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Proquest Dissertations INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. Bell & Howell Information and Learning 300 North Zeeb Road, Ann Artxar, Ml 48106-1346 USA 800-521-0600 UMI' 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 <t of G defined by a{g) = Jg~^^J where J is the unit anti­ diagonal. 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
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