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SOME EXAMPLES OF LIFTINGS FOR FINITE GROUPS OF LIE TYPE
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of the Ohio State University
By
Savva 0. Pavlov *****
The Ohio State University 2000
Dissertation Committee:
Stephen Rallis, Advisor Approved by Cary Rader Robert Stanton Advisor Bruce Walsh Department Of Mathematics UMI Number; 9994920
UMI
UMI Microform 9994920 Copyright 2001 by Beil & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
Bell & Howell Information and beaming Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 ABSTRACT
Weyl’s Unitarian trick is one of the cornerstones of the classical representation the ory of reductive Lie groups. Using a natural bijective correspondence between real compact groups and complex reductive groups, Weyl was able to prove a number of important properties of representations of reductive complex and real groups. For instance, he proved the complete reducibility of linear representations of complex reductive groups using the analogous fact for real compact groups and the correspon dence between the irreducible modules.
In this paper we study minimal representations of finite groups of Lie type. We are concerned with restricting such representations to large subgroups in order to find nontrivial occurences of irreducible modules. Specifically, for such large subgroups of non-split type we develop an analogoue of Weyl’s Unitarian trick to handle these cases.
Our methodology originates in Shintani lifting theory. Shintani proved in [Sh] th at if G = GLji, the general linear group, and a is a Frobenius isomorphism, there is a canonical bijection ^ between the Galois-stable irreducible representations of G*^”* and all irreducible representations of G“^, satisfying a fixed trace identity. His results were later extended by Kawanaka to include most classical groups ([Kwl], [Kw2]). In our constructions lifting theory serves as the analogue of complexification in classical
u theory; it allows us to deduce results about minimal representations in non-split cases from easily obtainable results in split cases.
The examples we consider are the liftings of linear Weil representations (4.2), non-traditional dual pairs in Sp^ (chapter 5) and the ^-correspondence for a D;] type group (chapter 6).
In chapter 7 we introduce a more general notion of lifting. We construct an analogue of Shintani theory for an isomorphism a which is not of Galois type. We examine the limits of lifting theory techniques by studying the case G = GL 2n+i(Fç) and the automorphism In Chapter 8 we present a new method for constructing a specific lifting from PGL2 SOz to SLz- 111 ACKNOWLEDGMENTS I thank Karl Rummeihart, Moshe Baruch, Pham. Huu Tiep, and Zhengyu Mao for many valuable recommendations and references. 1 thank Ju-Lee Kim for suggesting the proof of theorem 4.7. I am grateful to Cary Rader for the meticulous care with which he examined the dissertation. His numerous remarks allowed me to greatly improve the text. I consider all the remaining mistakes as my own. .A.bove all I thank Stephen Rallis, my advisor, for introducing me to the subject and for his infinite patience during our numerous conversations. IV VITA 1972 ...... Born in Chernovtsy, USSR 1992 ...... Diploma in Mathematics. Moscow State University. Moscow 1992 - present ...... Reseach in mathematics. FIELDS OF STUDY M ajor field: Mathematics Specialization: Representation theory TABLE OF CONTENTS A b stra c t ...... ii Acknowledgments ...... iv V i t a ...... V CHAPTER PAGE 1 Introduction ...... I 2 Preliminaries ...... 5 3 Technical lemmas ...... 13 4 The base change of Weil representations ...... 21 5 The Weil representation of Spi{Fq) ...... 35 6 The c a s e ...... 54 7 Symmetric liftings ...... 62 8 The symplectic correspondence ...... 76 Bibliography ...... 85 VI CHAPTER 1 INTRODUCTION Let p > 3 be a prime number, F, a finite field of characteristic p, and F, its algebraic closure. Let G be an algebraic group over F,. We will denote by Fr the geometric Frobenius endomorphism (raising the coordinates to the g-th power). Let a be a (twisted) Frobenius map that is a composition cr = F ro e of the geometric Frobenius and a finite order automorphism of G. commuting with Fr. Let m be an odd positive integer. Define the norm map N : 0'^"' C " . S{g) = This map induces a surjective map between the conjugacy classes of G‘^"' and the conjugacy classes in containing elements of G^- For an irreducible representation (0, V) of G‘^"' define —another irreducible representation acting on the same space by formula (0a-p)u = {Qg‘^)v. 0 is called Galois-stable if it is equivalent to Bq., i.e. there exists an intertwining operator : V —)■ V such that: o (0 ,9 ) = Q{g) o Aa- Lifting theory was originated by Shintani who proved in [Sh) that if G = GL^ is the general linear group there is a bijection k between the Galois-stable irreducible representations of G®^’" and all irreducible representations of G ,. This correspondence satisfies a twisted character formula (for a certain .4,): 1 tr(0(.V'(^))) = tr(bçiô)(g) o .4,), where N'(g) is possibly on an element of conjugate to jV(g) in Kawanaka extended this result to the case 6' = Un in [Kwl] and to the cases G = S 0 „, G = Spn, m odd in [Kw2|. In this paper we study the application of liftings to non-split groups. We develop a method to overcome the technical difficulties in these cases that is in a way analogous to Weyl’s unitary trick. The minimal representations of non-split groups will be lifted (via Shintani lifting) to a split situation and then the results will be deduced from well-known facts about standard cases and Shintani liftings. A typical situation which we will encounter will involve algebraic groups G D H m and L defined over a finite field F, such that H(F<,) is isomorphic to fj L(F,). Suppose 1=1 L is split over F, and H is obtained by twisting with the automorphism e of G of order m, commuting with the Fr action on L and cyclicly permuting the L s, e(ÿi, g2 , • ■ •. ffm) = (5 2 , • • • T ffm, ffi)- In 3.2 we establish a lifting theory in this general setting, which we then use to study the liftings of linear Weil representations (4.2), the exceptional case of the ^-correspondence for Sp^ (chapter 5) and the ^-correspondence for a type group (chapter 6). In Chapter 2 we introduce Shintani liftings and Weil representations, and some important notations. In Chapter 3 we prove several technical lemmas that will be central to our constructions. The first part of chapter 3 establishes the connection between the restriction of the Weil representation of Sp„ to the dual pair SL2 x 5 0 j, and the restriction of the minimal representation of 5 0 ^ to the dual pair 502t x SOom, m + k = n. The second part of chapter 3 provides the basic framework for our method in a general setting. In Chapter 4 we study the liftings of Weil representations. First we prove that the lifting of the Weil representation of is again the Weil representation of In section 4.3 we prove an analogous result for the Weil representations of 5p„ in case of odd-dimensional liftings (the proof suggested by Ju-Lee Kim). In section 4.4 we give a counterexample for the case of even-dimensional liftings of 5p„. In Chapter 5 we use the methods and results of chapters 3 and 4 to work out in detail the restriction of Weil representation of Sp.i(F,) to SL-i{Vqi)—a "twisted” version of the dual pair 5^2 x SO^ C Sp^. In Chapter 6 we apply our method to the minimal representation of D\ and study its restriction to the dual pair SLoi^q) x Using Chapter 5 and Lemma 2.2 we easily compute the “split case,” i.e. restriction of the minimal representation of D\ to SL2 X SL2 X SL2 X 5Io- Then we obtain the restriction for Dl as a. Shintani descent from the split case. In chapter 7 we study a more general notion of lifting. As our main example we consider G = GL2n+i{^q) and a the automorphism of G defined by a{g) = Jg~^^J where J is the unit anti-diagonal. We will show that in that case the norm map does not induce a bijection between the cr-conjugacy classes of G and the conjugacy classes of G containing elements of G®'. In this case one cannot follow the Shintani construction to establish lifting theory. We will show that it is still possible, under certain conditions, to establish a twisted character formula. We will also introduce a weaker notion of lifting, requiring the twisted character formula only on regular semisimple classes, and prove its existence for the principal series representations. In Chapter 8 we present a new method for constructing a specific lifting from PGLo ^ SO-K to SL-x- We will indure the representations of GLx to SP^ and then restrict them to the subgroup SH, where S ~ SLo and H is the Heisenberg group. We will show that this construction gives a correspondence between the representations of type E,[det g) and the principal series representaions, and also between cuspidal representations of GL2 and 5Lo. CHAPTER 2 PRELIMINARIES 1. Some basic definitions and notations. Hr is the group of r-th roots of unity in C. Fg is the finite field with g = p" elements, where p > 3 is a prime number, f, is the algebraic closure of Fg. N and T are the norm and trace maps correspondingly. In = diag{l, 1,..., 1)—the n x n unit matrix. If G is an algebraic group defined over Fg and denote the subgroup of G consisting of ^-stable points. S02n is the special orthogonal group. We use this notation only for the split form of the group and fix the standard bilinear form, i.e. S 0 2 n is the group of automorphisms of 2n-dimensional space W preserving the bilinear form defined in some basis by a symmetric form Q. We will fix the form Q = ' ° In 0 in chapters 2-6, and use 0 ... 