Open local periods

Proof. — By (2.5) and (A.4) we have

B^α

π x ,δ^π_e^∨

W(π ) (1K)= α^π_x(W^π₀)

[W^π₀,W^π₀^∨]^π = α^π_x(W^π₀) L(1, π×π^∨).

On the other hand, by (3.6), the defining property ofα^π_x, and (2.17) we have B^α

π x ,δ^π_e^∨

W(π ) (1K)=Bπ(1K)=L

1, π×π^∨−1

.

The Lemma follows.

4. Open local periods

The local periods that we are going to study in this section were introduced in [LR00] for principal series representations. They were further analyzed in [Off07]. Here we introduce analogous objects for more general induced representations of finite length.

Fix a standard parabolicP=MU. Note that P has an open orbit X^◦=X∩^tUMU onX. While X^◦is not a single P orbit in general, any P orbit in X^◦intersects X^M=X∩M in a single M-orbit. Therefore, X^◦/P is in bijection with the finite set X^M/M. The P-orbits in X^◦are the open orbits of P in X (in the usual topology).

Letν_M=ν_M^G be either the character of Mor the function on X^Mdefined by (4.1) ν_M(g)=

!t i=1

ηⁱ⁻¹(det gi), g=diag(g1, . . . ,gt) and set furtherν₀=ν_M0.

Let σ ∈R(M) and let α∈EM(X^M, σ^∗). For any x∈X, λ∈a^∗_M,C and ϕ∈I(σ ) consider the expression

(4.2) JM(ϕ:x, α, λ)=

y

ν_M(y)e^{12ρM+λ,H(y)}

P^y\G^y

α_y ϕ_λ

gι^x_y dg

where y ranges over a (finite) set of representatives of the M-orbits(M∩x•G)/M in M∩ x•G andι^x_y∈G is such that x=y•ι^x_y.(See Lemma4.3below for geometric motivation.) Proposition4.1. — Letσ ∈R(M)andα∈EM(X^M, σ^∗). In the Archimedean case, assume further thatσ is unitarizable. Then

(1) The sum of integrals (4.2) is well-defined and absolutely convergent forλ=(λ₁, . . . , λ_t)∈ a^∗_M,Csuch that Re(λi−λ_i+1)0,i=1, . . . ,t−1.

(2) For every x∈X the linear formϕ→J^G_M(ϕ:x, α, λ) admits a meromorphic continuation inλ∈a^∗_M,C(still denoted by J(x, α, λ)=J^G_M(x, α, λ)). In the p-adic case, it is a rational function in q^λ_F=(q^λ_F¹, . . . ,q^λ_F^t). In the Archimedean case, it has hyperplane singularities of the formλ_i−λ_j=c.

(3) The map x→J(x, α, λ)is an element ofEG(X,I(σ, λ)^∗)which we denote by J(α, λ).

(4) Suppose that J(x, α,·)is regular atλ. Then J(x, α, λ)≡0 if and only ifα_y≡0 for all y∈M∩x•G.

Proof. — We first check that the definition formally makes sense. If y = diag(y1, . . . ,yt)∈X^Mthen

P^y=M^y=

diag(g1, . . . ,gt):gi∈G^y_nⁱ

i,i=1, . . . ,t

is a product of unitary groups in the diagonal blocks of M. The measure on P^y is the product of the measures on G^y_nⁱ_i according to our convention. It follows from the descrip-tion of P^y that H(m)M=0 for all m∈P^y. Furthermore, sinceαy is an M^y-invariant linear form onσ we have

α_y ϕ_λ

mgι^x_y

=α_y

σ (m)ϕλ

gι^x_y

=α_y ϕ_λ

gι^x_y

, m∈P^y,g∈G

and therefore, the integrand in (4.2) is indeed M^y-invariant. Ifι∈G is another element is independent of the choice of a representative y of the M-orbit and whenever absolutely convergent, (4.2) is well-defined. Furthermore, for any h∈G we may chooseι^x_y^•h=ι^x_yh.

it is also formal from the definition that whenever defined by an absolutely convergent sum of integrals, we have

J(ϕ:x•g, α, λ)=J

I(g, σ, λ)ϕ:x, α, λ

, g∈G.

Next, we justify the absolute convergence and meromorphic continuation. The absolute convergence of integrals of the form

M^y\G^y

α_y ϕλ(gι)

dg

in some positive cone and their meromorphic continuation (to a rational function in q^λ_Fin the non-Archimedean case) was proved in a more general context of a reductive symmet-ric space. In the Archimedean case this was done by Brylinski, Carmona and Delorme ([CD94, Proposition 2 and Theorem 3], cf. [BD92]) using a result of Flensted-Jensen, Oshima and Schlichtkrull [FJ ¯¯ OS88]. In the non-Archimedean case we refer to Blanc-Delorme [BD08, Theorems 2.8 and 2.16].⁶The result in [loc. cit.] is stated under an ad-ditional assumption, but the latter is always satisfied thanks to a result of Lagier [Lag08,

6Note that in the terminology of [loc. cit.], any P is aθy-parabolic subgroup of G whereθy(g)=y⁻¹(^tg^τ)⁻¹y.

