parabolic induction and jacquet functors for metaplectic groups

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  • Journal of Algebra 323 (2010) 241260

    Contents lists available at ScienceDirect

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    00doJournal of Algebra

    www.elsevier.com/locate/jalgebra

    arabolic induction and Jacquet functors for metaplecticroups

    arcela Hanzer, Goran Muic

    iversity of Zagreb, Department of Mathematics, Bijenicka c. 30, Zagreb, Croatia

    r t i c l e i n f o a b s t r a c t

    ticle history:ceived 2 April 2009ailable online 16 July 2009mmunicated by Dihua Jiang

    ywords:etaplectic groupsrabolic inductionquet functors

    In this paper, we give a new formulation of a geometric lemmafor the metaplectic groups over p-adic eld F , analogous tothe existing one for classical groups. This enables us to givea Zelevinsky type classication of irreducible admissible genuinerepresentations of metaplectic groups. As an application of ourJacquet module technique, we explicitly calculate Jacquet modulesof even and odd Weil representations.

    2009 Elsevier Inc. All rights reserved.

    Introduction

    This paper is the rst in a series where we systematically study the representation theory ofdouble cover Sp(n) of a symplectic group Sp(n) of rank n over a local non-archimedean eld Fcharacteristic different from two. In this paper we establish fundamental results on parabolic in-ction and Jacquet modules relying on general results of BernsteinZelevinsky on the l-group theory,21,2], and general principles established in [9]. In the sequel to this paper [8] we give the descrip-n of rank-one cuspidal reducibility of Sp(n) via theta correspondence. We expect applications in theeory of automorphic forms where metaplectic groups play a prominent role.Now, we explain the content of the paper as well as the results that we establish. One of the niceoperties of classical groups such as Sp(n) is that all Levi subgroups have a particularly nice form:ery proper parabolic subgroup P of Sp(n) has a Levi subgroup M of the form:

    GL(n1, F ) GL(nk, F ) Sp(n), n = n (n1 + + nk). (1)

    Corresponding author.E-mail address: gmuic@math.hr (G. Muic).

    21-8693/$ see front matter 2009 Elsevier Inc. All rights reserved.

    i:10.1016/j.jalgebra.2009.07.001

  • 242 M. Hanzer, G. Muic / Journal of Algebra 323 (2010) 241260

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    bychis is used extensively in studying its representation theory (see for example [19,9]). The group Sp(n)no longer a linear algebraic group, and the notion of parabolic subgroups needs to be introduced.is is done in [11] (see also [13]) as follows. A parabolic subgroup P is the preimage of P undere canonical (surjective) map Sp(n) Sp(n). If we write P = MN for a Levi decomposition, whereis the unipotent radical, then Sp(n) splits over N and we have the following Levi decomposition= MN . The only problem is that the analogue of (1) is no longer true for M . But this does not causerious problems since we have the following epimorphism (see [11]):

    GL(n1, F ) GL(nk, F ) Sp(n) M, (2)

    hose kernel is easy to describe. Here GL(n, F ) is a very simple double cover of GL(n, F ) (see Sec-on 1) whose representation theory is easy to describe in terms of that of GL(n, F ). A Zelevinsky type

    assication for GL(n, F ) is described in Section 4.1 in Propositions 4.1, 4.2, and 4.3.

    Also, although M is not exactly the product GL(n1, F ) GL(nk, F ) Sp(n), it differs fromby a nite subgroup that enables to write every irreducible representation of M in the form

    1 k where all representations 1, . . . ,k, are genuine or not at the same time.is simple property enables us to set-up Tadics machinery [19,9] of parabolic induction and Jacquetnctors. This is obtained in Section 4.2 in Propositions 4.4 and 4.5. The results of Propositions 4.4d 4.5 and later results in the same section depend on a precise form of the composition of parabolicduction functor and Jacquet functor (i.e., the form of a geometric lemma (see Theorem 3.3), alsoction 2.1 of [3]). This ultimately depends on a non-trivial computation of the action of the action ofe preimage of the non-trivial element w0 in NSp(n)(M)/M (P a maximal parabolic subgroup) on M .is computation covers almost all Section 3. Surprisingly this was not done before except usingobal methods but for the special case of the minimal parabolic subgroup recently in a paper byvin and Loke [16]. As we said above, the reward for these tedious computations is a rather precisescription of irreducible representations and intertwining operators for Sp(n). The Zelevinsky typeassication is obtained in Theorem 4.7 using methods of our previous work [9]. The constructiond analytic continuation can be obtained directly from [14] since this approach uses a geometricmma and general principles established by Bernstein (see Remark at the end of 4.2).Finally, in Section 5 we give a sample of the possible applications of our results. In Theorem 5.1 ofction 5 we explicitly calculate Jacquet modules for even and odd Weil representations.