0 1 0 ... 1 0 Q = \ 1 0 ... 0 / in chapters 7 and 8. Keeping the same basis, denote by the space spanned by the first n basis vectors and W~ is spanned by the last n basis vectors. The decomposition W = © W~. will be called the polarization of W. Spn = Sp{W) is the symplectic group of a 2n-dimensional space IV = V' © V’*. In chapters 2 through 6. J will be fixed to be f J = — In 0 We will use 0 ... 0 I 0 ... 1 0 J = -1 0 ... 0 V / in chapters 7 and 8. We denote by the sum of all possible tensor products with n a’s and m b's, e.g. a®^6®^ = a® a^b + a®b®a + b®a®a. When it does not create confusion we denote bv 1 the trivial character. The quadratic character of the multiplicative group will be denoted by oq , and the quadratic character of Hp+i will be denoted by ujq . 2. Weil representations. Let V be an n-dimensional vector space over F,, W = V' © V'*. Fix 0, an additive character of F, and define < x.y > = w{T t{ ^xy)) f i x ) = complex-valued functions on V = F," on which the generators of Sp{W) act as follows: / a 0 f{x) = x(det(a))/(xa) V 0 'a - i fix) = w(Tr(6 ^ x i) ) /( x ) 0 1 \ 0 1 9 fix) = / ’ (i) V -1 0 It is well-known that if y = 1 (mod 4) then all Weil representations are isomor phic and if g = 3(mod 4) there are exactly two non-equivalent Weil represen tations over a finite field depending on the quadratic class of the character il). We suppose that the character ip is fixed unless specifically noted otherwise. 7 3. Shintani liftings. ([Sh], [Kwl], [Kw2]) Let G be an algebraic group defined over an algebraicly closed field F, and cr be a Frobenius map. Let m be a posi tive integer. Define the norm map N : G'^"' -> 0°'^, N{g) = .. .g'^g. This map induces a surjective map (see eg [Kw3], pg. 129) between the conju gacy classes of and the conjugacy classes in containing elements of 0°. For an irreducible representation (0, V) of G°^ define —another irreducible representation acting on the same space by the formula (O^g)o = (Og‘^)v. 0 is called Galois-stable if it is equivalent to 0^, i.e. there exists a unitary inter twining operator : V —> I' such that; Aa o {Q^g) = 0(ÿ) o --1er Definition 2.1. A cr-stable representation 0 of G‘^"' is a lifting (or base change) of the representation 9 of if for every g 6 G*^"* there is an element N'{g) € G®” conjugate to N{g) in G^"" such that Tr(g(:V(g))) = Tr(0(p)oA,). Shintani and Kawanaka proved that base change provides a bijection between the irreducible representations of G®' and cr-stable irreducible representations of G*^"* for G = GLn, G = t/„ and G = SOn, G = 5p„, m odd. Remark 2.2. The intertwining operators in [Sh], [Kwl], and [Kw2] can be as sumed to be unitary since they arise from the representations of the semidirect product AG, were A is the cyclic group of order m. 8 In this paper we will use a more general notion of base change. We will use our definition 2.1 without requiring 0 and 9 to be irreducible. This definition should be used with care, since it adds more freedom to the choice of A^. The following few simple lemmas, deduced from elementary properties of intertwin ing operators, demonstrate the extent to which this notion of lifting can be used. Let TT be a representation of G(F,) = and tt be the base change of ~ to G(Fqm) = Let A(r be a unitary intertwining operator for tt and rr‘^ such that lx{9{N'[g])) = tr(0(^) o .-l^-). We will consider the decomposition of tt into the sum of irreducible subspaces: where Q is the isotypic component of type Ç, i.e. the submodule of ü all of whose irreducible components are isomorphic to Set if^ ^ Q and Q- .4(T takes Q to Q,. In particular, stabilizes all Q for cr-stable E,, and if^. Lemma 2.3. Tr(^^ o ,4o-) = 0 i.e. ifo- is the only part that contributes to the twisted character formula, i. e. for any g G : Tr{iT{N{g)) = Tr{TT{g) o .4^) = Tr(if^(g) o .4<^) Proof Since .4g. takes Q to Q ,, its restriction to if^ will consist of the blocks of the form: 0 0 0 0 0 A 0 y ^ 0 0 ... 0 y In particular, restriction of to has all O’s on the diagonal. Since ttIg,"* stabilizes all Q, that means Tr(ir^ o A^) = 0 . □ Corollary 2.4. iTa is a base change of i: relative to A^. Lemma 2.5. Suppose = XI si rnultiplicitrj free . Then for any unitary I intertwining operator A : - tî there are unitary complex numbers q such that for any g € Tr{K{N'[g))=Tr{Y^c,^g)oA,) i Conversely, for any choice of unitary complex numbers Q there is a unitary intertwining operator A such that the formula above is satisfied. Proof. Notice that if tTo- is multiplicity free then all the a-stable Q’s are irre ducible. Therefore by Schur’s lemma the set of unitary intertwining operators is given by .A|(, = djA^,, |d,| = 1, where A^, is the restriction ofA* to the given Set Cj = d~^ For any g 6 T r ( ^ Ci^iig) o A) = T r ( ^ ° d.AçJ i i = T r(^^i(5) o AçJ = Tr(?r(ÿ) o A,,) = Tr(;r(jV'(g)) i □ 10 Lemma 2.6. Suppose multiplicity free and every is a lifting of i k an irreducible representation pi of G'^. Then ir = '£^pi- 1=1 Proof, f, being a lifting of o,; means there are unitary intertwining operators such that o A^_ and tv{pi{M'{g))) = tr((;(g)o A^J for every i.g. Just like in lemma 2.5 we must have = A,A^^, [Ail = 1, so Tr(7r(iV'(p)) = tr(7T^(^) o .4?) = T r (^ Ç .(^ ) o A^ArJ = T r ( ^ X,p,{N'{g)) I 1 = 1 Since the norm map is a surjeciion of conjugacy classes and pi are irreducible, k we must have tr = ^ A,pi. Since ~ is an actual represenation. we must have i=l Ai = 1. □ 4. Irreducible representations of 5Z2(F,), an overview. In this part we define some notations and give the classification. For proofs and details see [DM], 150-155. G = 5Z2(F,) contains 2 classes of tori: the split torus T = {diag{gi, p f | 6 FJ}, and the anisotopic torus = {diag{gi,gf) | gi 6 F,a, g^gf = 1}. For all (f) Ç. Ç. Pq+i, P^{(p), will denote the corresponding Deligne-Lustig characters. If c>, ip ±1 these characters are irreducible. For the rest we have: R^{1) = 1 + S t 11 -■^Tu,(l) - -1 + St ^ ( ' l ) = .Yao+Xâo = xlo + XJo, where 1 is the trivial representation, St is the Steinberg represenation, are representations of dimension and are representations of dimension ^ (ûo and wo denote quadratic characters, see pg. 7). 12 CHAPTER 3 TECHNICAL LEMMAS. 1. In this section n will be an even number, 5p„ the group of 2n x 2n symplectic matrices, 0„ its Weil representation, 50„ the split group of n x n orthogonal matrices with determinant 1. 50„ has a single minimal (lowest-dimension non trivial) representation of dimension this representation is unipotent ([TZ]). Let k, m be any two positive even integers such that m-\-k = n. The defining space on which SOn acts can be represented as a direct sum of two orthogonal spaces of dimensions m and k: V" = V'"' © V'*'. Correspondingly, we can consider the subgroup of SOn isomorphic to SOm x SOk, m + k = n. with natural embedding: 9i = E SOm, 92 E SOk / {9 1, 9 2) 0 ÿ2 0 c 0 d V / The goal of this part is to compute the restriction of to this dual pair SOm x SOk C SOn. 13 If 0, Ip are two representatioas of a group G we will write 0 x 0 = where G C G x G is embedded diagonally {g {g,g))- Note ch(0 x 0 ) = (ch0)(ch0). The following is an easy corollary of Frobenius reciprocity and Peter-Weyl theorem. F act 3.1. If é and 0 are irreducible, 1 occurs in ( 0 x 0 ) if and only if 0 = 0 *, and its multiplicity is at most 1. Consider now 5p„—the symplectic group over a finite field and a dual pair of type SL2 X SOn, where n is even and SLo is identified with Spi. We will write the restriction of the Weil representation to this dual pair as ©IsC.xSOn = ^ 0®Hn{0), 0 €lrr(S£,2) where Hn{4>) is some (not necessarily irreducible) representation of 50„. Lemma 3.2. = 1 -I- Proof. The ambient Spn space can be represented as a tensor product V = where Aisa 2-dimensional symplectic space, and W is an n-dimensional orthogonal space. Consider a polarization W = W'^ 0 W~ and a corresponding Lagrangian V'^ = .4 ® W'*’. Consider, relative to this polarization, the Weil representation, where the SLo action on V'*' is linear via its action on A. Our first goal will be determining the dimension of /fn(l). In our realization, the représentation space for iïn(l) is the space of complex-valued functions 14 on the space of 2 x n matrices over F,, invariant under the left SL^ action. Therefore, its dimension is the number of orbits of the left SLo action on the space of 2 X n matrices over F,. The action obviously preserves the rank, so we will carry out the computation in 3 cases: (a) The rank of the matrices is 0 . There is one orbit. (b) The rank of the matrices is 1 . Using elementary matrices we can make the second row all zeros: / \ * * ... * y 0 0 ... 