FOURIER COEFFICIENTS FOR THETA REPRESENTATIONS ON COVERS OF GENERAL LINEAR GROUPS

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1 FOURIER COEFFICIENTS FOR THETA REPRESENTATIONS ON COVERS OF GENERAL LINEAR GROUPS YUANQING CAI Abstact. We show that the theta epesentations on cetain coves of geneal linea goups suppot cetain types of unique functionals. The poof involves two types of Fouie coefficients. The fist ae semi-whittake coefficients, which genealize coefficients intoduced by Bump and Ginzbug fo the double cove. The coves fo which these coefficients vanish identically (esp. do not vanish fo some choice of data) ae detemined in full. The second ae the Fouie coefficients associated with geneal unipotent obits. In paticula, we detemine the unipotent obit attached, in the sense of Ginzbug, to the theta epesentations. 1. Intoduction Let F be a numbe field containing a full set of nth oots of unity. Let A be its adele ing. Let GL (A) be a metaplectic n-fold cove of the geneal linea goup. In thei pioneeing wok, Kazhdan-Patteson [23] constucted genealized theta epesentations Θ on GL (A) as multi-esidues of Boel Eisenstein seies. The local theta epesentations wee also constucted as the Langlands quotient of educible pincipal seies epesentations. They showed that (both globally and locally) the genealized theta epesentations ae geneic if and only if n ; and uniqueness of Whittake models holds if and only if n = o +1 (when n = +1, the uniqueness popety only holds fo cetain coves). The theta epesentations and thei unique models have been used to constuct Rankin-Selbeg integals fo symmetic powe L-functions fo the geneal linea goups; see Shimua [31], Gelbat-Jacquet [12], Patteson- Piatetski-Shapio [30], Bump-Ginzbug [4], Bump-Ginzbug-Hoffstein [5], and Takeda [32]. Suppose > n. Motivated by the above backgound, one may ask the following natual questions: (1) Does Θ suppot othe types of Fouie coefficients? (2) If Θ suppots a nonzeo Fouie coefficient, when does the uniqueness popety hold? (3) If the uniqueness popety holds fo cetain types of Fouie coefficients, can we use it to constuct Rankin-Selbeg integals that epesent Eule poducts? All the thee questions have affimative answes and this pape mainly addesses the fist two questions. We fist intoduce a genealization of the Whittake coefficients, which we call semi-whittake coefficients. Let λ = ( 1 k ) be a patition of. Let P λ be the standad paabolic subgoup of GL whose Levi subgoup M = GL 1 GL k. Let U λ be its unipotent adical. Let U be the standad unipotent subgoup of GL. Fix a nontivial Date: Octobe 10, Mathematics Subject Classification. Pimay 11F70; Seconday 11F30, 11F27. Key wods and phases. Theta epesentation, semi-whittake coefficient, Fouie coefficient, unipotent obit, unique functional, metaplectic tenso poduct. 1

2 additive chaacte ψ : F \A C. Let ψ λ : U(F )\U(A) C be the chaacte such that it acts as ψ on the simple positive oot subgoups contained in M, and acts tivially othewise. The λ-semi-whittake coefficient of θ Θ is defined to be θ(ug)ψ λ (u) du. U(F )\U(A) When the patition is λ = (), this ecoves the usual Whittake coefficients. Theoem 1.1. (1) If thee is an i > n, then θ(ug)ψ λ (u) du is zeo fo all choices of data. (2) If i n fo all i, then U(F )\U(A) θ(ug)ψ λ (u) du U(F )\U(A) is nonzeo fo some choice of data. (3) When = mn, i.e. when the ank is a multiple of the degee, and the patition is λ = (n m ), then global uniqueness of λ-semi-whittake models holds. We emak that the local vesion of the above theoem is also established (see Coollay 3.34, 3.36, and Theoem 3.44). Indeed, pats (1) and (3) ae poved by using the local esults, and pat (2) is poved by using a global agument. We also emak that when n = 2 and λ = (2 k ) o (2 k 1) (depending on the paity of ), such coefficients and thei uniqueness popeties wee aleady used in Bump and Ginzbug [4] in thei wok on symmetic squae L-functions fo GL(). The second type of Fouie coefficients we conside is the Fouie coefficients associated with unipotent obits. The unipotent obits of GL ae paameteized by the patitions of via the Jodan decomposition. Given a unipotent obit O, we can associate a set of Fouie coefficients; see Section 5 below. Roughly speaking, stating with a unipotent obit O, we can define a unipotent subgoup U 2 (O). Let ψ U2 (O) : U 2 (O)(F )\U 2 (O)(A) C be a chaacte which is in geneal position. The Fouie coefficient of θ Θ we want to conside is θ(ug)ψ U2 (O)(u) du. U 2 (O)(F )\U 2 (O)(A) When the unipotent obit is O = (), this also ecoves the usual Whittake coefficients. Thee is a patial odeing on the set of unipotent obits. Ou goal is to show that thee is a unique maximal unipotent obit that suppots nonzeo Fouie coefficients of Θ (see Definition 5.1 below). Let O(Θ ) be this obit. The main esults fo the Fouie coefficients associated with unipotent obits ae summaized as follows (Theoem 6.2, 7.4, and 6.11). Theoem 1.2. (1) Wite = an + b such that a Z 0 and 0 b < n. Then both locally and globally O(Θ ) = (n a b). 2

3 (2) Let v be a finite place such that n v = 1 and Θ,v is unamified. If = mn and O = (n m ), then dim Hom U2 (O)(F v)(θ,v, ψ U2 (O),v) = 1. This unique model is valuable and it aleady finds applications in Rankin-Selbeg integals fo coveing goups. In the eseach announcement by Fiedbeg, Ginzbug, Kaplan and the autho [6], the notion of Whittake-Speh-Shalika epesentation was intoduced (see Definition 7.5). Such epesentations ae ieducible automophic epesentations on GL (A) and they possess unique functionals. The Whittake-Speh-Shalika epesentations and thei uniqueness models ae used in the genealization of the doubling methods to coveing goups. The theta epesentations ae examples of such epesentations. Theoem 1.3 (Theoem 7.6). When = mn, Θ is a Whittake-Speh-Shalika epesentation of type (n, m). This unique functional also plays a ole in a new-way integal (Eule poducts with nonunique models) fo coveing goups; see Ginzbug [17]. We now descibe the ideas of the poofs. The poof of Theoem 1.1 is based on an induction in stages statement. We descibe it in the global setup. Such an agument was also used in Bump-Fiedbeg-Ginzbug [3] whee they studied the Fouie coefficients of theta epesentations on the double coves of odd othogonal goups. Fist of all, we can ewite the λ-semi-whittake coefficients as θ(ug)ψ λ (u) du = θ(vug) dv ψ λ (u) du. U(F )\U(A) U M(F )\U M(A) U λ (F )\U λ (A) The inne integal is actually a constant tem of the theta function. To compute it, we compute the constant tem of the Eisenstein seies and use the fact that the multi-esidue opeato and the constant tem opeato commute. By the standad unfolding agument, the constant tem of the Eisenstein seies is a sum of Eisenstein seies on M(A). Afte applying the multi-esidue opeato, only one tem suvives. This implies that the constant tem of a theta function is actually a theta function on M(A). This fact is also called peiodicity in [23] and [4]. Now we ae facing a difficulty which did not appea in [3]. In the double cove of the odd othogonal case, the constant tems of theta functions give ise to a epesentation on the cove of the Levi subgoup. In that case, diffeent blocks commute in M(A). Thus, one can take theta epesentations on each block and fom the tenso poduct. It is shown that the tenso poduct of theta epesentations on each block is the same as the theta epesentation on M(A). We would like to seek an analogous esult fo the geneal linea goup. Howeve, in the geneal linea case, when we estict the metaplectic cove to M, the blocks neve commute (except when n = 2). In fact, thee is even no natual map between M and GL 1 GL k. This means that, stating with epesentations on the GL i, thee is no diect way to constuct a epesentation of M. To ovecome this difficulty, a constuction called the metaplectic tenso poduct has been intoduced (see Section 3.4 and 4.4). The local vesion is developed in Mezo [25] and the global vesion is given in Takeda [33, 34]. Roughly speaking, the constuction goes as follows (both locally and globally). Let GL i be the subgoup of GL i, consisting of those elements 3

