Tel Aviv University School of Mathematical Sciences The Langlands-Shahidi Method for the metaplectic group and applications Thesis submitted for the degree "Doctor of Philosophy" by Dani Szpruch Prepared under the supervision of Prof. David Soudry Submitted to the senate of Tel-Aviv University October 2009 This PhD thesis was prepared under the supervision of Prof. David Soudry Dedicated with love to the memory of my grandfather Wolf Alpert (1912-2000) and to the memory of my aunt Jolanta Alpert (1942-2008) Acknowledgements I would like to take this opportunity to acknowledge some of the people whose presence helped me significantly through this research. First, and foremost, I would like to thank my advisor Professor David Soudry. Literally, I could not have found a better teacher. I am forever indebted for his continuous support and for his endless patience. I consider myself extremely privileged for being his student. Next, I would like to thank Professor Freydoon Shahidi for his kindness. Over several occasions Professor Shahidi explained to me some of the delicate aspects of his method. In fact, Professor Shahidi motivated some of the directions taken in this dissertation. I would also like to thank Professor James Cogdell for his warm hospitality and for introducing me to the work of Christian A. Zorn which turned out to be quite helpful. I would like to thank Professor William D. Banks for providing me his PhD thesis which contained important information. I would like to thank my fellow participants in the representation theory seminar conducted by Professor Soudry, Professor Asher Ben-Artzi, Ron Erez, Eyal Kaplan and Yacov Tanay, for sharing with me many deep mathematical ideas. I would like to thank my office mate Dr. Lior Bary Sorocker for many hours of conversations. Last, but definitely not least, I would like to thank my wife Norma Musih for her love and support. Contents 0 Introduction 3 0.1 Main results . 4 0.2 An outline of the thesis . 5 1 General notations 11 2 The metaplectic group 12 2.1 The symplectic group . 12 2.2 Rao's cocycle . 14 2.3 Kubota's cocycle . 17 2.4 Some facts about parabolic subgroups of Sp2n(F)............... 20 2.5 The global metaplectic group . 22 2.6 Extension of Rao's cocycle to GSp(X). 22 2.6.1 Computation of v¸(g).......................... 23 3 Weil factor attached to a character of a second degree 27 4 Some facts from the representation theory of Sp2n(F) 34 4.1 Genuine parabolic induction . 34 4.2 An application of Bruhat theory . 36 4.3 The intertwining operator . 37 4.4 The Knapp-Stein dimension theorem . 38 5 Uniqueness of Whittaker model 39 5.1 Non-archimedean case . 40 5.1.1 Proof of Theorem 5.2 . 41 5.1.2 Proof of Theorem 5.4 . 44 6 De¯nition of the local coe±cient and of γ(σ £ ¿; s; Ã) 49 7 Basic properties of γ(σ £ ¿; s; Ã) 52 7.1 Multiplicativity of γ(σ £ ¿; s; Ã)......................... 52 7.2 Computation of γ(σ £ ¿; s; Ã) for principal series representations . 65 8 Computation of the local coe±cients for principal series representations of SL2(F) 66 1 8.1 Non-archimedean case . 67 8.1.1 The local coe±cient expressed as an integral . 68 8.1.2 Some lemmas. 70 R ¡1 ¤ 8.1.3 Computation of F¤ γà (u)Â(s)(u)Ã(u) d u ............... 73 8.2 Real case . 88 8.2.1 Notations and main result . 88 8.2.2 Proof of Lemma 8.12 . 88 8.3 Complex case . 91 8.4 A remark about the chosen parameterization . 92 ¡1 8.5 A remark about the similarity between CÃ( ­ γà ; s) and the Tate γ-factor 93 ¡ ¢ Sp2m(F) 0¡1 8.6 Computation of Cà Pm;0(F); s; ¿; !m for principal series representations 97 9 An analysis of Whittaker coe±cients of an Eisenstein series 99 9.1 Unrami¯ed computations . 99 9.2 Crude functional equation. 103 ¡ ¢ Sp2m(F) 0¡1 9.3 Computation of Cà Pm;0(F); s; ¿; !m for generic representations . 108 10 Irreducibility theorems 110 10.1 Irreducibility of principal series representations of Sp2n(F) induced from uni- tary characters. 110 10.2 Irreducibility criteria for parabolic induction . 114 10.3 A comparison with SO2n+1(F)......................... 