0 y The stabilizer of such an element is the subgroup of upper-triangular unipo tent matrices in SLo, so its cardinality is q. The total number of matrices of rank one is (9 -f 1 ) (<7 " - 1 ), so the number of orbits is • (c) The matrices are rank 2. The cardinality of this subset is 9^" — 1 — (9 1 )(9" - 1 ) (all matrices except matrices of rank zero and one). The stabilizer of any point is trivial, so there are = q—1 orbits. Adding up the numbers in the three cases we get the total of 1 -t- 4- + 1 orbits, i.e. + 1 . Next we want to prove that iî„ is not a direct sum of ~ gl!." + 1 trivial representations. To see that, consider the restriction of 0„ to the subgroup 15 isomorphic to SL-i x GLn/ 2 - Since it is a subgroup of our dual pair SL2 x S 0 „, if Hn = + 1 ) 1 then the restriction of 0 „ to SL^ x GLn /2 must also contain + 1 copies of the trivial representation. That would mean that the GLn/-i action (which is linear action on the right) preserves the SL2 orbits. That is not true (e.g., it is easy to see that GLn/2 acts transitively on the set of rank one orbits). Now we know that Hn has dimension + 1 and it must contain nontrivial representations. Since dim On = is the smallest possible dimension for a nontrivial representation of 50„, we must have Hn(l) = 1 + On. □ Now consider dual pairs SL'j x SOm C Spm and SL 2 x SO/^ C Spt, m + k = n. and corresponding Weil representations and their restrictions to the dual pairs like in Lemma 3.2. Theorem 3.3. ^nlsOm^so^ ~ d- ® )■ Proof. Consider the restriction of 0„ to Spm x Spk- It is well-known that it is isomorphic to 0m ® 0jt (see e.g. [Si]). Restricting further to {SL2 x SOm) x {SL2 X SOk) we get {0 ® Hm{à)) ® ® Hk{ib)) Rewriting {SL2 x SOm) x {SL2 x SOk) as SL2 x SL2 x SOm x SOk we restrict the representation to S L f x SOm x SOk, where SL^ is the diagonal embedding 16 of SL2 in SL-i X 5 I-2 - Notice that the resulting restriction is equivalent to the restriction to the dual pair SLo x and then to SL^ x {SOm x SOk) (indeed, A ® (V i © V2 ) = (.4 ® V i) © (.4 ® V o)). So ^nli'OmxSOfc = i/(l) - 1 = - 1 + ^ Hrn{(p) ® Hk{(p’) 0 eln{SL 2) by Fact 3.1 and Lemma 3.2. □ 2. Let G and L be algebraic groups defined over a finite field F,. We will denote by Fr the geometric Frobenius endomorphism (raising the coordinates to the q-th m power). Suppose G contains a subgroup H(F,) isomorphic to f] L(F,). Let e 1=1 be an automorphism of G of order m commuting with the Fr action, stabilizing _ m H, and, under the isomorphism H(F,) ~ L(F,), cyclically permuting L’s i.e. 1=1 L ■ ■■■9m) {9 2, ■ ■■9m-9i)- Let F = Fro €. Notice F^ = Fr"^ 1 = 1 and via p : g \ If p is a representation of H^, we will denote by the Fr-twisting of p (p^""(p) = p{g^'')) and set p = p ® p^'' ® p^'’* ® • • • ® p^'"'" '. Lemma 3.4. Let (p, W) be a representation o /H ^ . Its Shintani lifting to is p. The p for irreducible p are the only F -stable irreducible representations of r f - Proof. Define a linear operator .4F : ->• .4f(i'i ® U 2 ® • • • ® Vjn) = (Utji ® ui ® Ü2 • • • ® Um_i). It is easy to see that it is an intertwining operator for p and p^. Indeed: 17 (.4p O p^{gi,g2, . . . , gm)){Vl ® ^2 ® ■ ® ^m) = i^F ° (p(pfn ® ® • • • 0 (pD))(^l ® Ü2 ® • • • ® Urn) = -4F(p(pf ")Ui ® 0 • • • ® (pf^)Um) = P^""'\9r)Vm 0 P(pflui 0-0 p^"'"”'(p^'’)u^_i = p(Pl)l^m 0 P^"(P2)U1 0 • • • 0 P^'’’"~\gm)Vm-l = (P(P1,P2, • • ••9m)) O -4f)(Ui 0 Ü2 0 • • • 0 l^m) Now if g = {91,9-2, •••,9m) € (H^"), then It is conjugate in (by - - -^D) to Finally we prove the twisted character identity: Tr(p® p^ 0 p ^' 0 - - • 0 p^’""‘ o.4f)(pi,P2, •••,Pm) t=l,2,...,m 18 i=l,2,...,m ~ ^2 Pjl,Jmi9l)Pj2jli92) ■ ■ ■ Pjm-lJm-iiPm-l )Pjmdm-\{9rn ) / y A^J2JmVa2 at//^J3j2Vi/3 J ' ' ’ ;/^Jm,Jm-l Vi/m > I ^_7i t“ 2,3^ ...,ni ~ Phdmig-i 92 9\)Pj^j3^9A ) ■ ■ ■ Pjm-jm-l(9m ) «=l,4,5,...,m C p m - l E’r’n-2 P, \ E Pjm,jj9m 9m-1 --^2 = = Tr(XjV(^i,^2,... □ Now let 7T be a representation of G(F,) = and if be the base change of rr to G(F,m) = G^"*. Let Ap be an intertwining operator for if and . We will restrict if to and consider its decomposition into the sum of irreducible subspaces: H Q ' where Q is the ^-isotypic submodule of iflnf"" • Set ifp = X) Q and n'p = ^ Q. A F takes Q to Qf. In particular, Ap stabilizes all Q for F-stable irp. and fp. Theorem 3.5. Triir'p o Ap) = 0 i.e. iff is the only part that contributes to the twisted character formula: for any g € ; Tr{-ir{N{g)) = Tr{T:{g) o Ap) = Tr{Trp{g) o Ap) k k In particular, if fp = Pi 'multiplicity free then 7t|H ^ = ^ Pi- i= l t= l 19 Proof. This is an easy corollary of Lemma 3.4 and Lemmas 2.3 and 2.6. □ 20 CHAPTER 4 THE BASE CHANGE OF WEIL REPRESENTATIONS. In this chapter we study the liftings of Weil representations. In section 1 we prove that the lifting of the Weil representation of GLn is again the Weil representation of GLn- In section 4 we prove analogous result for the Weil representations of Spn in case of odd-dimensional liftings. In section 5 we show that the results of section 4 are “the best one can get” by giving a counterexample in case of even-dimensional liftings of Spn- The additive character ip of F, will be fixed throughout the chapter. Most of the time the distinction between and us^-i will not be important and we will simply write w instead of We denote by id = f o the base change of ii! and by the Weil representation over the extention field. We will simply write Q instead of in most cases. 1 . Weil representations of Let Fq be a finite field with q elements. Following [G] we define the Weil representation a;„ of GL„(F,) as the representation in the space of complex functions on by ujn{g) ■ f{x) f{g~^x), where x G is a column vector. Let Qn be the Weil representation of GL„(F,m). 21 Theorem 4.1. is the Shintani lifting of uj^, i.e. is stable under the natural Galois action a and the following twisted character identity holds: Tr{uJn){N'{g)) = 7V(Q„(t/) o .4^), Vg E GL„(F,m), where Ag- is defined by [Agf){x) = f{x‘^). (See section 2.3 for a more detailed definition of Shintani liftings). Proof. Ag clearly is the intertwining operator between and Indeed, A,^(K9)U)( x) = (Ks)f{^n = = mg-'x)”) = {A,f]{g-'x) = n M o A ,{f){ To prove the twisted character identity choose as the basis the functions 6 ^ : Sx(x) = l,âx(y) = 0 if y # X. For clarity, N(g)~^ will always refer to the linear operator on the n-dimensional vector space over F,m. We know from the general theory (see [Sh]) that for any g 6 GLn(Fgm) there is an element ^ GLji(Fq), conjugate to iV(g)~^ in GLn(Fgm). Any two matrices that are conjugate to each other have the same set of eigenvalues, and the cor responding dimensions of the eigenspaces—they have the same characteristic polynomial and the same Jordan form over the algebraic closure. In particular, the dimension of the subspace of N(g)~^-fixed vectors (over Fqm) is equal to the dimension of the subspace of iV'(y)~^-fixed vectors (over F,). Let ki be that dimension. Then for the left side of the twisted character identity we have: 22 Tr(wJ(jV'(^)) = ^ < >= Z < > xeF" xei7 = # { x G F^|iV'(^)-^x = x} = gdim((%eF;|N'( 9) ‘x=i}) = qdim{{v€¥;m\i\{g)-^v=v}) _ ^ki For the right side of the twisted character identity note: (.■4.(jJi)(y) — dxiUa) — (^), so we have Tr(Q„(^) o .4„) = < gd^-i, 4 > xeF^m = #{x G ‘ = x} = #{x G F"„|x = F a c t 4.2. There is an element E, G F.m such that the matrix: ( -1 \ \ has a non-zero determinant. For such , ■ ■ ■ ) form a basis of Fqm as a vector space over F, (normal basis). Proof. This is a well-known fact from Galois theory. See e.g. [A], Theorem 19. □ 23 Now fix Ç as in Fact 4.2 and the corresponding normal basis. We can now view the space F"m as an mn-dimensional vector space V' over F,. Consider a linear operator on this space S : S(x) = Let k2 be the dimension over F, of the subspace of vectors fixed by 5. We have then. Tr(firi(g) o A^) = # { x 6 F"m|(ç7 "^x)‘^ = x} = q^- So, in order to prove the twisted character identity we need to show k\ = fco. Let ff{g)~^ = S^. It is easy to see that N{g)~^ is gN{g)~^g~^. viewed as an operator on V: (z) = = = • • • = = gN{g)-^g-^x The dimension of the space of N{g)~^-Rxed vectors is ki (over F<,m), i.e. the number of N{g)~^-f\xed vectors is In V, the subspace of N{g)~^-dxed vectors will therefore be mA:i-dimensional over F,. Lemma 4.3. Let x Ç.V be an N{g)~^-fixed vector. Then there are m S-fixed m vectors tj € V' and coefficients aj € F q m , 1 < j < m, such that x = ^ afij. j=i Proof. Let W be the F,m-span of x\ = x,X2 = S{x), Xz = 5^(x),...,Xm = m—1 5"*~^(x) {W is not necessarily m-dimensional). Let TS = ^ S'. Choose i=0 24 tj = T5((Ç‘^x), they are clearly 5-stable. Also, since 5 = {g~^xy and g is Fgm-linear, we have x) = .f^‘~''5‘(x) i.e. k = -I- ^ ‘’x o + H------h to — Ç'^Xi + X-2 -h ■ • • -f- Xm - 1 + ÇXr, (m = -K -K . . . + (""-'xm Now, since *) # 0, ') is a non-degenerate linear operator, so tj's span W , hence the Lemma. □ Now fix some basis i/i, I < i < ko over Fq of the subspace of the 5-fixed vectors in V'. Lemma 4.4. are linearly independent over F,m. k2 Proof. Let a, € F^m, 1 < i < fco be such that Y1 ^iVi = 0 at leeist one a, is i= l non-zero. Without loss of generality we can take Ci = 1 . Since 5 = {g~^xy and g is F,m-linear, we have S{aiiji) = a’y,. Applying (5 - 1) to 53 o-iVi = 0 we 1 = 1 *2 get 53(°i “ tii)vi = 0. Since y/s are linearly independent over F,, at least one i=2 üi ^ Fq and therefore at least one (a^ - a,) ^ 0. The Lemma now follows from the induction on the number of non-zero a/s. □ 25 We are now ready to prove ki = ko. Indeed, from Lemma 4.3 , the dimension of N{g)~^-üxed vectors in F”m is at most the dimension of 5-fixed vectors in V, i.e. 5 ^2 - From Lemma 4.4, there are at least k2 vectors fixed by N(g)~^ = and linr-rirlvearly in.4independent over Yqm, i.e. k-, < ki. u 2. The lifting theory for the subgroups GI„(Fqm) c GLnmi^q)- We start the construction by considering — the Weil representation of GLnmi^q’^) and its restriction to the subgroup of GZ,„„,(Fqm) isomorphic to (GL„(Fqm))'". Use the obvious embedding; / gi 0 ... 