4 whose deteminants ae nth powes. Let M be the subgoup of M consisting of those elements such that the deteminants of all the blocks ae nth powes. The GL i s commute in M, and M is the diect poduct of GL 1,, GL k with amalgamated µ n. Now stat with epesentations π i on GL i. We fist estict π i to GL i, and pick an ieducible constituent π i. Then we take the tenso poduct π 1 π k. This is a epesentation of M. We then use induction to obtain a epesentation of M. Exta cae must be taken in ode to establish the well-definedness and ieducibility of such constuctions. Theoem 1.4 (Rough fom). Both locally and globally, Θ M = Θ1 Θ k. The local vesion is given in Theoem 3.28, and the global vesion is Theoem 4.3. Once we have the induction in stages statement, Theoem 1.1 can be established by caefully analyzing the estiction and induction pocess. In the local setup, we give an explicit fomula fo the dimension of the twisted Jacquet module J U,ψλ (Θ ). Theoem 1.2 is poved in Sections 6 and 7. The poof consists of two pats. The fist pat is to show that any unipotent obit geate than o not compaable to (n a b) does not suppot any Fouie coefficients. The second pat is to show that (n a b) actually suppots a nonzeo Fouie coefficient. The idea is to build a elation between the semi-whittake coefficients and the Fouie coefficients associated with unipotent obits. Once we know enough infomation about the semi-whittake coefficients, the unipotent obit attached to the epesentation can be detemined. Two tools play cucial oles in the poof. The fist one is called oot exchange. This allows us to eplace the domain of integation with a slightly diffeent one. The second one is the Fouie expansion. This allows us to enlage the domain of integation if we know cetain coefficients vanish (this is usually elated to the vanishing of semi-whittake coefficients). When we combine these tools in a systematic way, vanishing and nonvanishing of Fouie coefficients associated with unipotent obits can be elated to the esults on the semi-whittake coefficients. Futhemoe, when n and b have the same paity, we actually establish an identity between these coefficients. In paticula, Theoem 1.2 pat (2) follows fom Theoem 1.1 pat (3). The emainde of this pape is oganized as follows. Section 2.1 intoduces notations and defines metaplectic coves of geneal linea goups. Cetain issues such as centes and maximal abelian subgoups ae also discussed. The local theoy of semi-whittake functionals is developed in Section 3. We fist eview the pincipal seies epesentations and theta epesentations of GL (F v ). In Section 3.2, we give an explicit desciption of the estiction of these epesentations to GL (F v ). These esults ae used to povide examples of the metaplectic tenso poduct in Section 3.5. We then caefully analyze the constuction and compute the dimensions of some twisted Jacquet modules in Section 3.6. Section 4 is devoted to the global theoy. The nonvanishing pat of Theoem 1.1 is poved in Theoem 4.4. In Section 5, we eview the association of Fouie coefficients to a unipotent obit. The unipotent obit attached to the theta epesentations is detemined in Sections 6 and 7. Section 6 intoduces the local agument. The elation between semi-whittake coefficients 4

5 and Fouie coefficients associated with unipotent obit is established in a seies of lemmas. Section 7 descibes the coesponding global pictue. Acknowledgement. The wok in this pape foms pat of my Boston College PhD thesis. I heatily thank my adviso, Solomon Fiedbeg, fo his guidance and suppot. I am also indebted to David Ginzbug fo many helpful discussions and comments. I would also like to thank the efeee fo caeful eading and vey detailed and helpful comments fo an ealie vesion of the pape. This wok was suppoted by the National Science Foundation, gant numbe Notations and Peliminaies 2.1. Notations. Fix a positive intege n and let µ n (F ) = {x F : x n = 1} be the goup of nth oots of unity in a field F. In this pape we always assume µ n (F ) = n. Fix, once and fo all, an embedding ɛ : µ n C. We always wite µ n fo µ n (F ), if thee is no confusion. We often invoke the convention of omitting ɛ fom the notation. All epesentations which we conside ae epesentations whee µ n acts by scalas by the embedding ɛ. Such epesentations ae called genuine. If F is a non-achimedean local field, we denote by o the ing of integes of F. Let val be the nomalized valuation on F. Let F be the nomalized absolute value on F. Let (, ) = (, ) F,n : F F µ n (F ) be the nth ode Hilbet symbol. It is a bilinea fom on F that defines a nondegeneate bilinea fom on F /F n and satisfies (x, x) = (x, y)(y, x) = 1, x, y F. When F is a numbe field, and v is a place of F, we denote by F v the completion of F at v. When v is non-achimedean, we let o v be the ing of integes of F v. Fo GL, let B = T U be the standad Boel subgoup with unipotent adical U and maximal tous T. The set Φ = {(i, j) : 1 i j } is identified with the set of oots of GL in the usual way. Let Φ + denote the set of positive oots with espect to B. Fo a patition λ = ( 1 k ) of, let P λ be the standad paabolic subgoup of GL whose Levi pat M λ is GL 1 GL k embedded diagonally (g 1,, g k ) diag(g 1,, g k ), g i GL i, and let U λ denote the unipotent adical of P λ. We usually wite M fo M λ when the patition is fixed. We usually wite m M by m = diag(g 1,, g k ) with g i GL i. Let Φ λ and Φ + λ denote the set of oots and positive oots contained in M λ, espectively. We also wite B M = B M and U M = U M. We sometimes add subscipt M to indicate the ambient goup. Fo example, T M (which is T ) is viewed as a maximal tous of M. We might still use T fo T M when thee is no confusion. Let W be the set of all pemutation matices. The Weyl goup of GL is identified with W. We also identify W as the goup of pemutations of {1, 2,, } via w = (δ i,w(j) ). 5