Theorem 4(i)]. (Alternatively, one could also prove the rationality using Bernstein’s mero-morphic continuation principle as in the proof of [LR00, Proposition 2]—cf. Remark6.8 below.) The formal steps at the beginning of the proof are now justified.

For the last part, let G^◦[x] = {g∈G:x•g⁻¹∈X^◦} be the union of open double cosets in P\G/G^x. One observes (cf. [CD94, Theorem 3, part 3]) that the restriction of J(x, α, λ)to

ϕ∈I(σ ):suppϕ⊆G^◦[x]

is non-zero if and only if there exists y∈X^M∩x•G such thatα_y≡0.

Corollary4.2. — Letσ ∈R(M),π=I(σ, λ)and x∈X. In the Archimedean case assume further thatσ is unitarizable. Suppose thatσ is M^y-distinguished for some y∈M∩x•G. Thenπ is G^x-distinguished.

Proof. — The Corollary follows by takingα∈EM(X^M, σ^∗)non-zero and supported (as a function on X^M) in the given M-orbit y•M and taking the leading term atλof the restriction of JM(x, α,·)to a complex line throughλin general position.

We may interpret and motivate the definition of the operation JMas follows. Recall the isomorphism

EG

X,I(σ, λ)^∗

→HomG

S(X),I(σ, λ)^∨

defined by (2.6). Similarly, forα∈EM(X^M, σ^∗)we may defineαby (α)(v)=

X^M

ν_M(y)(y)αy(v)dy, ∈S X^M

, v∈σ.

(Note that this is consistent with (2.6) when M=G.) Thenα→(→α)defines an isomorphism of vector spaces

EM

X^M, σ^∗

→HomM

S X^M

, σ^∨ . We also have a canonical isomorphism of vector spaces

EM

X^M, σ^∗

→EM

X^M, σ[λ]^∗ for anyλ∈a^∗_M,Cgiven byα→α[λ]where

α[λ]y=e^12λ,H(y)α_y, y∈X^M.

The mapα→JM(α, λ)(when defined) gives rise to a map JM(λ):EM

X^M, σ^∗

→EG

X,I(σ, λ)^∗

and therefore, via the identifications above, gives rise to a map By Frobenius reciprocity, we obtain a map

(4.3) HomM we identify S(X^◦)with the P-invariant subspace ofS(X)consisting of functions which vanish (together with all their derivatives in the Archimedean case) on the complement of X^◦. We get an M-equivariant map

The alternative description of JMis as follows.

Lemma4.3. — The composition of (4.3) with the restriction map HomM obtained by composition with the map (4.4) induced by→^M.

Proof. — Explicating the various identifications, the Lemma amounts to the relation JM(α, λ)

To show this, we compute the left hand side as

=

where the integrals are absolutely convergent for∈S(X^◦). The lemma follows.

It will be useful to normalize the functionals JM for σ ∈Rpi(M) (assumed uni-tarizable in the Archimedean case). The normalizing factor depends on an auxiliary σ∈Rpi(M) (as well as on σ itself which will be omitted from the notation, since in and define the normalized linear forms and equivariant maps

J^σ(x, α, λ)=n

By Proposition 4.1, these are meromorphic functions in λ (rational in q^λ_F in the non-Archimedean case) and lie in HomG^x(I(σ, λ),C)andEG(X,I(σ, λ)^∗)respectively.

Sup-pose further thatσ∈^M andσ =bc(σ). In this case, we may write

(4.5) n_M

σ, λ

= !

1≤i<j≤t

λ⁻_ψ^ninjωσ_j(−1)ⁿⁱγ

λi−λj, σ_i×σ_j^∨·η;ψ .

Note that it follows from (3.12) that in the notation of Lemma3.7, for anyσ∈^M_bc-gen we have

(4.6) Jσ[μ]

α^σ[μ], λ

=Jσ

α^σ, μ+λ

◦A(σ, μ) whereσ =bc(σ)andμ, λ∈a^∗_M,C.

For inductive arguments, it will be useful to consider the open periods in the relative situation for a pair L=Mγ ⊆M=Mκ of standard Levi subgroups in G. Let ν_L^M() be defined forin either Lor X^Lby

ν_L()=ν_L^M() ν_M().