0 0 go ... 0 y 0 ... 0 gm j It is clear from the definition of the Weil representation that ^C7Lnm(Fqm ) ^{GL„(F,m))'"' To construct an embedding GL„(F,m) ^ GL„m(F?) select ^ as in Fact 4.2 and let ..., be the basis of F^m as vector space over F,. This will give an embedding GTn(Fqm) GT„m(Fq) via its natural action on F^m a; F^" Note that GL„(Fq) is embedded ■‘diagonally’: 26 9 0 0 0 g 0 0 ... 0 3 ) Following now the construction from the section 3.2, let Fq be the algebraic closure and Fr the geometric Frobenius endomorphism. Let ( \ 0 /„ 0 ... 0 0 0 .. 0 0 /„ 0 0 V / be an automorphism of degree m. We define a twisted Frobenius automorphism F = Fr o e. Fr commutes with e. Since Glnm is a connected reductive group, we have H^{Gal,Glnm) = {1} ([La]). Therefore, ((GL„m(F,))^ = GLnmi^q) since e is an inner automorphism. For the subgroups we can easily see that if H(Fq) C GLnmi^q) IS the SubgtOUp of GLnrni^q) iSOmOtphiC to GLn{¥q)'^, then H(Fq)^" = H(Fqm) ^ (GL„(Fq-n))- and H(Fq)^ = H(Fq) ^ GL„(Fqn,). We will now study the lifting of the Weil representations of GLn (F^m) (identified with H(Fq)) to (GL„(Fqm))'" (identified with H(Fq„.)) Theorem 4.5. The lifting of Q„ is Qn ® ® Proof. Follows from lemma 3.4. □ 27 3. Weil representations of SP„. Lemma 4.6. Weil representations uj^ o/5P„(Fqm) are Galois stable for Galois- stable ip. Proof. Take A^{f{x)) = f{x‘^). Let us check that A„o = g o .4^ on all the generators: f{x) = x{det a‘^)f{a 0 ‘a / a 0 = xiidet ay)f{{a = \{det a)f{{a -I ^z)')w „\(T\ V 0 A^uj f{x) = .4<,î/;(Tr(6‘" ^ x x ))/(z ) yO IJ = t/;(Tr(6^ = v(T r( 6^zz)')/(z') / 1 6 = '0(Tr(6^xz))/(x°') = w A^f{x) 0 1 ^ 0 1 ^ ' A„uj / ( i ) = rix”) = g = ?"* >= 9 " X)/(%/') < ^ ,y > y€V y^V 28 = UJ □ Theorem 4.7. Let m be odd. If ü> is the additive character of Fq and 0 ia an additive character ofFqm that is a lifting of lii, then is the Shintani lifting of u)^^, i.e. is stable under natural Galois action a and the following twisted character identity holds: rr«J(yV'(a)) = Trin^-ig) o .4,). Vj € SP„(F,.) Proof. Recall the intertwining operator : .Aa{f{x)) = f{x‘^). Notice now that the twisted character formula holds for g = 1 . Indeed, let F^m be the set of unordered pairs (x, —x), x € F q m . Select as the basis of the even functions 5^ = Sx + 6 -x and the basis of the odd functions 6 ~ = — <5_i. Lemma 4.8. . Ifm is odd then the equation x'^~^ + 1 = 0 has no solutions in Fqm. If m is even, there are (ç — I) solutions. If x satisfies the equation, its square belongs to F’a \ F* Proof. F’m contains a cyclical subgroup with 2 (ç — 1 ) elements iff m is even. □ Corollary 4.9. Ifm is odd # { x 6 F^m = x} = 0 . If m is even, # { x G F^mix*^ = x} = Ç - 1 . Now we see easily: 29 Tr(<^)(N(l)) = T r«^)(l) = dim(C*(F;)) = Tr(Q^^^(l) o .4,) = #{(r, -x) € Fqm"|x°’ = x} +#{(a:, -x ) e F,m" \ Olx‘" = -x } = ^ 4- 0 = — ^ ^ TrK,)(iV(l)) = T r« ,)(l) = dim(C-(F;)) = Tr(fi“ ^(l) o ,4<^) = #{(x, -x ) G Fqm" \ 0|z'' = x} -#{(x, -x) G Fqm" \ 0|x' = -x} = -■ ^ - 0 = ^ ^ ^ There are only at most two representations of 5P„(Fq) of dimension (or 2^^)—the two odd (even) Weil representations ([TZ]). From [Kw2] it is known that every Galois-stable representation of SPn(Fqm) is a lifting of some irre ducible representation of SP„(Fq). From the calculation above we must have then that is a lifting of either or and, correspondingly is a lifting of either or Let us calculate now the character on the elements of type: ^ In B,_ ^ B[z) = In j where P- = diag{z, 1,..1). Notice that this is an element of the subgroup isomorphic to Spi x 5pn-i- The restriction of the Weil representation to a subgroup of this kind is the tensor product of the corresponding Weil represen tations: ^n, 30 and '-^n,t/)l5pixSp„_i — (gl ^n.^UpixSpn-i = ® ^n-l,(i; ® ^n- 1,^' Let B', ^ SPi, 1 z B', = 0 1 and let us calculate both sides of the twisted character identity for B', 1 T(z) '' 0 1 / Tr(fii ,^(Bl)o.4<,) = >= ^ é{zx‘) = ^ igF^m l€F,m,x=I<' xgF, Since 0 is the base change of 0 Tr(n,,i(Bl) o .4.) = Y , = E = TrW,.,(!V(B;)) xeF,, igF, Now notice B{z) = B', x /„_i € Spi x Spn-i, therefore Tr(n.^(B(z)) oA,)= Tr(n,^(B:| o A.) x Tr(n„_, o ,4J = f - ' E MTMo:;) = TrK,^(,V(B(z))) XieFq Notice that the value of the character is different for different Weil representa tions, therefore we must have - is the Shintani lifting of and Q~ - is the ’ 71,(6 ° "'W 71,76 Shintani lifting of cj~^. □ 31 4. Weil representations of Spn, even-degree liftings, a counterexample. In this section we will consider the case of n = 1 , m = 2 and prove that the twisted character formula does not hold. By Lemma 2.5 if fZ is a lifting of u then for our choice of the twisted character formula holds for Cifi^ - + coQ~ ■ and ui for some unitary Ci, Co. Let us start by computing both sides of the twisted character identity for the ele ment 1 € Spi for even and odd Weil representation separately. Using Lemma 4.9 Tr(w+J(!V(D) = Trlc.'+JCD =dim(C+(F,)) = ^ T r(n + ^( 1 ) O A„) = #{(x. -x) e = x} -h#{(x, -x) € Fq:|x' = -x} = = 9 Tr(w,-J(iV(D) = TrK -J(l) = dim(C-(F,)) = ^ Tr(rZ"^{l) o A^) = #{{x, - x ) G Fq: \ 0|x"^ = x} -# {(x , -x ) 6 Fq: \ Olx"^ = -x } = = 0 Therefore, we must have ci = 1. Repeating the computation for the elemenet gi = 6 Spi: V “ V ) = 1 2 V;(2 x^) xeF, 32 > + Y l Y 2 ip{x^) = y 2 0 (2 x^) = 0 {(i,-r)eF 2,z-eF,} {(i,-i)eF 2 ,i- 6 F,} Tr(Q^__^(^i)o.4<,) = Y 2 < u.'{x-)d^,-i,6^ > - Y 2 < > {(i,-i)eF^2} {(i.-r)eF^2} Y 2 Qq{x^ M x^) = Y 1 »o(x^)îA(2x^) =Tr(o;i,„,)(iY(5i)), {(x,-x)€F^ 2,i2çF,} {(x.-x)eF 2,x2eF,} so C2 = 1. Finally, letting / \ — 1 — 1 -1 0 1 1 92 = 0 -1 0 -1 0 1 V We have ,\ ■^(9 2) = \ ° 1 / By Corollary 4.9 : 0>l = = ~ Y 1 = - ^ V'(2 %^) = -Tr(wi,^(W(^)) x^eFq.xgF, leF, leF, i.e. is not a lifting of 33 Remark 4.10. Since Tr(Q^^(l) o .4j.) = q and Tr(Q“^(l) o .4^^) = 0 , and are irreducible Galois-stable representations of SPi which are not liftings of irreducible representations. Remark 4.11. One can also see that is not a lifting of by computing both sides of the twisted character identity on the same elements 1 , 5 1 , 5 2 - Remark 4.12. By considering the elements ( 5 , /„_i) G Spi x Spn-i C 5p„ we obtain a counterexamaple for quadratic liftings for any n. 34 CHAPTER 5 THE WEIL REPRESENTATION OF %(Fq) 1. The brief description of Kazhdan’s construction from [Ka]. In this section we describe Kazhdan's construction of an embedding of SL2 {E) Sp 4 (Fq) where E is a three-dimensional commutative semisimple algebra over ¥q. Denote by N, T the norm and the trace maps E —>■ F,, by Be the bilinear form ( 6 1 , 6 2 ) —>■ T'(6 i6 2 ) and by 6 e/f € F*/F*^ its discriminant. The algebra È = E F, is isomorphic to F^, the product of three copies of F,. The isomorphism z : Ë = F^ is unique up to a composition with an element of .Antg-^F^ = S 3 , the symmetric group on three letters. Consider the map F^ given by 0 :2 , 0 :3 ) = (a 20 !3 ,a iQ 3 , 0 1 0 2 ) and define 9 \ È Ë hy d = o i. Then 0 is a quadratic map of È into itself which does not depend on the choice of isomorphism z : Ë = and commutes with the action of 0 = Gal{Fg : Fq). Therefore the restriction of ^ to E defines a quadratic map E E, such that 9{e) = e~^N{e) for all invertible 6 6 E. Let V = Fq © E, and for any t E E denote by Qt the quadratic form on V given by Qt{xo,x) = N{t)xl xoT{9{t)x) + T{t9{x)). Let V = Hom{V, Fq) be the dual space and W = V © V a symplectic vector space over F, where the 35 symplectic form [•, •] is defined by: [(Ai: A^), (A2 , Ao)] = A2(Ai) — A'^(A2), Ai, A2 G V, Aj, Ao € V' We will write elements v e V' as n = (xn. x. x'. xi,). xn Ç F^. x GE. xf, Ç IF^, x' G E'. Denote by 0 : V V the linear map such that d{xQ,x){yo, y) = —xoyo — T{xy) for all xq, i j q G F,, x .y G E. N o w for any t G E define: Qt : V' -> V, symmetric morphism, such that (Qt(A))(A) = 2 Qt(A) at : V -> V by Qc(xo,x) = (xq.x + Xot) Qj : V V as dual to Qt ft : y y by ft(A, A') = («.