6 The action of W on Φ is given by w(i, j) = (w(i), w(j)). Fo a Levi subgoup M λ, let W (M λ ) be the subset of pemutation matices contained in M λ. The goup W (M λ ) is identified with the Weyl goup of M λ (as sets). Let w M λ =... I 2 I 1 W. I k The element w M λ sends GLk GL 1 to M λ. Fo any goup G and elements g, h G, we define g h = ghg 1. Fo a subgoup H G and a epesentation π of H, let g π be the epesentation of g H defined by g π(h ) = π(g 1 h g) fo h g H. Let F be a local field. Let ψ be a nontivial additive chaacte on F. In this pape we need to conside seveal chaactes on vaious subgoups of U. We make the following convention. Fo a patition (p 1 p k ) of, let = {i : 1 i }\{p 1, p 1 +p 2,, p 1 + +p k }. Let V be a subgoup of U such that V contains all the oot subgoups associated to α = (i, i+1) fo i. Let ψ (p1 p k ) : V C be a chaacte such that ψ (p1 p k ) acts as ψ on the oot subgoups associated to α = (i, i + 1) fo i, and acts tivially othewise. Thus ψ () and ψ (1 ) ae the usual Whittake chaacte and the tivial chaacte on U, espectively. When F is a numbe field and ψ is a nontivial additive chaacte of F \A, these chaactes can be defined analogously. Let F be a non-achimedean field. Let ψ V be a chaacte on a unipotent subgoup V of U. We use J V to denote the Jacquet functo with espect to V. The functo J V,ψV is the twisted Jacquet functo with espect to (V, ψ V ). Thoughout the pape, the induction Ind is not nomalized The local metaplectic cove GL (F ). Let F be a local field of chaacteistic 0 that contains all the nth oots of unity. Associated to evey 2-cocycle σ : GL (F ) GL (F ) µ n (F ), thee is a cental extension GL (F ) of GL (F ) by µ n satisfying an exact sequence 1 µ n ι GL (F ) p GL (F ) 1. We call GL (F ) a metaplectic n-fold cove of GL (F ). As a set, we can ealize GL (F ) as GL (F ) = GL (F ) µ n = {(g, ζ) : g GL (F ), ζ µ n }. Notice that GL (F ) is not the F -ational points of an algebaic goup, but this notation is standad. We use GL to denote GL (F ). This abuse of notation is widely used in this pape, especially in the local setup. The embedding ι and the pojection p ae given by ι(ζ) = (I, ζ) and p(g, ζ) = g whee I is the identity element of GL. The multiplication is defined in tems of σ as follows, Fo any subset X GL, let (g 1, ζ 1 ) (g 2, ζ 2 ) = (g 1 g 2, ζ 1 ζ 2 σ(g 1, g 2 )). X = p 1 (X) GL. 6

7 We also fix the section s : GL GL of p given by s(g) = (g, 1). Then fo g 1, g 2 GL, s(g 1 )s(g 2 ) = (g 1 g 2, σ(g 1, g 2 )). In [23], Kazhdan-Patteson povided 2-cocycles σ (c) paameteized by c Z/nZ. They ae elated by σ (c) (g 1, g 2 ) = σ (0) (g 1, g 2 )(det g 1, det g 2 ) c, g 1, g 2 GL. (1) In this pape, we use the 2-cocyles constucted in Banks-Levy-Sepanski [1]. The 2-cocycles in [1] satisfy a block compatibility popety. Let σ (0) = σ (0), and σ (c) = σ (c) be elated to σ (0) by Eq. (1). Block compatibility means the following. If = k and g i, g i GL i fo i = 1,, k, then σ (c) (diag(g 1,, g k ),diag(g 1,, g k)) [ k ] [ ] = σ (c) i (g i, g i) (det g i, det g j) c+1 (det g j, det g i) c. i=1 Thoughout the pape we fix the positive integes and n and the modulus class c Z/nZ and let σ = σ (c). Note that the estiction of σ to T is given by [ ] σ(t, t ) = (t i, t j) (t i, t j) c i<j fo t = diag(t 1,, t ) and t = diag(t 1,, t ). The goup U splits in GL. In fact s U is an embedding of U in GL ([24] Poposition 2). Let K = GL (o). When n F = 1, K also splits in GL ([24] Theoem 2). Thee is a map κ : K µ n such that g κ (g) = (g, κ(g)) is a goup homomophism fom K to GL. We denote its image by K. We shall fix κ such that κ is what Kazhdan-Patteson efe to as the canonical lift of K to GL. It is chaacteized by the popety that s T K = κ T K, s W = κ W, and s U K = κ U K. ([23] Poposition 0.1.3). The topology of GL as a locally compact goup is detemined by this embedding Centes. The following lemma is Takeda [33] Lemma 3.13, which is vey useful fo us. Lemma 2.1. Let F be a local field. Then fo each g GL and a F, i<j σ (g, ai )σ (ai, g) 1 = (det(g), a 1+2c ). Lemma 2.2. Let n 1 = gcd(n, 2c + 1), and n 2 = n n 1. Then the cente of GL is i,j Z GL ={(zi, ζ) : z 2c+ 1 F n } ={(zi, ζ) : z F n 2 }. The fist pat is poved in [23] Poposition 0.1.1, and the second pat is poved in Chinta- Offen [7] Lemma 1. The cente of T is also detemined in [23]. Let T n = {(t n, ζ) : t T }. 7

8 Lemma 2.3. The cente of T is Z GL T n. GL Let := {g GL : det g F n }. We ae inteested in this goup since it contols the epesentation theoy of GL. Moeove, it plays a ole in developing tenso poducts and paabolic inductions fo metaplectic goups; see Section 3.4. Let T := centes of GL and T behave bette than the centes of GL and T. Lemma 2.4. The cente of Z GL GL = Z is GL ={(ai, ζ) : a F n } ={(ai, ζ) : a F n gcd(n,) }. GL T. The Poof. The fist equality is immediate fom Lemma 2.1. Fo the second equality, the poof is exactly the same as in [7] Lemma 1. The poof of the following lemma is also staightfowad. Lemma 2.5. The cente of T is Z GL T n Maximal abelian goups. Maximal abelian subgoups of T play an impotant ole in the epesentation theoy of T. Let T be a maximal abelian subgoup T. In Section , we assume that T T is a maximal abelian subgoup of T, unless othewise specified. We biefly explain why such a goup exists. Fist we stat with a maximal abelian subgoup T of T. If T is not a maximal abelian subgoup of T, then we can choose x T T such that the goup geneated by T and x is abelian. Notice T / T is finite. Thus we can epeat this pocess until we obtain a maximal abelian subgoup of T. A concete constuction of maximal abelian goups is given in Section The global metaplectic cove GL (A). Let F be a numbe field that contains all the nth oots of unity and A be the ing of adeles. To constuct a metaplectic n-fold cove of GL (A) of GL (A), we follow [33] Section 2.2. The adelic 2-cocycle τ is defined by τ (g, g ) = τ,v (g v, g v), v fo g, g GL (A). Hee, the local cocycle is obtained fom the block-compatible cocycle, multiplied by a suitable cobounday. It can be shown that thee is a section s : GL (F ) GL (A) such that GL (F ) splits in GL (A). The cente Z GL(A) of GL (A) can be easily found by using the local esults. As in the local case, we define GL (A); = {g GL (A) : det g A n }. The goup GL (A) can also be descibed as a quotient of a esticted diect poduct of the goups GL (F v ). Fist we conside the esticted diect poduct GL v (F v ) with espect to K v fo all v with v n and v. Denote each element in this esticted diect poduct by v (g v, ζ v ) so that g v K v and ζ v = 1 fo almost all v. Then ρ : GL (F v ) GL (A), (g v, ζ v ) ( g v, ζ v ) (2) v v v v 8