116 2 0 Introduction Over a period of about thirty years Freydoon Shahidi has developed the theory of local coe±cients and its applications. Nowadays this method is known as the Langlands-Shahidi method. The references [46], [47], [48], [49], [50], [51], [52], [53], [55] are among Shahidi`s works from the ¯rst half of this period. These works are used in this dissertation. The applications of this theory are numerous; see the surveys [16], [54], [56] and [30] for a partial list. Although this theory addresses quasi-split connected reductive algebraic groups, our aim in this dissertation is to extend this theory to Sp2n(F), the metaplectic double cover of the symplectic group over a p-adic ¯eld F, which is not an algebraic group, and present some applications. We shall realize Sp2n(F) as the set Sp2n(F) £ f§1g equipped with the multiplication law ¡ ¢ (g1; ²1)(g2; ²2) = g1g2; ²1²2c(g1; g2) ; where c(¢; ¢) is Rao's cocycle as presented in [42]. The properties of Sp2n(F) enable the extension of the general representation theory of quasi-split connected reductive algebraic groups as presented in [60] and [67]. A great part of this extension is already available in the literature; see [29], [11], [1], [2] and [72] for example. An analog to Bruhat decomposition holds in Sp2n(F). If F is a p-adic ¯eld, Sp2n(F) is an l-group in the sense of Bernstein and Zelevinsky, [8]. Since Rao's cocycle is continuous, it follows that there exists U, an open compact subgroup of Sp2n(F), such that c(U; U) = 1. Thus, a system of neighborhoods of (I2n; 1) is given by open compact subgroups of the form (V; 1), where V ⊆ U is an open compact subgroup of Sp2n(F). Furthermore, in Sp2n(F) the analogs of the Cartan and Iwasawa decompositions hold as well. If F is not 2-adic then Sp2n(F) splits over the standard maximal compact subgroup of Sp2n(F). Over any local ¯eld (of characteristic di®erent than 2) Sp2n(F) splits over the unipotent subgroups of Sp2n(F). For a subset H of Sp2n(F) we denote by H its pre-image in Sp2n(F). Let P = M nN be a parabolic subgroups of Sp2n(F). P has a "Levi" decomposition. P = M n ¹(N), where ¹ is an embedding of N in Sp2n(F) which commutes with the projection map. During a course by David Soudry (2008-2009) dedicated to Waldspurger's La formule de Plancherel pour les groupes p-adiques (d'apr`esHarish-Chandra) which is a remake of Harish-Chandra's theory as presented in [60], the author checked that the general theorems regarding Jacquet modules, L2-representations, matrix coe±cients, intertwining operators, Harish-Chandra's c-functions etc. extend to the metaplectic group. Same holds for Harish- Chandra's completeness theorem and the Knapp-Stein dimension theorem which follows from this theorem. In fact many of the geometric proofs that are given in [8] and [9] apply word for word to the metaplectic group. We note that many of the properties mentioned in the last paragraph are common to general n-fold covering groups of classical groups. However, the following property is a special feature of Sp2n(F). Let g; h 2 Sp2n(F). If g and h commute then the pre-images in Sp2n(F) also commute. In particular, the inverse image of a commutative subgroup of Sp2n(F) is commutative. This implies that the irreducible representations of TSp2n (F), the inverse image of the maximal torus of Sp2n(F), are one dimensional. As noted in [7] this is the reason that a Whittaker model for principal series representation of Sp2n(F) is unique. 3 Furthermore, in Chapter 5 of this dissertation we prove the uniqueness of Whittaker model for Sp2n(F) in general. We emphasize that this uniqueness does not hold for general covering groups; see [15] and [1]. It is the uniqueness of Whittaker model that enables a straight forward generalization of the de¯nition of the Langlands-Shahidi local coe±cients to the metaplectic group (see [5] for an application of the theory of local coe±cients in the context of non-unique Whittaker model). 0.1 Main results Let F be a local ¯eld of characteristic 0. Let à be a non-trivial character of F. We regard à also as a genuine non-degenerate character of the inverse image of the unipotent radical of a symplectic group. Theorem A. Let ¼ be an irreducible admissible representation of Sp2n(F). Then, the dimension of the space of Ã-Whittaker functionals on ¼ is at most 1; see Chapter 5. Theorem B. Let ¿ be an irreducible admissible generic representation of GLn(F). Let P (F) be the Siegel parabolic subgroup of Sp2n(F) and let w be a particular represen- tative of the long Weyl element of Sp2n(F) modulo the long Weyl element of P (F ). Let ¡ ¢ Sp2n(F) Cà P (F); s; ¿; w be the metaplectic analog to the Langlands-Shahidi local coe±cient. Then, there exists an exponential function cF(s) such that ¡ ¢ 2 Sp2n(F) γ(¿; sym ; 2s; Ã) Cà P (F); s; ¿; w = cF(s) 1 : γ(¿; s + 2 ;Ã) Furthermore, cF(s) = 1 provided that F is a p-adic ¯eld of odd residual characteristic, ¿ is unrami¯ed and à is normalized.