(A), (a;)-'(A' + Qt(A)) For e G E* define Xg : V V by re(xo,x,y,f/o) = (;V(e)xo, 0(e)x, ey, i/o), Xo, 1/0 G F„ x, y G E F act 5.1. There exists a homomorphism r : GL2 {E) -> GSp{W) such that: = It, t G E = r-, e G E* 0 1 V 0 -1 0 - 3-^ 0 36 Proof. See [Ka], p.134. □ Remark 5.2. Kazhdan’s proof establishes the homomorphism using the sym plectic form: 0 0 0 0 0 0 0 - 1 0 0 0 0 I 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 Jk ~ 0 - 1 0 0 0 0 0 0 0 0 - 1 0 0 0 0 0 0 0 0 - 1 0 0 0 0 \ I 0 0 0 Q Ü 0 0 In order not to create confusion we will always use our standard form J. One can switch from one form to the other by conjugating by: / \ 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 2. The split case. 37 Fact 5.1 gives an embedding r : SL^iE) Spi{¥q). Our goal will be to decompose the Weil representation of SpJ^{Fq) restricted to SL2 {E) where E = F,m and E = Starting with the case E = (F,)^, denote by H(F,) the subgroup Sp^iFq) iso morphic to SL2 {fqY. If Fr is the geometric Frobenius endomorphism, H(Fq) = Theorem 5.3. The restriction of the Weil representation uj^,of Sp^{¥q) toH(Fq) is either (tT ® 7T ® 7t) r€lrr(SL2(F,,)) xJg (case 1 ) or (tT 0 7T ® 7T) T€lrr(S£,2(F,))T?ixi(j. xig (case 2 ). (For a®* 6®^ notation see page 7.) Proof. We will use Srinivasan’s formula for the uniform part of the restricion 38 of the Weil representation of Sp^ to the dual pair Spi x SO4 ([Si]). Notice Spi = SL2 - Use embeddings Ei,Eo : SLo ^ SO^ f \ a b 0 0 c d ÜÜ Ex 0 0 d —c 0 0 -b a a 0 0 b 0 a - 6 0 Eo c d 0 —c d 0 \ c 0 0 d Now we get the uniform part of our restriction from [Si]: W6lrr(F") It is easy to check that for the uniform representations the answer given by Srinivasan’s formula coincides with the formula in our theorem. Indeed, for 0 in general position (0^ ^ 1) the Lustig characters are irreducible and equivalent for 0 and 0~U Therefore for these we have easily equivalence between (i?^^^(0))®^ and {— (each of these is counted twice with the coefficient 5 ), and (?r (g) 7T (gl tt) in the formula. The only other irreducible uniform representations are the unipotent representations 1 and St. Their contribution to the formula above is: i(i4^^(i))®^ + ^(--R^t'(i))®^ + = ( 1 + st)®^ 39 + (S t - 1)®^ + 1®^ = St®2 + 1®3 + l® ^st® \ just like in the theorem. We will denote wi = ^ (tt ® tt ® tt) + l®^St®^ (the contribu- T€Irr(SLo(F,)) .Voq tion of the irreducible uniform representetions of SIolF?) to our decomposition). Set w_i = u) — wi. The only irreducible representations of 51-2(F,) which are not uniform are the representations X qo and . The uniform projection of X qq is and the uniform projection of is From Srinivasan's formula the uniform projection of = a; - wi is g(J^^-(o!o))®^ + and therefore w_% consists of irreducible components 3i ® do ® where d, are isomorphic to x ^ or x^q- To find out the decomposition of we will first notice that the symplectic map: ^looooooo'^ 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 e = 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 00000001 0 0 0 0 0 1 0 0 40 permutes the A spaces from Fact 5.1 and therefore the conjugation by e gives a cyclical permutation of the three SLo’s. Since e is symplectic it defines an inner automorphism on Sp4. The restriction from Sp^ of any representation to r(jL g X SL2 X oLo) is stable under t and therefore stable under the cyclical permutation of the three SLo's. To finish the proof we compute the value of the character on some unipotent classes. Let / be the unit element of SL^ and a = We will compute the value of on three classes: ui — r{u,I,I), uo = r{u,u,I), U3 = r(u,ii, u) by evaluating both the uniform part of the repre sentation (which we already know) and the character of the Weil representation (which is known from [G|) on these classes. As lemma ?? will show, there are only two non-equivalent representations of SLo x SL2 x SL2 whose uniform projection is |(iî^^^(ao))®^ + and which are stable under the cyclical permutation: the ones given in the theorem. Lemma 5.4. cui(ui) = q~ - 2q, uji{u2) = q^ - q, ^ 1(^3) = 0. Proof. Notice that i?^^*(0)(u) = 1, {&){!) = ç+1 for any 6 , —E^^J{9){u) = - 1, = Ç - 1 for any 9, l(n) = 1(7) = 1, St(u) = 0, St(/) = q. Therefore: 0eIrr(F:),flVl 41 ôelrr(/i,+i),fl25ti +St®^(ui) + l®^(ui) = ^ —(9+l)(ç+l) - ^ — (9 —l) ( ç —l ) + 2ç + l = q^ — 2q M u 2) = E R i‘-H6 ){u)i 4 ‘--{e){u)i^‘-^e)(i) 56lrr(F;),flVl fl6lrr(p,+ i),fl25ii +St®=(l.2) + 1®=(U3) = ^ { , + 1) + 1 ^ ( , - 1) + , + 1 = _ Ô6lrr(F;),Ô=?‘l + E 4 t’W(“)4t’W(“)4fw (“) + i®'st®‘(«3) (?€lrr(/i,+ i),e-5!il +St®^(U3) + l®^(î^3) = ^ ^ + 0 + 1 = 0 □ Lemma 5.5. uj{ui) = 9 ", ^(uo) = 9 ^, ^(223) = ± i^q ^, where iq = I if 9 = l(m od 4) and iq = i if 9 = 3{mod 4). Proof. The calculation of the character of w is carried out in [G]. We only need to identify the u,- with the elements of Sp^. From Kazhdan’s construction ([Ka], p.l34), r(u, 1,1) takes xqCi ® eg ® 63 + x i/i ® 62 ® 63 + ® A ® 63 + xaci ® 42 ^2 ® /s + yi^i ® A ® /s + 1/2A ® ^2 ® A + ys/i ® A ® 63 + j/oA ® A ® A to (lo + Xi)ei ® 62 62 0 A + 2/1610 A 0 A + 2/2 A 0 62 0 A + 2/3A 0 A 0 63 + (2/0 + 2/1) A 0 A 0 A- Changing to our symplectic fonxi J we get: 1 1 0 0 0 0 0 0 ' 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 ri = r(n , 1. 1) = 0 0 0 0 1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 V Similarly: 1 0 1 0 0 0 0 0 ^ 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 T2 = r(l,u, 1) = 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 1 > 43 and ^ 1 0 0 1 0 0 0 0 ^ 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 f 3 = r(l, l,u) 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0-1 0 0 1 So. / 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 ui{u\) = oj(ri) = cj 0 0 0 0 1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 44 Using [G] theorem 4.9.1, part (b) with E+ as the first coordinate line: ^110 0 0 0 0 0 ^ 0 1 0 0 0 0 0 0 ^10 0 0 0 0^ 0 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 10 10 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 - 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 V / Now using part (b) with being the third coordinate line: ( 1 0 0 0 0 0 0 1 0 0 0 1 u 0 0 0 ^ 0 0 1 0 1 0 0 1 0 0 W3 = •■Jjl = 9" 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 V 0 0 0 0 0 1 ; 45 Similarly, 1 1 1 0 0 0 0 1 ^ 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1-1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 - 1 1 0 0 0 0 0 0 - 1 0 1 0 0 0 0 0 0 0 0 1 Again using [G] theorem 4.9.1, part (b) with E+ first as the first coordinate line and then as the last coordinate line we get 1 0 0 0 0 1 0 0 u{u 2) = U,'2 = 9 0 0 1 0 0 0 0 1 46 Finally, 1 1 1 1 -1111 0 10 0 -10 11 n n 1 n -110 1 0 0 0 1-1110 !jj{u 3) = toiriTors) = w 0 0 0 0 1 0 0 0 0 0 0 0-1100 0 0 0 0-1010 0 0 0 0-1 0 0 1 \ / Using [G] theorem 4.9.1, part (b) with E+ as the first coordinate line and then as the third coordinate line we get ^ 1 0 0 0 1 1 ^ 0 10 10 1 0 0 1 1 1 0 UJ{U3) = W3 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 47 Conjugating by: 1 I 0 0 0 0 1 0 - 1 / 2 0 0 0 n 1 - 1 / 2 0 0 0 0 0 0 1/2 1/2 1 0 0 0 1/2 - 1 / 2 - 1 0 0 0 - 1 / 2 1/2 - 1 we get: 1 0 0 2 0 0 0 1 0 0 -1 0 0 0 1 0 0 -1 u{U3) = CÜ3 0 0 0 1 0 0 xeF„ xeF. 0 0 0 0 1 0 0 0 0 0 0 1 □ Combining the two lemmas we get £j_i(ui) = 2g, w_i(u 2) = g, w_i = ± i^q ^. We are now ready to finish the proof of Theorem 5.3 by matching these values with the character tables of and x^q- We have (see [DM], pg 155) on I and u 9 + 1 48 x‘c„(“) = i( l + £i,\/?) x î ‘(/) = ^ Xnni^) = %( —1 + ^iq\/q) Notice also that since the uniform projection ofw_i is and the uniform projections of and are jiî^^-(Qo) and cor respondingly, each part contains exactly 4 irreducible representations. We start with computation on uy. Since the values of Xao X^o ^ do not depend on e, we must have: ^-l(U l) = XaW Xao(^)xfA^) ^ Xao(^)Xaoi^)X^qi^) +XZi^)Xaii^)Xaoi^) + XaM)xS{I)xtiiI) +Xwo(^)xS(^)XJo (^) + XwoWXwo (^)XJo U) +Xwo(“)xJJ(^)x*o(^) + Xuo(“)xS(/)xJo(^)) = 4 ( ^ ^ ) ^ ( X a o ( “ ) + X%)W + X qo (w) + X ^(^)) + 4(^y^)-(x^(u) -h X%(u) 4- X%(n) 4- Let Tiy be the number of positive e,’s z = 1, . . . 4 and ng be the number of positive e/s z = 5,.. . 8. The y/q term in the above expression is then: {2iq{q 4- l)^(rzi — 2) 4- 2iq{q — l) ‘ (n 2 — 2))\/g Since, as we already know cj_i(ui) = 2q, it does not contain a y/q term and therefore rzi = rzg = 2. In other words, there are exactly the same number of xto 49 and Xqo (and, similarly, the same number of and xZq) representations in the decomposition ofw_i_ Since is stable under cyclical permutations and has exactly 4 irreducible pieces coming from R^^'{ao), the part must be + ■KAi.rCv;.')”'. or + Similarly, the part must be + { x u n x : ; r , or 2((x-‘)®“ + ix u r ^ y Calculating now the value of the character on uo, we see that ((xi.)®’ + (x;.‘)“ (xi.)®‘)(«2) = ((x;.')®= + (xL)®'(x;.')®')("2) = i((l±i,V ^)“(l + ?)+(lT i,v^f(l + ,)+2(lTi,v/5)(l±i,y^)(l+,)) = 1 ^ , 2((x;.')®' + (xL)®')(«2 ) = i ( 2(l ± + «) + 2(1 T + ,)) = ^ ( 1 + i=,). Similarly, O ® ' + (xi')®"(xij®'(u2) = (xQ')®' + (xi„)®"(x;„‘)®‘(u2) = ? - i . 2((x;.')®" + (xij® ’)(a2) = ^ ( 1 + ij?)- Since w_i(u 2) = Q we see that we cannot have 2((%jj)®^ + (xio)®^) ^ the i4^*(û!o) component or 2((%J^)®^ + ixlo)®^) as the B^^J{uq ) component. Finally, making the calculation for u^: ((xio)®'+(x«)®"(xi.)