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Jacquet Modules of Parabolically Induced Representations and Weyl Groups

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Jacquet Modules of Parabolically Induced Representations and Weyl Groups JACQUET MODULES OF PARABOLICALLY INDUCED REPRESENTATIONS AND WEYL GROUPS DUBRAVKA BAN Abstract. The representation parabolically induced from an ir- reducible supercuspidal representation is considered. Irreducible components of Jacquet modules with respect to induction in stages are given. The results are used for consideration of generalized Steinberg representations. 1. Introduction Jacquet modules of parabolically induced representations can be applied to some problems in representation theory, for example, the question of reducibility of parabolically induced representations ([T3], [J3]). They are also used for important work on description of dis- crete series for classical p-adic groups ([MT], [J1], [J2]). These articles concern the classical groups Sp(n, F ) and SO(2n + 1,F ). For calcu- lating Jacquet modules of parabolically induced representations, they are using a structure on representations of the groups Sp(n, F ) and SO(2n + 1,F ), described in [T1]. The purpose of this article is to find new techniques for calculating Jacquet modules for any connected p-adic group. Consequently, we also describe the structure of parabolically induced representations, their irreducible subrepresentations and irreducible subquotients. Let G be a connected reductive p-adic group, P = MU a standard parabolic subgroup of G and σ an irreducible supercuspidal represen- tation of M. The geometric lemma ([BZ], [C], here Theorem 2.1) de- scribes composition factors of rM,G ◦ iG,M (σ), where iG,M denotes functor of parabolic induction and rM,G the Jacquet functor ([BZ]). After eliminating all zero components in rM,G ◦ iG,M (σ) ([BZ], [C], here Theorem 2.2), we prove some interesting facts about the structure of iG,M (σ) and rM,G ◦ iG,M (σ) (Lemma 3.1 and Corollary 4.3). 1991 Mathematics Subject Classification. 22E50. 1 2 DUBRAVKA BAN In Section 5, we consider an intermediate standard Levi subgroup N such that M<N<G and describe how information about iN,M (σ) and rM,N ◦ iN,M (σ) can be used in determining iG,M (σ) and rM,G ◦ iG,M (σ) (Theorem 5.1, Corollaries 5.2 and 5.3). Using results of Sections 3, 4 and 5, choosing appropriate inter- mediate Levi subgroups, we can deduce information about paraboli- cally induced representations and their Jacquet modules. In Section 7, we carry this out for some examples of representations of Sp(m,F ), SO(2m + 1,F ) and SO(2m,F ). More precisely, we consider the repre- sentation να+nρ ×···× ναρ ⋊ σ, where ρ, σ are supercuspidal and ναρ ⋊ σ is reducible (see Section 7 for notation). For α> 0, this representation is of length 2n+1 (Proposition 7.2) and it has the unique irreducible subrepresentation. This subrep- resentation is square integrable and it is the unique square integrable subquotient of να+nρ ×···× ναρ ⋊ σ (Propositions 7.1 and 7.2). We shall call it a generalized Steinberg representation (see Remark 7.1). I would like to thank David Goldberg, Chris Jantzen, Freydoon Shahidi, Marko Tadi´cand the referee for valuable suggestions. The first part of this article was done during post-doctoral fellowship at ICTP, Trieste, and I would like to thank them for their kind hospital- ity and support. 2. Preliminaries In this section, we introduce notation and recall some results from [BZ] and [C] on Jacquet modules of parabolically induced representa- tions. Let F be a p-adic field and G the group of F -points of a reductive algebraic group defined over F . Fix a minimal parabolic subgroup P0 and a maximal split torus A0 ⊂ P0. Let ∆ be the corresponding set of simple roots. If Θ ⊂ ∆, then we write PΘ = MΘUΘ for the standard parabolic subgroup determined by Θ. Let P = MU be a standard parabolic subgroup of G. If σ is a smooth representation of M, then we denote by iG,M (σ) the represen- tation parabolically induced by σ. For a smooth representation π of G, rM,G(π) is normalized Jacquet module of π with respect to M ([BZ]). For a smooth finite length representation π we denote by s.s.(π) the semi-simplified representation of π. It is the direct sum of the irreducible components of π. Let ≤ denote the natural partial order on the Grothendieck group of the category of all smooth finite length INDUCTIONANDJACQUETMODULES 3 representations of G. For smooth finite length representations π1 and π2, we write π1 ≤ π2 if s.s.