For a representation∈R(L)(unitarizable in the Archimedean case), y∈X^M andα∈ EL(X^L, ^∗), we define the linear form J^M_L(y, α, μ)∈HomM^y(I^M_L(, μ),C)as the function inμ∈a^∗_L,Cgiven by the meromorphic continuation of the sum of integrals

J^M_L(ϕ:y, α, μ)=

z∈(L∩y•M)/L

ν_L^M(z)e^{12ρLM+μ,H(z)}

L^z\M^z

α_z ϕ_μ

mι^y_z dm

forϕ∈I^M_L(σ ), where ι^y_z∈M is a choice such that z•ι^y_z=y. We denote by J^M_L(α, μ)∈ EM(X^M,I^M_L(σ, μ)^∗)the associated equivariant map. Assume that∈Rpi(L)and let= ₁⊗ · · · ⊗_t∈^L where_i is a representation of the Levi M_γi of G_ni (see Section 1for the notation). Letμ=(μ₁, . . . , μt)∈a^∗_L,Cwithμi∈a^∗_M

γi,Cand let w∈W^M(L).Writing w=diag(w₁, . . . , w_t)wherew_i∈W^Gni(Mγ_i), we set

C^M_L

w:, μ;ψ

=

!t i=1

C^G

ni

M_γ

i

w_i:_i, μ_i;ψ ,

C^M_L(w:, μ;ψ )=

!t i=1

C^G_Mⁿⁱ_γ

i(w_i:_i, μ_i;ψ ), and

n^M_L

, μ

= C^M_L(w_M^L :, μ;ψ ) C^M_L(w^L_M:, μ;ψ).

It follows from the multiplicativity of γ-factors (1.8) that for any w₁∈ W(M), w₂ ∈ W^M(L)we have

(4.7) CL

w₁w₂:, μ+λ;ψ

=C^M_L

w₂:, μ;ψ CM

w₁:I^M_L

, μ

, λ;ψ as meromorphic functions inμ∈a^∗_L,Candλ∈a^∗_M,C. On I^M_L()we set

JL,^M(x, α, μ)=n^M_L , μ

J^M_L(x, α, μ).

We now show that the open local unitary periods are compatible with transitivity of induction.

Lemma 4.4. — Let L ⊆M be Levi subgroups of G, ∈R(L), ∈^L and α ∈ EL(X^L, ^∗). In the Archimedean case, assume further that is unitarizable. Then as a meromor-phic function inλ+μ(in the p-adic case, rational function in q^μ_F^+λ) forμ∈a^∗_L,Candλ∈a^∗_M,C,we have

(4.8) J^G_L(x, α, μ+λ)=J^G_M

x,J^M_L(α, μ), λ

◦_L,M(, μ) and if∈Rpi(L)then similarly

(4.9) JL,^G(x, α, μ+λ)=J^G

M,I^M_L

,μx,JL,^M(α, μ), λ

◦_L,M(, μ).

The right-hand sides of (4.8) and (4.9) are defined by an absolutely convergent sum of integrals (also in the Archimedean case) wheneverλandμare sufficiently positive and by meromorphic continuation in general.

Proof. — Sincew_L=w_Mw^M_L,(4.9) follows from (4.7) and (4.8). To prove the iden-tity (4.8) we may assume that Reμand Reλare sufficiently regular in the corresponding cones. A set of representatives for(L∩x•G)/L can be chosen by first fixing a set{y}of representatives for (M∩x•G)/M and then taking the union over all such y of a set of representatives for(L∩y•M)/L.By Proposition4.1for anyϕ∈IL(), J^G_L(ϕ:x, α, μ+λ) is equal to

(4.10)

y∈(M∩x•G)/M

z∈(L∩y•M)/L

ν_L(z)e^{12ρL+μ+λ,H(z)}

L^z\G^z

α_z ϕ_μ+λ

gι^x_z dg

where the sum-integral is absolutely convergent. Since z∈y•M, we have (4.11) ν_M(z)=ν_M(y).

Letι^y_z∈M andι^x_y∈G be chosen so thatι^x_z=ι^y_zι^x_y.Since z•ι^y_z=y, we have

(4.12) (

ρ_L+μ+λ,H(z))

=(

ρ_L^M+μ,H(z)) +(

ρ_M+λ,H(y)−2H ι^y_z)

.

Since(ι^y_z)⁻¹M^zι^y_z=M^y and(ι^y_z)⁻¹G^zι^y_z=G^y,by integrating in stages and performing the

Plugging this into (4.10), taking into account (4.11) and (4.12), and changing the order of summation over z with integration over M^y\G^ywe obtain that

J^G_L(ϕ:x, α, μ+λ)=

where the right-hand side is absolutely convergent. Note that the term in curly brackets equals J^M_L(ξ_λ(gι^x_y):y, α, μ)and therefore, this gives the identity (4.8).

Remark 4.5. — Under the interpretation of Lemma 4.3, Lemma4.4 reflects the relation^L=(^M)^L.

For future use, we define a meromorphic family of normalized relative Bessel dis-tributions associated with a representationσ∈^M_bc-gen (unitarizable in the Archimedean case) by

Dans le document On representations distinguished by unitary groups (Page 35-44)