®‘)(«3) = i((l+i,vÆ )=+3(l+l,,Æ )(l-i,v/S") = 1+ 4,V;, 50 and similarly (Xao )®^ + (Xao)®^(Xao )®^ = 1 ~ iqQ\/Q, (Xwo )®^ + (Xwo)®"(xJo)®^ = -1 + iqqy/q, and (xio)®^ + (Xwo^)®^(xio)®^ - -1 - i^qy/q. We see that the only two ways to combine these get the value ±iqQy/q are the one in the statemnet of the theorem. □ 3. T h e tw iste d case. Let F, be the algebraic closure of F, and Fr the geometric Frobenius endo morphism. We identify W = ® ® Az where .4, is a two-dimensional subspace with the standard SLoiFq) action. Just like in the proof of Theo rem 5.3, e \ W W, e(u ® v ® w) = v ® w <2) u. It is easy to see that e is a symplectic map W —^ W ^ 1 0 0 0 0 0 0 0 ^ 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 e = 00001000 0 0 0 0 0 0 1 0 00000001 0 0 0 0 0 1 0 0 51 and therefore induces an inner automorphism of Sp{W). Fr commutes with e and we can define a twisted Frobenius map on Sp{W): F = Fr o e. We have to have {{Sp^iFq))^ = Spi{¥g) since e is an inner automorphism. In other words, we get the construe tiun from section 3.2 with G = Sp^, L = 6X 2, H(F,) = r(SL2(Fg) X SL2(Fg) x SL2(Fq)) C Sp4(Fg), = r(SL2(Fg3) x SL2(Fg3) X S l 2(F,3)) C Sp4(Fgm), = r(SL2(Fg3)) C = Sp 4(F j. Theorem 5.6. The restriction of the Weil representation of G ^ = 5p 4(F,) to = SL2{Fq3) is either E T€lrr(SL2(F^3)), T2!îr«,ir?iXu,ob iCoq or E TeIrr(S£-.(Fj3)), T2!x<',!r 5sxi(j, xig Proof. After noticing that the Weil representation of G^' is the lifting of the Weil representation of G^ (Theorem 4.7), the theorem follows directly from Theorem 5.3 and construction in the section 3.2. From Lemma 3.4 the only F-stable representations are of the form tt ® ® Studying the decom position of w|gf 3 in Theorem 5.3 we notice that representations of the form a <81 /3 ® a, Q 7^ /3 or its cyclical permutations can not be F-stable, since that would imply a = o:^’’ = 0. Therefore all F-stable representations in the de composition of c j |j j i r 3 are of the form ( tt ® tt ® tt) , where tt is a Galois-stable ±1 representation of 5X2 Notice that by theorem 4.7 lifts to Xqq and Wo 52 lifts to . Therefore, by Lemma 3.5, the restriction of the Weil representation of = Spi{¥q) to = SLoiFqi) is either E :r6lrr(5£,2(F,3)), ,ît^ X uo ’ or E 7reIrr(SL2(F^3)), TT^Tr<'xig depending on the decomposition in Theorem 5.3. □ 53 CHAPTER 6 THE CASE 1. The split case. Let OslF,) be the group of automorphisms of 8-dimensional space preserving the bilinear form: ^ 0 0 0 0 1 0 0 0 ^ 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 Q 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 \ Let SOs = Og n SLg. This is a group of type D^. Consider now the Weil representation of 5pg(F,). and its restriction to the dual pair of type x 50g. Let 9 be the minimal unipotent representation of 50g. Lemma. 6. 1. The representation of S0% in the space of the functions on which are invariant under the SL 2 action is the sum of the trivial representation and 9. 54 Proof. This is Lemma 3.2 with n = 4. □ Let Qi, 0:2, 0:3,y9o be the simple positive roots (a/s being orthogonal to each other) and 7 be the highest root. We have the following embeddings of SL2 in SOs: (compare with the embeddings in the theorem 5.3 ) / \ a 6 0 0 0 0 00 cdOOO 0 00 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 = A : c d 0000 d -cOO / 0 0 0 0 -6 a 00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 V 1 0 0 0 0 0 0 0 0 a 6 0 0 0 0 0 0 c d 0 0 0 0 0 0 0 0 1 0 0 0 0 c d 0 0 0 0 1 0 0 0 0 0 0 0 0 d —c 0 0 0 0 0 0 -b a 0 0 0 0 0 0 0 0 1 00 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 n a h n n n 0 0 0 c d 0 0 0 0 la-i = h : 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 d -c 0 0 0 0 0 0 -b a 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 a 0 0 0 0 b a b 0 0 0 a 0 0 —b 0 /as = h : c d 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 —c 0 0 d 0 0 0 c 0 0 0 0 d 56 ^ d 0 00 0 c 0 0 ^ 0 d 00 - c 000 0 0 i 0 n n 0 0 0 0 0 1 0 0 0 0 I—, = h c 0 - 6 00 a 000 / 6 000 0 a00 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 V / Theorem 6.2. The restriction of 9 to Ii{SL2) x 72(51/2) x 73(57,2) x 74(57,2) can be decomposed into a sum of irreducible representations in the following way: S4 ^\li{SL2)xh{SL2)xh{SL2)xU{SL2] = TT Tr€lrr(SL2),7r^Xao, xÙq + ( , x ' + (x' +(xi.)®"(xi‘)®‘ + (x:,)®‘(x;,‘)®“ (See sections 2.1 and 2.4 for notations.) 57 Proof. Follows from Theorem 5.3 and Theorem 3.3. Indeed, after identifying SL2 X SL2/{± I} = SO4 as in the proof of Theorem 5.3, it follows from Theo rem 5.3 for 7T = or 7T = where t'- ^ I, th at = tt®^. We also have i^4(l) = l®- + l®‘St®^ = 1®2 -fSt®“ % (xL) = (xi.)®" + (x„-‘)“ H . ( , 0 = where e is either 1 or -1, depending on whether the first case or the second case in Theorem 5.3 applies. Since all representations of SL2 are self-dual, except Xao = Xâo^ xll = XJo, by Theorem 3.3 ^|/,(S£,2)x/2{SL2)x/3(Si,2)x/4(SL2) = ^ ® TT€lVV{SLn) = _1® 4+ ^ 7T®-'+ Y i -^(1®2 ^ i®lSt®l) 0 (1®2 + l®lgt®l) + (1®2 + St®2) 0 (1®2 + St®^) +((xi,)®’ + (x;.')®') ® ((xij® ‘(x;,')®‘) +((xij®" + (X:.')®') ® ((xij® ‘(x i‘)®') ^ JT®-* + l®=St®‘ + l®=St®* jreIrr(S£,2),ff#Xîo> Xwo 58 □ Corollary 6.3. The restriction of the minimal representation 9 of SpirisiFt,) to the subgroup generated by 5Lo(qi), SL^iao), SL-ziaz), 5Z-2(-7) is d\h{SLi)xh(.SL 2 )xh(.SLn)xh(SLi) = ^ TT®"* ff6lrr(S£,3),ff5£xoo. xSq ^l®3st®l+ l®2st®2 Proof. Since 0 is a unipotent representation, it is trivial on the center. □ 2. The tw isted case. In this section we will follow the construction from 3.2 to obtain results for the restriction of the minimal representation of the group of type Df to the dual pair SLziFqi) x SLoiFg). 59 Let G = Spins, Fr the geometric Frobenius endomorphism. Let e be the triality automophism and F = Froe. The group is a group of type. We will denote by 9^ the minimal unipotent representation of G^. It has dimension / .* O # \ /rr* q(q- -q--r L) ippj;. Theorem 6.4. The lifting of 03 is 6 . The restriction of 63 to the dual pair SL^iFq) X SLiiFqi) can be written as a sum of irreducible representations in the following way: ^3|s/:2(F,)x5L2(F,3) = S ^ ® 6c(:r') + St ® 1 Tr€lrr(S£.2 (F,)) Proof. It follows from [Lee] that there is an irreducible unipotent representation ^3 of G^ such that the lifting of 03 is 9, the minimal representation of G^^ = Spins{Fq3). By lemma 3.4 the F-stable represenations in the decomposition of Corollary 6.3 are: 9p = ^ 7T ® ;r ® ?r ® 7T 7r6lrr(5i,2),îr=’r^’’îr?^xto' Xwq + S t® 1 0 1 ® 1 -1 +Xwo ® x2,o ® xlo ® x!,o + xlo ® x j ® Xw,^ ® X Wo It follows from lemmas 2.3 and 2.6 that the decomposition of the restriction of 03 to SL2{Fq) X SL2{Fq3) Can be obtained by just taking the irreducible 60 representations of SL2{Fq) x SL2{Fqi) that lift to the components in the above decomposition of Op. By lemma 3.4 those are: ^3ls£,2(F,)x5£,2(F^3) ~ 0 ® bc(çi)*) + St ® 1 In particular, ^ 3( 1) = lxH-çxç^ + ^ ( < 7 + l)(ç^ + 1) + - l){q^ - 1) + 2 2 ^ 2^ + + Ç X 1 = q{q'^ - 4- 1 ) and therefore = 83 □ 61 CHAPTER 7 SYMMETRIC LIFTINGS 1. Liftings and induction: technical lemmas. Let G be a linear algebraic group over a finite field F,. Let B be a subgroup of G and a be an automorphism of G of finite order m stabilizing B. We define the norm map N to he N : g We will say that E G is cr-conjugate to ^2 if there exists an element x 6 G such that = x‘^g2X~^. It is clear that cr-conjugacy is an equivalence relation. We will denote by ZQ{g) ‘‘twisted centralizer” of ^ in G i.e. the set of x € G such th at x‘^gx~^ = g. We will denote by Zo{g) the usual centralizer of g in G i.e. the set of x E G such that xgx~^ = g. Lemma 7.1. If gi is a-conjugate to g2 then N{gi) is conjugate to N{g2)- Con versely, for any gi E G, if N{gi) is conjugate to an element §2 ^ 0 then gi is a-conjugate to some g2 such that N{g2) = §2- Proof. If x ‘^^2 = gi^ then by taking cr‘ of both sides we get = 5 f ‘x°'‘, so ^Y (^ 2 ) = JV(gi)x. Conversely, if xg2 = N{gi)x then 52 = will satisfy the conditions of the lemma. □ 62 We will say that a representation tt of G (or B) is a lifting of the representation 7T of G '(B '') if (a) TT is cr-stable. If V is the representation space for tt then we can define a new representation tt°' on the same space by Tt‘^(g)v = Tr{g‘^)v. We will say that tt is cr-stable if there is an intertwining operator such that AffOg^^ = go Aff for all g. (b) TT satisfies the twisted character formula: if N{g) is conjugate to some element N'{g) E then tr(7r(iV'(p))) = tx{TT{g) o A^), otherwise tr(ir(^) o .4^) = 0. We will say that a representation if of G (or B) is a weak lifting of the representation tt of G ‘^(B‘^) if ft is cr-stable and the twisted character formula is satisfied for all g such that N{g) is a regular semisimple element of G. Let i/> be a representaion of B*^, 0 its lifting to B, tt = and ft = Ind^Tp. Lemma 7.2. ft is a-stable. Proof. Let U be the representation space for tp and U be the representation space for ip. Let A : Ü —> be an intertwining operator for ip and Tp^. Let V be the space of functions on G®' with values in U satisfying f{xb) = Tp{b)f{x) for any b € B'^, i.e. the representation space for tt. Let V be the space of 63 functions on G with values in Ü satisfying f{xb) = ï){b)f{x) for any 6 € B, i.e. the representation space for tt. We will define the intertwining operator .4i for 7T and 7T°' by for any / E V. Since a is an automorphism and A is a linear automorphism, 4i is an injective linear operator. 