9 is sujective goup homomophism. (Notice that v ζ v is a finite poduct.) We have GL (F v )/keρ GL (A). v Thus we have the notions of automophic epesentations and automophic foms on GL (A). We now explain how to wite an ieducible automophic epesentation π on GL (A) as the metaplectic esticted tenso poduct v π v in the sense of [34] Section 2. Fist of all, we view π as a epesentation on the esticted diect poduct GL v (F v ) by pulling it back by ρ in Eq. (2). By the usual tenso poduct theoem fo the esticted tenso poduct, we obtain π ρ vπ v, whee each π v is genuine. We call π v the ieducible constituent of π at v. Fo almost all v, π v is K v -spheical. The epesentation vπ v descends to a epesentation GL (A). Thus we wite π v π v. Notice that the space of v π v is the same as v π v Metaplectic cove of Levi subgoups. Let λ = ( 1 k ) be a patition of. Let M := M λ be the Levi subgoup of GL descibed in Section 2.1. This section discusses metaplectic coves M, both locally and globally. The 2-cocycle τ does not satisfy blockcompatibility. To get ound it, an equivalent cocycle τ M was intoduced in [33] Section 3. We use this cocycle to define M. Notice that the blocks GL i do not commute with each othe. Let R = F if F is local and R = A is F is global. Define M (R) = {(diag(g 1,, g k ), ζ) : det g i R n }. Let T M be the maximal tous consisting of diagonal matices. We wite T i = T GL i, whee GL i is embedded in GL via g diag(i 1,, g,, I k ). The tous T i can be viewed as a maximal tous of GL i. Define T M following esults ae poved in [33] Section 3. We omit the details. = T M M. The Lemma 2.6. The cente of M(R) is 1 I 1 Z M(R) = a... : a 1+2c i R n and a 1 a k mod R n. a k I k Remak 2.7. Notice that Z GL T n n = Z M T M and Z GL M = Z M M. Lemma 2.8. The cente of M is 1 I 1 Z M = a... Lemma 2.9. The cente of T M is Z M T n., ζ : a i i R n. a k I k 9

10 Let T M, be a maximal abelian subgoup of T M. We again assume T M, T M is a maximal abelian subgoup of T M. One can also talk about automophic foms and epesentations on M(A) as well. 3. Local Theoy In this section, F is a non-achimedean local field. Recall that we use GL to denote GL (F ) The pincipal seies epesentations. The pincipal seies epesentations of GL wee studied in [23]. Fo the genealization to metaplectic coves of othe eductive goups, see McNamaa [24]. We stat with the epesentation theoy of T. In geneal, T is not abelian, but it is a two-step nilpotent goup. The ieducible genuine epesentations of T ae paameteized in the following way ([24] Theoem 3): stat with a genuine chaacte χ on the cente of T and extend it to a chaacte χ on any maximal abelian subgoup T, then the induced epesentation i(χ ) := Ind T T χ is ieducible (see [24] Theoem 3). This constuction is independent of the choice of T and of the extension of chaactes. We extend i(χ ) to a epesentation i B(χ ) on B = T U by letting U act tivially. Let δ B be the modula quasichaacte of B. Then Ind GL B i B(χ )δ 1/2 B is the pincipal seies epesentation. This epesentation is denoted by I(χ ), although its isomophism class only depends on χ. Thee is an altenative way to descibe the pincipal seies epesentations. We can extend the chaacte χ to B = T U, and then induce it to GL. The epesentation Ind GL χ B δ 1/2 B is isomophic to I(χ ). The epesentation I(χ ) is ieducible when χ is in geneal position. Fo a positive oot α, thee is an embedding i α : SL 2 GL. Define χ n α(t) = χ Theoem 3.1. Suppose that χ n α ±1 F (i α ( t t 1 ) n ). fo all the positive oots α. Then I(χ ) is ieducible. This is poved by the theoy of intetwining opeatos; see [23] Coollay I.2.8. If χ n α = F fo all the positive simple oots α, we call χ exceptional. In this case, I(χ ) is educible, and we ae inteested in the unique ieducible subquotient of I(χ ). Recall that the intetwining opeato T w : I(χ ) I( w χ ) is defined as (T w f)(g) = f(w 1 ug)du. U(w) whee U(w) is the subgoup of U coesponding to oots α > 0 such that w 1 α < 0. If this conveges fo all f I(χ ) and is non-tivial, then it is a geneato of Hom GL (I(χ ), I( w χ )). Fo geneal χ, the intetwining opeato can be defined via analytic continuation. Theoem 3.2. Let χ be exceptional. Let Θ(χ ) = Im(T w0 : I(χ ) I( w 0 χ )), 10

11 whee w 0 is the longest element of W. Then (1) Θ(χ ) is the unique ieducible subepesentation of I( w 0 χ ). (2) Θ(χ ) is the unique ieducible quotient epesentation of I(χ ). (3) The Jacquet module J U (Θ(χ )) = Ind T w 0 ( w 0 χ δ 1/2 T B ). This is [23] Theoem I.2.9. Θ(χ ) is called exceptional. The Whittake models of exceptional epesentations ae studied in [23] Section I.3. These authos have shown the following esults. Poposition 3.3. Suppose that n F = 1. (1) The epesentation Θ(χ ) has a unique Whittake model if and only if n =, o othewise n = + 1, and 2(c + 1) 0 mod n. (2) The epesentation Θ(χ ) does not have a Whittake model if n 1. (3) The epesentation Θ(χ ) has a finite numbe of independent nonzeo Whittake models if n + 1. Remak 3.4. In the above poposition, pats (1) and (3) ae also tue when n F 1. This is shown in [23] Section II by using global aguments. Pat (2) is expected to be tue when n F 1, but this is known only when n = 2; see Kaplan [22] Theoem 2.6 and Flicke- Kazhdan-Savin [9]. Remak 3.5. When = 1, we take Θ(χ ) to be Ind T T χ. This fits into the metaplectic tenso poduct pefectly Restictions. We study the estiction functo Res GL in this section. We obtain an GL explicit desciption of the estiction of the pincipal seies epesentations and exceptional epesentations fom GL to GL. This is useful in Section 3.5 whee we give explicit examples of the metaplectic tenso poduct. Notice that GL is an open nomal subgoup of GL, and GL / GL = F /F n is finite and abelian. By Gelbat-Knapp [13] Lemma 2.1, if I(χ ) is ieducible, and π is an ieducible constituent of I(χ ) GL, then I(χ ) GL = g m g π. The multiplicities m depend only on I(χ ), and the diect sum is ove cetain elements of GL. Fom now on, we assume T := T T is a maximal abelian subgoup of T. Let B = T U and B = T U. Poposition 3.6. I(χ ) GL GL = Ind ( x χ δ 1/2 x B B ) x B. (3) x 1 T \ T / T Poof. This follows fom Benstein-Zelevinsky [2] Theoem 5.2. We ae woking with epesentations of GL. Let us choose tiples B, T, U with tivial chaacte on U on the induced 11