    Preliminaries

    1. Central extensions

    The groups we study are two-sheeted central extensions of linear, orthogonal and symplectic p-ic groups. More precisely, we study the groups of F -points of F -split symplectic, general linear and-(quasi)-split orthogonal groups and their two-sheeted central extensions. In each case we have thellowing central extension (with 2 = {1,1})

    1 2i

    Gp

    G 1 (3)

    here G is any of the groups mentioned above. These extensions will be topological, i.e., the topologyG is such that i and p are continuous, p is open and i is closed. We will recall of the topologyG in the section dealing with the geometric lemma. To conclude, G will be a Hausdorff, l-group.

    om the construction of the topology on G, it will be obvious that there exists an open compactbgroup K of G such that (3) splits on K .We note that Schurs lemma holds in the setting of l-groups (which are countable at innity)[4, Chapter 1, Section 4.2]. So, for an irreducible smooth representation of G, there exists aaracter of 2 (which is a restriction of the central character of to the central subgroup 2).

  • M. Hanzer, G. Muic / Journal of Algebra 323 (2010) 241260 243

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    We use the same realization of the symplectic group as in [10]. So, for every n Z0, letn, ,) be a non-degenerate symplectic vector space of dimension 2n over F . We have a com-

    ete polarization Wn = W n + W n , where {e1, . . . , en} is a basis for W n and {e1, . . . , en} for W n withi, ej = i j . Let Sp(Wn) be the unique two-fold central extension of Sp(Wn), i.e., the following holds

    1 2 Sp(Wn) Sp(Wn) 1.

    e multiplication in Sp(Wn) is given by Raos cocycle cRao [15,10]:

    [g1, 1][g2, 2] =[g1g2, 12cRao(g1, g2)

    ],

    here gi Sp(Wn), i 2, i = 1,2. If G is a subgroup of Sp(Wn), we denote by G its inverseage in Sp(Wn). Now, we follow [11, Section 1.4] very closely. We x a diagonal maximal split torusSp(Wn) (having in mind the basis introduced above) and we x a Borel subgroup stabilizing aaximal isotropic ag consisting of vectors {e1, . . . , en}. Every standard Levi subgroup Ms of Sp(Wn)isomorphic to GL(n1, F ) GL(nk, F ) Sp(Wn|s|), where s = (n1,n2, . . . ,nk) is a sequence ofsitive integers with n1 + + nk = |s| n. There is a natural splitting from the unipotent radicalNs of the corresponding standard parabolic subgroup Ps to its cover [13, Lemma 2.9, p. 43]; let

    s be the image of that homomorphism. We then have P s = MsN s. The subgroups of Sp(Wn) of thisrm we call the parabolic subgroups of Sp(Wn) and this decomposition is their Levi decomposition.We can explicitly describe Ms as follows: There is a natural epimorphism

    : GL(n1, F ) GL(nk, F ) Sp(Wn|s|) Ms

    ven by

    ([g1, 1], . . . , [gk, k], [h, ]) [(g1, g2, . . . , gk,h), 1 . . . k], (4)ith = i< j(det gi,det g j)F (ki=1(det gi, x(h))F ), where x(h) is dened in [15, Lemma 5.1], and,)F denotes the Hilbert symbol of the eld F . We may think of GL(ni, F ) as the two-fold coverGL(ni, F ) with multiplication [g1, 1][g2, 2] = [g1g2, 12(det g1,det g2)F ]. Of course, this agreesith Raos cocycle under an identication of GL(ni, F ) and its diagonal embedding in the Siegel Levibgroup of Sp(Wn) (we will explain (4) more thoroughly in the subsection dealing with the actionthe Weyl group on maximal, non-Siegel parabolic subgroups).We have to calculate Raos cocycle several times, so we write it down explicitly. For g1, g2

    (Wn) the following holds:

    cRao(g1, g2) =(x(g1), x(g2)

    )F

    (x(g1)x(g2), x(g1g2))F (1,det(2q))tF (1,1) t(t1)2F (2q). (5)l the quantities in this expression are explained in the rst chapter of [10]; we will further de-ribe q in Section 3.1.Having in mind the denition of the parabolic subgroups in the two-sheeted metaplectic cover

    (Wn) of the symplectic group (Section 1.3 of [11]), we can choose essentially the same Weyl group

    ere as in the symplectic group [13, Section 4, p. 59].

  • 244 M. Hanzer, G. Muic / Journal of Algebra 323 (2010) 241260

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    This section gives some technical results which enable us, in many instances, to give the sameoof of some representation-theoretic statement for the metaplectic group as for the symplecticoup.

    1. The Siegel case

    Let M be the (standard) Levi subgroup of the Siegel parabolic of Sp(Wn). We want to calculatee actio

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