4% is stabilizes V, since for any b Ç. B: (4i/)M = 4(/(z'6')) = 4(^(6n/(zl) = t^(6)4(/(z')) = V(6)(4./)(z), so it is an automorphism of V'. .A.nd it is an intertwining operator since; (/ll O 7T(/))/(%) = 4X(7T(f ) . /)W) = 4(/(g'%')) = 4(/( W)) = = 7r(^)(4i/)(x) □ Theorem 7.3. Let Og be the set of all elements a-conjugate to g and N{Og) be the conjugacy class (in Q) of N{g). Then the twisted character formula for TT{g) and TT{g) is satisfied whenever the following conditions hold: (i) If two elements o /G ° 'fl N{Og) are conjugate in G, they are conjugate in G*^. If two elements o/B'^ nN{Og) are conjugate in B , they are conjugate in B<^. (ii) For a„j, 6 e B n 0 „ 64 In particular, tt is a lifting of tî if conditions (i), (ii) are satisfied for all g and a weak lifting of ir if conditions (i), (ii) are satisfied for all g such that N{g) is regular semisimple. Proof. First suppose N{g) is not conjugate to any element of Then by Lemma 7.1 it can only be cr-conjugate to such elements 6 of B that iY(fa) is not conjugate to any element of Since ip is a lifting of ip, tr(fr(g) o A^) = 0. Now let g be an element of G such that N{g) is conjugate to iV'(^) = i-o7V(^)xo an element of G'^. Define by h the action of B on G x B given by h{bi){x, b) = (6ix, This action preserves the sets of pairs (x, 6) € G x B such that x'^g = bx. Indeed, if x'^g = bx then b1bb(^ x bix = b^bx = b^x'^g = {bixYg. Let Gg = {{x,b) € G X B\x‘^g = bx}/h. Similarly, h‘' will be the action of B*^ on G*^ X B°^ given by h‘^{bi){x,b) = {bix,bibbf^) and G^/(g) = {(x. 6) E G X B|xiV'( 5 ) = bx}/h‘^. From the definition of induction we have a well-known trace formula: tr(7r(7V'(5))) = tr(tA(6) The same calculation gives us: tr(7r(^) o ,4i) = Y ] tr(^(6) o .4) (i ,ù) € G , Notice that we can always choose our representatives (x, b) in such a way that N{b) 6 B*^. By Lemma 7.1 and (i) N'{g) is conjugate to N{b) in G°^ whenever 65 g is (T-conjugate to b. Since ^ is a lifting of ^ Theorem 7.3 will be proved if we can prove the next lemma Lemma 7.4. For a fixed g, N'{g), and b, #{(x, 6) € G^} = H^{{x,N{b)) € Proof. From Lemma 7.1 we know that the first set is empty iff the second set is empty. Whenever they are non-empty, #{(r,6) 6 Gg} = } |||^ = = #{(z,,V(6))EG^,(,)}by (ii). O □ Finally, let us state a few simple facts in case of commutative group G. F a c t 7.5. Let G be a commutative finite group. Then the norm map G is a homomorphism whose image N{G) is a subgroup inside G^. For any character -ijj of N{G) there is a lifting xb defined by w{g) = tp{N{g)). 2. Liftings S02n+i -> GLsn+i» In this section we let G = GL 2n+i(F,) and B be the subgroup of upper-triangular matrices. Let J E G be the element ( \ 0 0 1 0 1 0 J = 1 0 66 Denote by a the automorphism of G defined by cr{g) = J V . It is easy to see that a stabilizes B . Now we will start by considering the split torus T consisting of diagonal matri ces. A (T-fixed element of T is always of the form a = diag{ai,..., a„, 1, , af^). Lemma 7.6. Let g = diag{ai), i = 1 ,..., 2n -I- 1 be an element of T such that its norm N{g) is a regular semisimple element ofG. Then g satisfies the conditions (i), (ii) of Theorem 7.3. Proof, (i) If two elements of are conjugate in GL^n+i, they have the same eigenvalues. Therefore, if they are conjugate in GL^n+i, they are conjugate in 02n+i- Two distinct elements of T can not be conjugate in B since for any upper-triangular unipotent u. is unipotent iff ti = t?- (ii) If N{g) G T is a regular semisimple element of G , its centralizer in G is T . By lemma 7.1, Zg(g) C ZG{N{g)) = T. Since T is abelian, whenever = t we must have f[tÿ^ = 1. So, Zq(^) = T®’ = ZG<'{N{g)), and Z^ig) = T ' = ZB^{N{g)) hence (ii). □ Lemma 7.7. If N[g) € is regular in G then g E T . Proof If N{g) = diag{a\,..., a„, 1, \ ..., of ^) then since ^{g)9~^ = J ^9~^J we get the following relations for the matrix elements of 67 ^ii.9 )ij — {9 )2n+2-j,2n+2-i Therefore (^2n+2-jO‘i{9 )ij ~ '^2n+2-j{9 )2n+2-j,2n+2-i ~ {9 )ij Since N{g) is regular a2„+2-jQt 7^ 1 whenever i ^ j, so {g~'-)ij = 0 whenever i # j, hence g is diagonal. □ n Let 0 be a character of 0(diag(ai,..., a„, 1, a~ \..., af^)) = n(^i(“t))- i=l Define 6 as the character of T given by 9(g) = 0(N(g)), i.e. the lifting of 9 to T. Theorem 7.8. . R^(9) is a weak lifting of R^^{9). Proof. Let g be an element of G such that N{g) is a regular semisimple element of G. If N{g) is not split (i.e. not conjugate to an element of T), then both sides of the twisted character formula are 0. Indeed, in this case :V'(^) is not split, so the left-hand side is zero. .A.nd g is not cr-conjugate to an element of T by Lemma 7.1, so the right-hand side of the formula is zero by the proof of Theorem 7.3 (see the induction trace formula). If N{g) is split (i.e. conjugate to an element of T) then, since N is surjective on T , g is cr-conjugate to an element of T by Lemma 7.1 and Lemma 7.7. By Lemma 7.6 g satisfies the conditions of Theorem 7.3. □ 68 3. GLz ex am p le. In this section we let G = GLz{F^) and B be the subgroup of upper-triangular matrices. Let J G G be the element ^001^ J = 0 1 0 0 Oy Denote by a the automorphism of G defined by a{g) = Jg It is easy to see that a stabilizes B . In this section we give the full classification of a-conjugacy classes of GLz. Our main tool is Lemma 7.1: we will study the a-conjugacy classes of elements of GLz by calculating the usual conjugacy classes of their norms. For most classes these conjugacy classes intersect SOz- We will parametrize most cr-conjugacy classes of GLz by the 5 0 3 -conjugacy class of the norm, and call the a-conjugacy classes whose norm does not intersect SOz exceptional. Notations: G = GLz, G ‘^ = O3 , B, T, U, T' are correspondingly the standard (upper-triangular) Borel subgroup, the split (diagonal) torus, the unipotent (upper-triangular) subgroup and an anisotropic torus (a detailed description will be given in part (e)) of SOz, Zg,Tg are the center, the split (diagonal) torus in GLz, Tq is a maximal torus of GLz isomorphic to x F*. We will fix an element of F, that is not a square and call it t. Lemma 7.9. a-conjugation preserves the square class of thedeterminant. For 69 every a-conjugacy class in GLz with determinants which are squares there is one of the same size whose determinants which are not squares, obtained by multiplication by diag{t,t,t). Proof. Clear. □ Now let us list all the cr-conjugacy class in GLz with determinants which are squares. (a) (Trivial.) Representative: ^ 1 0 0 ^ g = \ 1 0 0 -^{g) = n 1 0 10 0 1 / Since g belongs to the center, the stabilizers are: Zc{N{g)) = G, Zc<>{N{g)) C , Stable {g) = G '. (b) (M inus one.) Representative: 0 1 0 0 0 1 70 / -1 0 0 ^ N{g) = 0 1 0 ^ 0 0 - I / Stabilizers: Zc{N[g)) = GLo x G 'li, ZG<'[N[g)) = T x {±1}, Stab^lg) SL2 X {±1}. (c) (Regular unipotent.) Representative; g= 0 1 - 1 0 0 1 / ^ 1 2 -2 ^ N{g) = 0 1 - 2 ^ 0 0 1 Stabilizers: ZG{N{g)) = Zc x U, Zc‘'{N{g)) = U x {±1}, Stab^ig) Ux{±l}. (d) (Regular semisimple split.) Representative: g = 71 where a EFq, ± 1. ^ a 0 0 ^ = Stabilizers; Zc{N{g)) = Tc, Zc’{N{g)) = T x {±1}, Sta 6^(p) = T x {± 1}. There are ^ classes of this type (a e F,, a 7^ ± 1, a = a~^). (e) (Regular semisimple anisotropic.) Let M(a, 6, c) be the matrix: ^ a - 2bt 2ct ^ M{a,b,c) = —6 a — c 2bt \ 2i ^ ° The determinant of M{a,b,c) is d = ((a — c)~ —4b'^t)(a + c). its eigenvalues are (o+c), a-c+26>/t and a-c+26v^. We have M(ai,bi,ci)M(a2, 6 2 , c^) = M (oiQj + 26162^"bC 1C2 , bi(û2 — C2 ) + 6 2 (0 1 ~Cl),giC 2 — 2 6 1 6 2 ^"bciUa)- There fore the set of all M(a, b, c) with d 7 ^ 0 makes up a maximal torus in G that is not split. Lemma 7.10. The set of all M{a, b, c) with a + c = d = 1 makes up an anisotropic maximal torus T' in SO^. The norm map defines a surjective map between the tori. Proof. M{a,b,cy = Suppose M(a, 6 ,c) € G°'. Since det{M{a,b,cY) = d”L we must have d = 1 . If 6 = 0 , we must 72 have a = = a{a — c) and similarly c = c(c — a). Since we can’t have a = 6 = c = 0 (it would imply d = 0), we must have (a — c)^ = 1, so a 4- c = ~ If 6 # 0, 6 = ■■"j'- implies a + c = 1. It is easy to check that a -r c = d = i implies M{a, b, c)~ = M{a, b, c). It is easy to see that N{M{a,b,c)) = Since is commutative, N{Tq) C T'. The cardinality of T ' is = q+1. To prove that the norm map is surjective we will simply show there are Ç + 1 elements in its image. If (a - c) = 0 0-10 0 0 If 6 = 0 then ^ 1 0 0 ^ N{M{a, 6, c)) = 0 1 0 V 0 0 1 For (a - c), 6 7 ^ 0 the image of the map (a, b, c) contains all ^ elements of F, that are not squares. The image of the map (a, b, c) —> 1 — then contains ^ non-zero elements and so does (a, 6, c) -> (1 — = l2z£lî[5±£l_ Changing the sign of b does not affect d but it changes the sign of Therefore, the image of the norm map therefore contains 14-14-2- ^ = g 4-1 distinct elements and therefore i IS surjecive. □ 73 The centralizer of a regular element of the torus (in SOz or GLz) is the torus itself. Since the torus is commutative the cr-stabilizer of any element of Tq with (a — c), 6 7 ^ 0 is the set of the elements of the torus fixed by cr i.e. T' X { il} . There are classes of this type—the uuuiber uf regular conjugacy classes of T'. (f) (Exceptional.) Representative: 9 = 0 1 0 0 0 1 / -10 2 •^{g) = 0 1 0 0 0-1 - ( 0 1 ^ 9 = 0 1 0 V 0 0 t J ^ - 1 0 2/t ^ 0 1 0 0 0-1 N{g) is a unipotent element of GLz that is not regular and therefore is not conjugate to anything in SOz, which has only three unipotent classes: one trivial and two regular unipotent classes. 