12 GL GL, {1} with tivial chaacte on {1} on the Jacquet functo side. functo side, and, The Jacquet functo in this case is the estiction functo. GL. The esulting functo is glued by functos indexed by the double coset space B\ GL / This double coset space is a singleton since T GL = GL. Theefoe, the functo is the composition of the induction functo fom T Res T T. GL to GL and the estiction functo By [13] Lemma 2.1, Ind T T χ T is a diect sum of ieducible T -epesentations. On the othe hand, it has a Jodan-Holde seies whose composition factos ae Ind T x T x χ, x 1 T \ T / T. Notice that T is also a Heisenbeg goup and T x T = x ( T ) is again a maximal abelian subgoup of T. This implies Ind T x T x χ is ieducible. Thus, (Ind T T χ ) T = Ind T x T x χ. x 1 T \ T / T Now the poposition follows. Remak 3.7. Notice that Eq. (3) depends on the choice of maximal abelian subgoup. Indeed, when χ is in geneal position, the condition that T T = T implies each component is ieducible. Without this condition, we get a simila decomposition, but the components ae educible. Next we show that, when χ is in geneal position, the components in Poposition 3.6 GL ae ieducible. Let us wite V ( x χ ) = Ind ( x χ δ 1/2 x B B ) x B fo x 1 T \ T / T. Thus Poposition 3.6 becomes I(χ ) GL = V ( x χ ). x 1 T \ T / T Definition 3.8. A chaacte of Z GL T n w W, w I. Lemma 3.9. o Z GL T n is called egula if w χ χ fo all (1) The T -module J U (V (χ )) has a Jodan-Holde seies whose composition factos ae Ind T w T ( w χ δ 1/2 ) (w W ). (2) If χ is egula, then fo any extension χ, χ T Z GL n J U (V (χ )) = w W B Ind T w T ( w χ δ 1/2 B ). is egula. Moeove, Poof. The fist pat follows fom [2] Theoem 5.2. Fo the second pat, we only need to show that χ T is egula. Indeed, if χ is egula, then fo any w W, thee exists Z GL n 12

13 x Z GL T n such that χ(w 1 xw) χ(x). Without loss of geneality, we may assume x T n. This implies that χ T is egula fo any extension χ of χ. Z GL n Lemma Let χ 1, χ 2 be two quasichaactes of Z GL T. Suppose χ 1 is egula. Then dim Hom GL (V (χ 1), V (χ 2)) 1. The equality holds if and only if χ 2 = w χ 1 fo some w W. T n and let χ 1, χ 2 be extensions to Poof. This is an immediate application of Lemma 3.9, Fobenuis ecipocity, and the fact that Ind T w T ( w χ δ 1/2 ) is ieducible. B Lemma The estiction of the intetwining opeato T w : I(χ ) I( w χ ) to Eq. (3) gives T w : V ( x χ ) V ( wx χ ). Poof. Recall that I(χ ) is the space of smooth functions I(χ ) = {f : GL C : f is smooth and f(bg) = χ (b)δ B (b) 1/2 f(g) fo b B }. The embedding of V ( x χ ) into I(χ ) is given as follows. Let f V ( x χ ). Define { f(xg) if x GL f(g) =, 0 othewise. Then it is staightfowad to check that f I(χ ). Now given f V ( x χ ). We can see that T w ( f) is in the image of V ( wx χ ) in I( w χ ). Poposition If χ n α ±1 fo all positive oots α, then V (χ ) is ieducible. Poof. Unde the assumption, T w : I(χ ) I( w χ ) is an isomophism, and hence its estiction T w : V (χ ) V ( w χ ) is again an isomophism. ieducible. Aguing as in [23] Coollay I.2.8, we can show that V (χ ) is Similaly we can deduce esults fo exceptional epesentations. Theoem Let χ be exceptional. Let whee w 0 is the longest elements of W. Then V 0 (χ ) = Im(T w0 : V (χ ) V ( w 0 χ )), (1) V 0 (χ ) is the unique ieducible subepesentation of V ( w 0 χ ). (2) V 0 (χ ) is the unique ieducible quotient epesentation of V (χ ). (3) J U (V 0 (χ )) = Ind T w 0 T ( w 0 χ δ 1/2 B ). 13

14 Poof. The map T w0 : I(χ ) I( w 0 χ ) esticts to T w0 : V ( x χ ) V ( w0x χ ). x 1 T \ T / T x 1 T \ T / T This implies that Θ(χ ) GL = V 0 ( x χ ). x 1 T \ T / T We fist show pat (3). Fom the exactness of the Jacquet functo, J U (V 0 (χ )) is a subepesentation of both J U (V (χ )) and J U (Θ(χ )). Theefoe, J U (V 0 (χ )) = Ind T w 0 T ( w 0 χ δ 1/2 B ). The epesentation Θ(χ ) GL is a diect sum of ieducible constituents, which ae conjugate to each othe. Thus V 0 (χ ) is a diect sum of some of these components. This implies that J U (V 0 (χ )) is also a diect sum of the coesponding Jacquet modules which ae conjugate to each othe. Thus V 0 (χ ) is ieducible since J U (V 0 (χ )) is ieducible. If π is anothe ieducible quotient epesentation of V (χ ), then its Jacquet module is a quotient of J U (V (χ )), and hence thee is a nonzeo homomophism J U (π) Ind T w T ( w χ δ 1/2 B ) fo some w W. By Fobenius ecipocity, thee is a nonzeo intetwining map π V ( w χ ). The composition V (χ ) π V ( w χ ) is nonzeo and it must be a constant multiple of T w. Theefoe, the composition V (χ ) π V ( w χ ) T wow 1 V ( w 0 χ ) is T w0 and its image is V 0 (χ ). We see that V 0 (χ ) is a quotient of π, and since π is ieducible, they must be the same. This poves pat (2). Pat (1) follows fom pat (2) by duality. As a coollay, we descibe the decomposition of Θ(χ ) when esticted to Coollay Θ(χ ) GL = x 1 T \ T / T V 0 ( x χ ). GL Pincipal seies of Levi subgoups. Let λ be a patition of and wite M fo M λ. Recall B M = B M, and U M = U M. The pincipal seies epesentations and exceptional epesentations can be similaly defined on M. Recall we may identify GL i as a subgoup of M via the embedding g i diag(i 1,, g i,, I k ). Let B i be the standad Boel subgoup of GL i and δ Bi be the modula quasichaacte of B i in GL i. n Let χ be a genuine chaacte of Z GL T M and χ be a chaacte of T M, extending χ. The genuine epesentation π TM (χ ) := Ind T M T M, χ is ieducible. The pincipal seies epesentation I(χ ) is the induced epesentation Ind M BM π T (χ ) δ 1/2 M, whee δ M = δ B1 δ Bk. Thee is an altenative way to descibe it as in the geneal linea case. 14

15 The theoy of intetwining opeatos applies just as the geneal linea case. Theefoe, I(χ ) is ieducible when χ is in geneal position. Theoem Suppose that χ n α ±1 F ieducible. fo all the positive oots α in M. Then I(χ ) is If χ n α = F fo all positive simple oots α in M, we call it exceptional. Theoem Let χ be exceptional. Let Θ(χ ) = Im(T wm,0 : I(χ ) I( w M,0 χ )), whee w M,0 is the longest element of W (M). Then (1) Θ(χ ) is the unique ieducible subepesentation of I( w M,0 χ ). (2) Θ(χ ) is the unique ieducible quotient epesentation of I(χ ). (3) J UM (Θ(χ )) = Ind T M ( w w M,0 TM, w 1 M,0 χ δ 1/2 M ). M,0 We also want to study I(χ ) Z GL M, and Θ(χ ) Z GL M. The aguments in Section 3.2 apply in this case without essential change. We only state the esults hee. Poposition I(χ ) Z GL M = x 1 T M, \ T M /Z GL T M Ind Z GL M x (Z GL M B M, ) Poposition If χ n α ±1 fo all positive oots α in M, then is ieducible. Ind Z GL M x x M (Z GL B χ δ 1/2 M, ) M As in the geneal linea case, wite V ( x χ ) = Ind Z GL M Poposition Let χ be exceptional. Let x (Z GL M B M, ) V 0 (χ ) = Im(T wm,0 : V (χ ) V ( w M,0 χ )), whee w M,0 is the longest elements of W (M). Then (1) V 0 (χ ) is the unique ieducible subepesentation of V ( w M,0 χ ). (2) V 0 (χ ) is the unique ieducible quotient epesentation of V (χ ). (3) J UM (V 0 (χ )) = Ind Z GL T M Poposition Z GL w M,0 T M, w 1 M,0 Θ(χ ) Z GL M = ( w M,0 χ δ 1/2 M ). x 1 T M, \ T M /Z GL T M V 0 ( x χ ). x χ δ 1/2 M. x χ δ 1/2 M. Lastly, let χ be an exceptional chaacte fo GL. Let P be the paabolic subgoup of GL with Levi subgoup M, and R be its unipotent adical. Let δ P be the modula quasichaacte of GL with espect to P. Recall we have δ M δ P = δ GL and w 0 = w M,0 w M. 15