(π1) ≤ s.s.(π2) in the Grothendieck group. Let W be the Weyl group of G. For Θ ⊂ ∆, we denote by WΘ ⊂ W the Weyl group of MΘ. Let Ω ⊂ ∆. Set ([C]) −1 [WΘ\W/WΩ]= {w ∈ W | wα > 0, ∀α ∈ Ω, w β > 0, ∀β ∈ Θ}. Theorem 2.1 (Geometric lemma). ([BZ], [C]) Let G be a connected reductive p-adic group, P = PΘ = MU, Q = PΩ = NV parabolic subgroups. Let σ be an admissible representation of M. Then rN,G ◦ iG,M (σ) has a composition series with factors −1 iN,N ′ ◦ w ◦ rM ′,M (σ) ′ ′ −1 where M = M ∩ w(N), N = w (M) ∩ N and w ∈ [WΘ\W/WΩ] . For Θ, Θ′ ⊆ ∆, we define W (Θ, Θ′)= { w ∈ W | wΘ′ =Θ }. IfΘ=Θ′, then we set W (Θ) = W (Θ, Θ), and this is a subgroup of W. Let Θ, Θ′ ⊆ Ω ⊆ ∆. Define ′ ′ WΩ(Θ, Θ )= { w ∈ WΩ | w Θ =Θ }, WΩ(Θ) = WΩ(Θ, Θ). Theorem 2.2. ([BZ], [C]) Let σ be a supercuspidal representation of M = MΘ. Then s.s.(r ◦ i (σ)) = w−1σ = wσ. M,G G,M X X w∈W (Θ) w∈W (Θ) Define W (σ)= { w ∈ W (Θ) | wσ =∼ σ }. We call σ regular if W (σ) = {1}. Otherwise, we say that σ is non- regular. Theorem 2.3. ([C], Proposition 6.4.1) Let σ be a regular supercuspi- dal representation of M = MΘ. Then r ◦ i (σ) ∼= w−1σ = w σ. M,G G,M M M w∈W (Θ) w∈W (Θ) 4 DUBRAVKA BAN 3. Regular case Lemma 3.1. Let σ be a regular irreducible supercuspidal representa- tion of M. Then: 1. iG,M (σ) has a unique irreducible subrepresentation. 2. All irreducible subquotients of iG,M (σ) are mutually inequivalent. 3. For any w1,w2 ∈ W (Θ), dimC HomG(iG,M (w1 σ),iG,M (w2 σ))=1. 4. Let p be the unique irreducible subrepresentation of iG,M (σ). Then ∼ .(wσ ≤ rM,G(p) ⇔ p֒→ iG,M (wσ) ⇔ iG,M (wσ) = iG,M (σ Remark 3.1. For σ a regular character, Lemma 3.1. is proved in [R]. Proof. 1. is well-known ([C]), 2. and 3. can be proved using Jacquet modules, Frobenius reciprocity and Theorem 2.3. 4.) The first equivalence follows from Frobenius reciprocity and Theorem 2.3. Now, suppose that p ֒→ iG,M (wσ). By 3.), we have a unique (up to a scalar) non-trivial intertwining operator ϕ between .(iG,M (σ) and iG,M (wσ) . Then Kerϕ ֒→ iG,M (σ), Imϕ ֒→ iG,M (wσ If Kerϕ =6 {0}, then p ֒→ Kerϕ, so, by 2.), Imϕ = {0}. But this contradicts the assumption that ϕ is non-trivial. We conclude that ∼ Kerϕ = {0}, so iG,M (wσ) = iG,M (σ). Let P = MU be a standard parabolic subgroup of G. Denote by P − the opposite parabolic subgroup of P , i.e., the unique parabolic subgroup intersecting P in M. Let P¯ = M¯ U¯ be the unique standard parabolic subgroup conjugate to P − ([C]); we can have either P¯ = P or P¯ =6 P . Proposition 3.2. Let P = MU be a standard parabolic subgroup of G, and let σ be an irreducible supercuspidal regular representation of M. Take w ∈ W such that w(P −) = P¯. If q is an irreducible subrep- resentation of iG,M (σ), then q˜ is a subrepresentation of iG,M¯ (wσ˜). (Hereσ ˜ denotes the contragredient representation of σ.) In the proof of Proposition 3.2, we shall use non-standard parabolic induction, in the notation of [BZ]: if P = MU is a parabolic subgroup of G, and σ is a representation of M, we denote by iU,1(σ) the repre- sentation parabolically induced by σ from P = MU. If P is a standard parabolic subgroup, then iU,1(σ)= iG,M (σ). The following proposition can be proved directly: INDUCTIONANDJACQUETMODULES 5 Proposition 3.3. Let P = MU be a parabolic subgroup of G. Let h : G → G be an automorphism of the topological group G. For smooth representations σ of M and π of G, we have ∼ ih(U),1(h(σ)) = h(iU,1(σ)), ∼ rh(U),1(h(π)) = h(rU,1(π)). (The representation h(π) is given by h(π)(g)= π(h−1(g)). Corollary 3.4. Let P = MU be a parabolic subgroup of G, and w ∈ W . For smooth representations σ of M and π of G, we have ∼ iw(U),1(wσ) = iU,1(σ), ∼ rw(U),1(π) = w(rU,1(π)). Proof of Proposition 3.2. If π is an admissible representation of G, then we have from [C], Corollary 4.2.5. ^ ∼ (**) rU,1(π) = rU −,1(˜π). Let q ֒→ iG,M (σ), q irreducible. Then HomM (rM,G(q),σ) =6 {0}, ^ ^ so HomM (˜σ, rM,G(q)) =6 {0} and HomM (rM,G(q), σ˜) =6 {0}, because ^ rM,G(q) is a direct sum of irreducible representations. Now, using (**), Frobenius reciprocity and Corollary 3.4, we get ^ ∼ {0}= 6 HomM (rM,G(q), σ˜) = HomM (rU −,1(˜q), σ˜) ∼ ∼ = HomG(˜q,iU −,1(˜σ)) = HomG(˜q,iG,M¯ (wσ˜)). 4. Non-Regular case Recall some notation from [C].
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