74 The centralizer of N{g) in G consists of matrices of type: / a 0 . c \ u = 0 6 0 0 0 a For those u’s we have I \ u‘^gu ^ = \ It follows th at u‘^gu ^ = p iff a = ±1,6 = ±1 and that there are two different classes of this type, one containing g and another containing g' By lemma 7.1 the a-conjugacy classes listed above are distinct. If we sum the number of elements of all those classes we get: SO our list is indeed a complete classification of cr-conjugacy classes of G. Remark 7.11. One can easily see that condition (ii) of theorem 7.3 is not satisfied for the class “minus one” . It follows from the proof of that theorem that the twisted character identity is not satisfied on that class. 75 CHAPTER 8 THE SYMPLECTIC CORRESPONDENCE. 1. Let ^ 0 0 0 1 ^ 0 0 10 J = 0-100 V -10 0 0/ Let G = Sp2[^q) be the symplectic group corresponding to J, i.e. the group of all linear transformations of the space stabilizing J. Let W be its Weyl group with respect to the split torus T ,and U the subgroup of upper-triangular unipotent elements. Let L be the Levi subgroup of G consisting of the elements of type: \ / a b 0 0 c d 0 0 a —6 0 0 ad—be ad—be n n ~c—c à ad—be ad—be / 76 L is clearly isomorphic to GL2. We will denote by P = LU the corresponding parabolic group. Let S be the subgroup of elements of type: ^ 1 0 0 0 ^ 0 a 6 0 0 c d 0 V 0 0 0 1 / (we must have ad — be = 1) and H be the subgroup of elements of type: 0 1 0 y 0 0 1 -X 0 0 0 1 Let Z be the center of H. Let ^ 1, Qq be a multiplicative character of F’ and let Ç be the representation of L ~ GL2 given by ^{g) = Ç(det(y)). In this section we study the restriction of 7T = to the semidirect product SH. The following is a well-known fact (see e.g. [G]). F act 8.1. For any nontrivial additive character'll; o /Z ~ F, there is exactly one irreducible representation o /H such that Res^{p^) = ip o Id. This repre sentation has a unique extention to the semidirect product S H : the metaplectic representation 9^. 77 Decompose the restriction 7 t|sh as a sum of irreducibles with respect to Z: •^^^s h (^) — Ci/Ji where Q is such that R es^{Q ) = ib o Id. As a corollary of Fact 1, it follows that we have: Resf^{îT) = ® 7T^ + 7Ti, where ir^ is a representation of SH that is trivial on H (in the following we will just treat tt^ as a representation of S) and tti is a representation of SH that is trivial on Z . Consider now the Bruhat decomposition of G: hi 6Wl\W where Wi, = VF D L, w is the representative of w of minimal length, and Uu, = U n where wq is the Weyl element of maximal length. More explicitly, the w's are: / \ 10 0 0 0 10 0 Wi 0 0 10 0 0 0 1 78 ^ 1 0 0 0 ^ 0 0 1 0 W o = 0-100 \ o 0 0 1 y ^010 0 ^ 0 0 0 -1 Wz = 10 0 0 ^001 0 ^ 0 0 10^ 0 0 0 1 = -10 0 0 V 0-100 U i = 1 , U 2 = C^2e2 ) ^ 3 = ^< 2ei,ei-e2>? U 4 = C/<2e2,ei+e2,2ei>- Our representation space is the space of functions on G satisfying f{pg) — ^{p)f{g) for every p € P, with right G-action. The above Bruhat decomposition renders a description of this space as a direct sum C(Ui) © C(U 2 ) © C(U 3 ) © C(U 4). Since Z = U 2ei, it commutes with f/ 2ej, so the Z-action on the first 2 subspaces is trivial. Z acts on the last two subspaces by multiplication in Uzd component. Fix a nontrivial tjj. Z acts transitively on U2ei, so the orbits of Z on U 3 = U<2ei,ei-e2> &re parametrized by the Ci — 62 component which is fixed by Z action. Therefore, we can select as the basis for the Q-subspace of C(U 3 ) 79 the “5-functions” in second variable, i.e. functions fa, a 6 F, such that /a (u 2ei,Uei-ei) = 0 if f a and /a(u 2epfl) = ^(u 2ei)- Similarly we’ll let /a,i be a function on U4 = U<2eî,ei+e2,2ei> such that = 0 if ( “ 2621 ^61+62) A (“ }“) 3,nd “2ei) — FuiiCtioUS /a alld /a,ft fur III a basis of Ci/i- Adding up the dimensions of Q ’s we get: L em m a 8.2. Res^{TT) = (q + l)'^l + '^ q { q + l) ^ >Plàl Next we want to describe the action of the elements of type: 0 a 0 0 dfi — 0 0 n -i 0 0 0 0 1 a 7 ^ ± 1 , on the space Q, 0 7 ^ 1 . L e m m a 8.3. tr(7r(da))lc^ =Ç(a)-i-Ç(a ) Proof, da commutes with Z, and its right action on Q has only two basis eigenfunctions fo and /o,o- Therefore, tr(7r(da))lc,, = iiw^daW^^) + ^{WidaWf^) 80 f a 0 0 0 \ f 0 0 0 \ 0 1 0 0 0 1 0 0 = ( 0 0 1 0 + i 0 0 1 0 = Theorem 8.4, = R^(ao^), where Oq is the quadratic character. Proof. It follows from Lemma 8.2 th at the dimension of is g + 1. From Fact 8.1, ÇxD = 9^® so tr{Q) = tr{-K^)tr{9^). The character of the Weil representation of SL^ is well known: we have tr{9^){da) = oo(o): tr{9^){u) = ± \/îç (the signs depend on q and 0). Therefore, from Lemma 8.3: tr{Tv^{da)) = O!o{a){^{a) + ^ ( a “ ^)), tr(7r^(u)) = 1. The rest of the theorem follows from the character table of SLo (see e.g. [DM], pg 155). Indeed, if Ç # l,a o , B^(ao^) is the only representation of S of dimension q+ 1 such that tr{îT^{da)) = ao(a)(Ç(a) +^(a~^)), so = ft^{aoO (and it is irreducible). □ 2. Let ^2 9^ 1, Wo be a character of F’m such that ^ 2 (2^’"^^) = 1- Then I 2 = /?t„(^ 2) is trivial on the center. Let K2 = (&)- Jf f 1, It follows from lemma 4.9 [S2] that 7T2 = -X e(^ 2 ) + X7 («f2), where —Xe(^ 2 ) is a representation of dimension {q - l)(g^ -h 1) and x?(^ 2 ) is a representation of dimension q{q - l)(g^ + 1). In this section we will study the restriction of —Xe(^ 2 ) to SH . 81 Fix ^2 and set % = —X 6 (^2)- Just like in the first section, we use Fact 8.1 to obtain decomposition: where X 0 is a representation of SH that is trivial on H and Xi is a representation of SH that is trivial on Z . First, we consider elements of type Ci{i), C2 i(i), C22(i), --l2 i(i), and .492 (using Srinivasan’s notations): / 1 0 0 0 0 r?' 0 0 Ci(z) = 0 0 0 0 0 0 1 1 0 0 1 0 f?' 0 0 C2i{i) = 0 0 T/-‘ 0 0 0 0 1 I 0 0 7 0 77* 0 0 <^22(î) = 0 0 77-* 0 0 0 0 1 82 10 0 1 0 10 0 .421 = 0 0 10 0 0 0 1 10 0 7 0 10 0 ,422 — 0 0 10 V 0 0 0 1 / where vj 6 F*2 is q + I’st root of unity and 7 is not a square. It follows from [S2| that x(Ci(f)) = (9 - 1)(6(9‘) + 6(9"‘)), xiC2i{i)) = x{C22{i)) = - ( 2 ( f ) - (^2 (9 “*), and x(' 4 2 i) = x(- 4 2 2 ) = 9—1- From the last value one easily deduces the decomposition of the restriction to Z; Res^ix) = (9 ^ - 1)1 4- 5Z9(9 - 1)0 ipjii In particular, for any 0 # 1, since dim{d^)'= 9 , dim{xxii) = 9 — 1. The elements of type Cl are exactly the regular elements of anisotropic torus Tu, C S. Now consider the restriction of x to Z x Tu, (the two clearly commute): The values of the character above allows to compute the character easily: OL^{rt) =< ^ 0Q,^(9‘),0 > 0 83 = -[^(-6(77') -6(/7 '))^(z) + (g - +6W '))^(i)] ^ z#0 = -[J^(-6(^‘) - + g(6(^') + ^2(^"‘))^(i)] ^ z = - ■ ?(6(^‘) + '^2(r7"‘))0(l) = + S2(^"‘) # Since = 9^ ® Xt 0,/,(??‘) = (-1 )'"^ \ Xvhu, = - « 0 (^2 + ‘^2’^)- There is only one representation of S of dimension (9 - 1) whose restriction to T„, is -Oio{^2 + ^2 ^), that is -/? f^ (a o ^ 2 ) i.e. we have proven; Theorem 8.5. Xth = ~R t (0:0^2 ). 84 BIBLIOGRAPHY [A] Adamson. Introduction to field theory. University mathematical texts, 1964 [DM] Digne, Michel. Representations of finite group of Lie type. London Mathemat ical Society Student Texts 21, 1991. [G] Gerardin. Weil representation associated to finite fields. J. of Algebra 46 (1977), 54-101. [Gan] Gan, Wee Teck. Exceptional Howe correspondences over finite fields. Compo- sitio M athem atica 118 (1999), 323-344. [GP] Geek, Pfeiffer. Unipotent characters of the Chevalley groups D.;(g), q odd. Manuscripta Math. 76 (1992), 281-304. [Ka] Kazhdan. Minimal representation of D^. Operator algebras, unitary represen tations, enveloping algebras and invariant theory. Actes du colloque en l’honneur de Jacques Dixmier, 1990, 125-158. [Kwl] Kawanaka. On the irreducible characters of the finite unitary groups. J. Math. Soc. Japan 29 (1977), 425-450. [Kw2] Kawanaka. Liftings of irreducible characters of finite classical groups. J. Fac. Sci. Univ. Tokyo Sec. lA Math. 28(1982), 851-861 and 30 (1983), 499-516. [La] Lang. Algebraic groups over finite fields. Amer. J. Math. 78(1956), 555-563. [Lee] Lee. Liftings of irreducible characters of ^D^{q). Ph. D. dissertation, Yale, 1986. [51] Srinivasan. Weil representations of finite eiassical groups. Inventiones Math. 51 (1979), 143-153. [52] Srinivasan. The finite symplectic group 5p(4, ç). Trans. Amer. Math. Soc. 131 (1968), 488-525. 85 [Sh] Shintani. Two remarks on irreducible characters of finite general linear groups. J. Math. Soc. Japan 28 (1976), no. 2, 396-414. [Sp] Spaltenstein. Caractères unipotents de ^£> 4 (F,). Comment. Math. Helvetici 57 (1982), 676-691. [TZ] Tiep, Zalesskii. Minimal characters of the finite classical groups. Comm. .A.lge- bra 24 (1996), no. 6, 2093-2167. [Wl] Weyl,H. Theorie der Darstellung kontinuieflicher halb-einfacher Gruppen durch lineare Transformationen I. Math. Zeitschrift 23 (1925), 271-309. [W2] Weyl,H. Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen II. Math. Zeitschrift 24 (1926a), 328-376. [W3] Weyl,H. Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen III. Math. Zeitschrift 23 (1926b), 377-395. 86