16 Poposition The chaacte wm χ δ 1/2 P is exceptional fo M, and J R (Θ GL (χ )) = Θ M( wm χ δ 1/2 P ). Poof. The Weyl element w M pemutes blocks of M, and thus the chaacte wm χ δ 1/2 P is exceptional fo M. To pove the isomophism of Jacquet modules, we apply J UM ( ) on both sides. The left-hand side is while the ight-hand side is J UM (J R (Θ GL (χ ))) = J U (Θ GL (χ )) = Ind T w 0 T ( w 0 χ δ 1/2 GL ); J UM (Θ M( wm χ δ 1/2 P )) w = Ind 0 wm,0 T w χ δ 1/2 1 P δ1/2 M = Ind T w M,0 0 ( w 0 χ δ 1/2 T GL ). This implies that J R (Θ GL (χ )) and Θ M( wm χ δ 1/2 P ) ae both ieducible subepesentations of I( wm χ δ 1/2 P ). Thus they ae isomophic The metaplectic tenso poduct. One of the basic constuctions in the epesentation theoy of GL (F ) is paabolic induction. Let = k be a patition of, and let M = GL 1 GL k be a Levi subgoup. We stat with a list of epesentations, one fo each of GL 1,, GL n, and then fom thei tenso poduct to obtain a epesentation of M. Howeve, since M is not simply the amalgamated diect poduct of the vaious GL i, this constuction cannot be genealized diectly to the metaplectic case. Fotunately, we have a eplacement, which is defined in Mezo [25]. We eview the constuction in this section. The two-fold cove case was outlined in Bump and Ginzbug [4], and studied in full detail in Kable [21]. Fo the global setup and futhe popeties see Takeda [33, 34]. We obseve that any element m M may be witten as diag(g 1,, g k ), such that p(g i ) GL i fo 1 i k. Recall M = {m M : det g 1,, det g k F n } and GL i = M GL i. Let π 1,, π k be ieducible genuine epesentations of GL 1,, GL k, espectively. The constuction of the metaplectic tenso poduct takes seveal steps. Fist of all, fo each i, fix an ieducible constituent π i of the estiction π i GL of π i i to GL i. Then we have π i GL i = g m i g (π i ), whee g uns though a finite subset of GL i, m i is a positive multiplicity and g (π i ) is the epesentation twisted by g. Then we constuct the tenso poduct epesentation π 1 π k 16

17 of the goup GL 1 GL k. Because the epesentations π 1,, π k ae genuine, this tenso poduct epesentation descends to a epesentation of the goup M, i.e. the epesentation factos though the natual sujection GL 1 We denote this epesentation of M by GL k M. π := π 1 π k, and call it the metaplectic tenso poduct of π 1,, π k. Let ω be a chaacte on the cente Z GL such that fo all (ai, ζ) Z GL() M whee a F, we have ω(ai, ζ) = π (ai, ζ) = ζπ 1 (ai 1, 1) π (ai k, 1). Namely, ω agees with π on the intesection Z GL M. We can extend π to the epesentation := ωπ π ω of Z GL M by letting Z GL act by ω. The last step is cucial. If we induce π ω to M, the esulting epesentation is usually To get an ieducible epesentation, we extend the epesentation π ω to a educible. epesentation ρ ω of a subgoup H of M so that ρ ω satisfies Mackey s ieducibility citeion and the induced epesentation π ω := ρ Ind M H ω is ieducible. It is always possible to find such H and moeove H can be chosen to be nomal. The constuction of π ω is independent of the choices of π i, H and ρ ω, and it only depends on ω (see [25] Section 4). We wite π ω = (π 1 π k ) ω and call it the metaplectic tenso poduct of π 1,, π k with the chaacte ω. The metaplectic tenso poduct π ω is unique up to twist. Poposition 3.22 ([25] Lemma 5.1). Let π 1,, π k and π 1,, π k be genuine epesentations of GL1,, GL k. tenso poducts with a chaacte ω, i.e. (π 1 π k ) ω = (π 1 π k) ω if and only fo each i thee exists a chaacte ω i of GL i, tivial on ω i π i. They give ise to isomophic metaplectic GL i, such that π i = Remak Notice that the metaplectic tenso poduct geneally depends on the choice of ω. If the cente Z GL is aleady contained in M, we have π ω = π and hence thee is no actual choice fo ω and the metaplectic tenso poduct is canonical. This is the case, fo example, when n = 2 o n. 17

18 A epesentation of M is always a metaplectic tenso poduct ([33], Lemma 4.5). Moeove, we have the following useful lemmas. Lemma 3.24 ([33] Lemma 4.6). Let π and π be ieducible admissible epesentations of M. Then π and π ae equivalent if and only if π Z GL M and π Z GL M have an equivalent constituent. Lemma 3.25 ([33] Poposition 4.7). We have Ind M M π ω = mπ ω Z GL fo some finite multiplicity m, so evey constituent of Ind M M π ω = mπ ω is isomophic Z GL to π ω. Indeed, we can veify that m = [ H : Z GL M ]. Lemma We have ( ) Ind M M π = m π ξ whee m is [ H : Z GL M ] and ξ is ove the finite set of chaactes of Z GL M that ae tivial on M. Poof. The poof is the same as in [33] Poposition 4.7; see also [34] Poposition Examples. We give some examples of the metaplectic tenso poduct in this section. The key ingedient in the poof is Lemma This allows us to compae ieducible smooth epesentations of M by esticting to Z GL M. Let χ be a genuine quasichaacte on Z GL T n, and ω = χ Z GL be the cental quasichaacte. Fo each i, let T,i be a maximal abelian subgoup of T i. Let T be the diect poduct of T,1,, T,k with amalgamated µ n. Then T is a maximal abelian subgoup of T. Let T be a maximal abelian subgoup of T such that T T = T. Let χ be an extension of χ to T. We may decompose χ T as χ 1 χ k, whee χ i is a genuine chaacte on T,i. Let T,i be a maximal abelian subgoup of T i such that T i T,i = T,i. We still use χ i to denote an extension of χ i to T,i (this extension is not unique). When χ is in geneal position, so ae χ i s. Theefoe the pincipal seies epesentations I(χ i) on GL i ae ieducible. Theoem Assume that χ is in geneal position. Then the metaplectic tenso poduct (I(χ 1) I(χ k )) ω is independent on the choices of χ i. Moeove, as epesentations of M, I(χ ) = (I(χ 1) I(χ k)) ω This esult shows that, fo pincipal seies epesentations, the metaplectic tenso poduct can be viewed as an instance of Langlands functoiality on coveing goups; see Gan [11]. 18 ξ

