in q^{−s_{0}}. The coefficient of q^{−ls_{0}} is equal to ∑_{d} W_{v}^{∘}(d)W′_{v}^{∘}(d) where d runs through the set of matrices of determinant q^{−l}. In [JPSS1, p212], the value of W_{v}^{∘}(d) is computed using the value of nonramified Whittaker models. In [Shi] it is proven that this value is a polynomial with integral coefficients in the eigenvalues of the Hecke operators thus are elements of F (π). It follows that Z_{v} ∈ F (π, π′)[[q^{−s_{0}}]]. In [JPSS2, p392], it is proven that Z_{v} is a rational fraction with denominator ∏(1 − α_{i}q^{−s_{0}}) where the α_{i}’s depend only from π and π′; with this setup, L_{v}(s_{0}; π ×π′) = ∏(1−α_{i}q^{−s_{0}})^{−1}. But now, from [JPSS2] and in particular proposition 8.1, theorem 8.2 and proposition 9.4 with the following remarks, each α_{i} ∈ F (π, π′). So that summing up, \frac{Z_{v}}{L_{v}(s_{0};π×π′)} ∈ F (π, π′)[q^{−s_{0}}]. Note that if all representations are unramified, this polynomial is 1. But q^{−s_{0}} ∈ Q thus \frac{Z_{v}}{L_{v}(s_{0};π×π′)} ∈ F (π, π′). We remove the restriction on ψ_{v} as we did in subsection 7.3.1.

We now build the global Schwartz function by multiplying all these Φ_{v}’s together: Φ_{t} = Φ. ∏_{v∈Σ_{f}} Φ_{v}.

Let S be the set of finite places where (at least) one of the representations π, π′ or ψ ramifies. As a local L-factor is never zero, we have

Z(s_{0}, W ⊗ W′ ⊗ Φ_{t}) = Z_{∞}(s_{0}, φ ⊗ φ′ ⊗ Φ)L_{f}(s_{0}; π × π′) ∏_{v∈S} \frac{Z(s_{0}, W_{v}^{∘} ⊗ W′_{v}^{∘} ⊗ Φ_{v})}{L_{v}(s_{0}, π × π′)}.

We now come back to our trivial K_{∞}-type. The finite part of all vectors are equal so that they factor out of the sum. As for the infinite part, they add up correctly to an element of the trivial K_{∞}-type. Since diagram (5) is commutative and from the definition of C(π), C(π′) and C_{s_{0}}(η), the integral

∑ Z (s_{0}, \frac{φ_{t}}{C(π)} ⊗ \frac{φ′_{t}}{C(π′)} ⊗ \frac{Φ_{t}}{C_{s_{0}}(η)})

is an element of F (π, π′). □

References

[BW] Armand Borel and Nolan R. Wallach. Continuous cohomology, discrete subgroups and representations of reductive groups, volume 94 of Annals of Mathematics Studies. Princeton University Press and Tokyo University Press, 1980.

[Clo] Laurent Clozel. Motifs et formes automorphes : application du principe de fonctorialité. In Clozel and Milne [CM], pages 77–159.

[CM] Laurent Clozel and J. S. Milne, editors. Automorphic forms, Shimura varieties and L-functions, Proceedings of the Ann-Arbor conference, volume I, II. Academic Press, 1990.

[FH] William Fulton and Joe Harris. Representation theory. A first course, volume 129 of Graduate texts in Mathematics. Springer-Verlag, Berlin, 1991.

[Gre] Loïc Grenié. Valeurs spéciales de fonctions L de GL(r) × GL(r). PhD thesis, Université Paris 7–Denis Diderot, Jan 2000.

[Har1] Günter Harder. Eisenstein cohomology of arithmetic groups. The case GL_{2}. Inventiones Mathematicae, 89:37–118, 1987.

[Har2] Günter Harder. Some results on the Eisenstein cohomology of arithmetic subgroups on GL_{n}. In Jean-Pierre Labesse and Joachim Schwermer, editors, Cohomology of arithmetic groups and automorphic forms, volume 1447 of Lecture Notes in Mathematics, pages 85–153. Springer-Verlag, 1990.

[Hid] Haruzo Hida. On the critical values of L-functions of GL(2) and GL(2) × GL(2). Duke Mathematical Journal, 74(2):431–529, May 1994.

[JPSS1] Hervé Jacquet, I. I. Piatetski-Shapiro, and Joseph A. Shalika. Conducteur des reprsentations de groupe linaire. Mathematische Annalen, 256:199–214, 1979.

[JPSS2] Hervé Jacquet, I. I. Piatetski-Shapiro, and Joseph A. Shalika. Rankin-Selberg convolutions. Amercian Journal of Mathematics, 105:367–464, 1983.

[JS1] Hervé Jacquet and Joseph A. Shalika. On Euler products and the classification of automorphic representations I. American Journal of Mathematics, 103:499–558, 1981.

[JS2] Hervé Jacquet and Joseph A. Shalika. On Euler products and the classification of automorphic representations II. American Journal of Mathematics, 103:777–815, 1981.

[Sha] Freydoon Shahidi. Automorphic L-Functions: A survey. In Clozel and Milne [CM], pages 415–437.

[Shi] T. Shintani. On an explicit formula for class-1 “Whittaker functions” on GL_{n} over p-adic fields. Proceedinds of Japan Academy, 52:180–182, 1976.

[Vog] David A. Vogan. Representations of Real Reductive Lie Groups, volume 15 of Progress in Mathematics. Birkhuser, 1981.

[Zuc] G. Zuckerman. Tensor products of finite and infinite dimensional representations of semisimple Lie algebras. Annals of Mathematics, 106:295–308, 1977.