19 Poof. Indeed, the choice of the chaacte χ i on T,i is up to a chaacte of T,i / T,i. Thus the esulting pincipal seies epesentations diffe by a chaacte that is tivial on GL i. By Poposition 3.22, the metaplectic tenso poducts ae still in the same isomophism class. This poves the well-definedness. Fo the second assetion, let us follow the constuction of metaplectic tenso poduct. Fo I(χ i) GL, we choose one ieducible constituent Ind GL i Z GL M, B,i i χ iδ 1/2 B i. Then as epesentations of GL ω(ind 1 χ B 1δ 1/2,1 B 1 Ind GL k χ B kδ 1/2,k B k ) = ω Ind M χ B δ 1/2 M. This is an ieducible constituent of On the othe hand, (I(χ 1) I(χ k)) ω Z GL M. ω Ind M B χ δ 1/2 M = Ind Z GL M χ δ 1/2 B Z GL is also an ieducible constituent of I(χ ) Z GL M. By Lemma 3.24, we ae done. Next, we tun to exceptional epesentations. We stat with an exceptional chaacte χ on Z GL T n, and fom the exceptional epesentation Θ M(χ ) as the ieducible quotient of Ind M B χ δ 1/2 M. The chaactes χ is ae defined as in the pevious case. Theoem The metaplectic tenso poduct (Θ(χ 1) Θ(χ k )) ω is well-defined. As epesentations of M, Θ M(χ ) = (Θ(χ 1) Θ(χ k)) ω. Poof. Again, we want to show that both sides have an equivalent ieducible constituent when esticted to Z GL M. Fo the left-hand side, we choose V 0 (χ T ). This is the unique ieducible subepesentation of Ind Z GL M V 0 (χ T ) is J UM (V 0 (χ T )) = Ind Z GL T Z GL w M,0 B w 1 M,0 Z GL w M,0 T w 1 M,0 M. w M,0 χ δ 1/2 M. The Jacquet module of ( w M,0 χ δ 1/2 M ). On the ight-hand side, we choose ω(v 0 (χ 1) V 0 (χ k )), whose Jacquet module is ω(ind T 1 w GL1,0( T ( w GL 1,0 χ 1, 1 δ 1/2 )w 1 B 1 ) Ind T i GL 1,0 w GLk,0( T ( w GL,0 k χ k, k δ 1/2 )w 1 B k )) GL,0, k = Ind Z GL T Z GL w M,0 T w 1 M,0 ( w M,0 χ δ 1/2 M ). Thus ω(v 0 (χ 1) V 0 (χ k )) can be also ealized as the unique ieducible subepesentation of Ind Z GL M w M,0 χ δ 1/2 w Z GL M,0 B w 1 M. M,0 19

20 Theefoe, as epesentations of Z GL M, By Lemma 3.24, we ae done. V 0 (χ T ) = ω(v 0 (χ 1) V 0 (χ k)). Example Conside the patition (1 ). In this case, M is just T and the metaplectic tenso poduct is just the epesentation theoy of T. The exceptional epesentation on GL 1 is Ind GL 1 A χ, whee A is a maximal abelian subgoup of GL 1, and ( χ is an ) extension of χ : F n C to A. Notice that χ is an ieducible constituent of Ind GL 1 F A χ Let χ n. 1,, χ be chaactes of F n. Thus the metaplectic tenso poduct of Ind GL 1 A χ 1,, Ind GL 1 A χ is Ind T T (χ 1 χ ), whee (χ 1 χ k ) is an extension of χ 1 χ k to T Semi-Whittake functionals. Fix a nontivial additive chaacte ψ : F C. Fo a patition λ of, let M = M λ be the coesponding Levi subgoup of GL. We define a chaacte ψ λ : U M U M /[U M, U M ] C as follows. Let α is a positive simple oot in U M and x α (a) be the one-dimensional unipotent subgoup in U coesponding to the oot α. We define ψ λ (x α (a)) = ψ(a). We extend this chaacte to ψ λ : U C via the naive pojection U U M. Notice this chaacte agees with the chaacte defined in Section 2.1. Fo a smooth epesentation (π, V ) of GL, a linea functional L : V C is called a λ-semi-whittake functional if L(π(u)v) = ψ λ (u)l(v) fo all u U, v V. When λ is fixed, we simply call it a semi-whittake functional An explicit fomula. We study semi-whittake functionals of exceptional epesentations. Fist, we have the following obsevation fo Whittake functionals of exceptional epesentations on GL. Let Θ(χ ) be an exceptional epesentation of GL. Recall ψ () : U C is defined as GL ψ () (u) = ψ( 1 i=1 u i,i+1). Let d = dim J U,ψ() (Θ(χ )). If we estict Θ(χ ) to, we still have d = dim J U,ψ() (Θ(χ ) GL ). By the exactness of Jacquet functo and Coollay 3.14, d = dim J (V U,ψ() 0( x χ )) x 1 T \ T / T = dim J U,x ψ (V () 0(χ )). x 1 T \ T / T Theefoe, x T \ T dim J U, x ψ () (V 0 (χ )) = d[ T T : T ] = d[ T : T ] Now let us etun to the setup of the metaplectic tenso poduct. Let Θ M(χ ) = (Θ(χ 1) Θ(χ k)) ω be an exceptional epesentation of M. Let d i = dim J UGLi,ψ (i ) Θ(χ i). We now choose epesentatives fo T i \ T i, and combine them togethe. This gives a set of epesentatives of 20

21 T \ T. Thus, x T \ T Poposition dim J UM, x ψ λ (V 0 (χ 1) V 0 (χ k)) = dim J UM,ψ λ (Θ M(χ )) = k d i [ T,i : T,i]. i=1 k i=1 d i[ T,i : T,i] [ H : M. ] Poof. Wite π = V 0 (χ 1) V 0 (χ k ). We have J UM,ψ λ (Ind M M π ) = J UM, x ψ λ (π ) = x M \ M By Lemma 3.26, the dimension of the left-hand side is [ H : M ] dim J UM,ψ λ (Θ M(χ )). x T \ T J UM, x ψ λ (π ). The dimension of the ight-hand side is k i=1 d i[ T,i : T,i]. This poves the esult. Now we poceed to simplify this fomula. Fom now on till the end of this section, we dop the subscipt M when the ambient goup is M to avoid buden on notations. The subscipt i indicates the subgoup consideed is in the i-th block GL i. Let π be an ieducible constituent of the epesentation Θ M(χ ) Z GL M. Recall Θ M(χ ) Z GL M = x π = x π. Apply the induction functo Ind M M Z GL gives x T \ T /Z GL T x T T \ T and use Lemma 3.25 on the ight-hand side. This Ind M Z GL M (Θ M(χ ) Z GL M ) = [ T : T T ][ H : Z GL M ]Θ M(χ ). Apply the Jacquet functo J UM ( ). This gives Ind T Z GL M T J U M (Θ M(χ ) Z GL M ) = [ T : T T ][ H : Z GL M ]J UM (Θ M(χ )). Compaing the dimensions and using [ T : Z GL M T ] = [ M : Z GL M ] gives Thus Theoem [ M : H] = [ T : T T ]. [ H : M ] = [ M : M ] [ T : T T ] = [ T : T ] [ T : T T ] =[ T T : T ] = [ T : T ]. dim J UM,ψ λ (Θ M(χ )) = 21 k i=1 [ T,i : T,i] [ T : T ] k d i. i=1

22 Remak We can see that the same calculation is tue fo the pincipal seies epesentations. Let us mention some immediate coollaies. Coollay Suppose n F = 1. If i > n, fo some i, then J UM,ψ λ (Θ M(χ )) = 0. Poof. This is because when i > n, d i = 0. Coollay Suppose n F = 1. Let Θ (χ ) be an exceptional epesentation of GL. If i > n fo some i, then J U,ψλ (Θ (χ )) = 0. In othe wods, thee is no semi-whittake functional on Θ (χ ). Poof. In fact, J U,ψλ (Θ (χ )) = J UM,ψ λ (J R (Θ (χ ))) = J UM,ψ λ (Θ M( wm χ δ 1/2 P )) = 0. The following coollaies ae tue without n F = 1. Coollay When i n fo all i, J UM,ψ λ (Θ M(χ )) 0. Coollay When i n fo all i, J U,ψλ (Θ (χ )) Constuction of maximal abelian goups. We now discuss the constuction of maximal abelian subgoups. This helps us simplify the fomula futhe. Given a maximal isotopic subgoup Ω of the Hilbet symbol, [23] Section 0.3 povides a way to constuct maximal abelian subgoups of T unde cetain assumptions. When n F = 1, F n o is a maximal isotopic subgoup of the Hilbet symbol. Let T o = {diag(a 1,, a ) T : val(a i ) 0 mod n}. Then Z GL To is called the standad maximal abelian subgoup of T, in the sense of [23] Section I.1. We use T st to denote this subgoup. T st Remak Notice that T is usually not a maximal abelian subgoup of T, even fo n = 2. When n = 2, c = 0, a canonical maximal abelian subgoup was intoduced in Bump-Ginzbug [4]. The intesection of thei maximal abelian subgoup and T is a maximal abelian subgoup of T. Let T o = T o GL. The following poposition can be poved by imitating the agument in [23] Section 0.3. Poposition The goup Z GL T o is a maximal abelian subgoup of T. Remak Ou calculation in Section 3.6 elies on the index [ T : T ], which is an invaiant of T st. We can compute it by using the standad maximal abelian subgoup T. 22

23 Remak When n F = 1, we give an example of maximal abelian subgoup such that its intesection with T is Z GL T o. Let Z = {(zi, ζ) Z : z o F n and Then Z Z GL T is Z GL T o. T (n ) gcd(n,(2c+ 1)) } o = {a T o : det(a) F gcd(n,) }. T (n ) o = Z (n T ) o is a maximal abelian subgoup of T and its intesection with Remak When n F 1, it is usually difficult to constuct maximal abelian subgoups of T. Howeve, when n, the situation is still nice in the following sense. Let Ω be an isotopic subgoup of the Hilbet symbol. Then by the constuction in [23] Section 0.3, is a maximal abelian subgoup of T. {(diag(t 1,, t ), ζ) : t i Ω, ζ µ n } We now discuss maximal abelian subgoups of T M. Fo 1 i k, let T,i be a maximal abelian subgoup of T i. Let T be the diect poduct of T,1,, T,k with amalgamated µ n. Then T is a maximal abelian subgoup of T. Let T M,o = T o M. Lemma The goup Z M T M,o is a maximal abelian subgoup of T. The othe maximal abelian subgoup we conside is the standad maximal abelian subgoup T M, st An explicit fomula, continued. We now continue the calculation of ou explicit fomula. Thoughout this section, the ambient goup is M. We again use the convention afte st the poof of Poposition Thus T ( T st o, esp.) is T M, ( T M,o, esp.) in the pevious st section, while T,i and T o,i ae the coesponding subgoups in the i-th block GL i. Theoem When n F = 1, Poof. Indeed, dim J UM,ψ λ (Θ M(χ )) = k i=1 st [ T,i : T o,i ] [ T st : T o ] k d i. [ T : T ] = [ T : T ] [ T : T ] = [ T : T ][ T : T ] [ T : T st ] = [ T : T ][ T : T st ][ T : T o ] [ T : T. o ] Notice that k i=1 [ T i : T i k ] i=1 [ T : T = [ T i : T k,i] i=1 ] [ T : T = [ T i : T o,i ] ] [ T : T = 1. o ] Combining with Theoem 3.31, we obtain the desied fomula. 23 i=1

24 When is a multiple of n, we have the following uniqueness esult. This holds even without the assumption n F = 1. Theoem If = mn, and λ = (n m ), then J U,ψλ (Θ (χ )) is one-dimensional. Poof. When n F = 1, this follows fom Theoem Indeed, in this case we have gcd(n, 2c + 1) = 1. Theefoe Z GL T st o, and [ T : T o ] = 1. Similaly, Z GLn T o,i, and st [ T,i : T o,i ] = 1. By Poposition 3.3, d i = 1 fo all i. Theefoe dim J U,ψλ (Θ (χ )) = 1. We now assume n F 1. Let Ω be a maximal isotopic subgoup of the Hilbet symbol. Then T := {(diag(t 1,, t ), ζ) : t i Ω} is a maximal abelian subgoup of T, and T := Z M ( T T ) is a maximal abelian subgoup of T. Notice Z M = Z M. Moeove, [ T : T ] = [ T : T ] [ T : T ] and k i=1 [ T i : T k,i ] i=1 [ T : T = [ T i : T,i] ] [ T : T = 1. ] Combining this unifom desciption with Theoem 3.31, we ae done. Remak Recall that the metaplectic cove GL depends on an implicit choice of the modulus class c Z/nZ. Ou esults ae tue fo all c Z/nZ. This is clea fo the vanishing esult (Coollay 3.34) and nonvanishing esult (Coollay 3.36). Fo the uniqueness esult, notice that when = mn, Z GL = {z n I : z F }. This fact is independent of c. Thus the poof of Theoem 3.44 is independent of c. Remak When n = 2, this is [4] Poposition 1.3 (i). Indeed, when = 2k and the patition is (2 k ), this follows fom Theoem When n = 2, = 2k+1, and M coesponds to the patition (2 k st 1). In this case, d i = 1 fo all i and [ T : T o ] = [F : F 2 o ]. Moeove, st [ T,i : To,i ] = 1 if i = 2; and = [F : F 2 o ] if i = 1. The twisted Jacquet module of Θ M(χ ) is again one-dimensional. 4. Global Theoy 4.1. Theta epesentations. Let n 2. Let F be a numbe field containing a full set of nth oots of unity µ n, and let A denote the adeles of F. Fo 2, let GL (A) denote an n-fold metaplectic cove of the geneal linea goup, as in Section 2.5. We ecall the definition of the global theta epesentations. These epesentations wee constucted in [23] using the esidues of Eisenstein seies as follows. Let B be the standad Boel subgoup of GL, and T B denote the maximal tous of GL. Let s C be a multicomplex vaiable, and define the chaacte µ s of T (A) by µ s (diag(a 1,, a )) = i a i s i. Let Z( T (A)) denote the cente of T (A). Let ω s be a genuine chaacte of Z( T (A)) such that ω s = µ s p on {(t n, 1) t T (A)}, whee p is the canonical pojection fom T (A) to T (A). Choose a maximal abelian subgoup A of T (A), extend this chaacte to a chaacte of A, and induce it to T (A). Then extend it tivially to B(A) using the canonical pojection fom B(A) to T (A), and futhe induce it to the goup GL (A). We abuse the notation slightly and 24