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Construction of Series of Degenerate Representations for Gsp(2) and Pgl(N)

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Construction of Series of Degenerate Representations for Gsp(2) and Pgl(N) CONSTRUCTION OF SERIES OF DEGENERATE REPRESENTATIONS FOR GSP(2) AND PGL(N) DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Martin Bozhidarov Nikolov, M.Sc. ***** The Ohio State University 2008 Dissertation Committee: Approved by Professor Yuval Flicker, Advisor Professor James Cogdell Advisor Professor Wenzhi Luo Graduate Program in Mathematics ABSTRACT The relative trace formula is a tool for studying automorphic representations on symmetric spaces. In this work two relative trace formulas are studied: One for GSp(2) relative to a form of SO(4) and one for PGL(n) relative to GL(n − 1). The first trace formula is used to study the Saito-Kurokawa lifting of automor- phic representations from PGL(2) to PGSp(2), thus characterising the image as the representations with a nonzero period for the special orthogonal group SO(4, E/F ) associated to a quadratic extension E of the base field F , and a nonzero Fourier coefficient for a generic character of the unipotent radical of the Siegel parabolic subgroup. The second trace formula is used to study the representations π of PGL(n) which have a nonzero linear form invariant under the Levi subgroup GL(n−1), and a certain nonvanishing degenerate Fourier coefficient, over local and global fields. The result 0 is that these π are of the form I(1n−2, π ), namely normalizedly induced from the parabolic subgroup of type (n − 2, 2), trivial representation on the GL(n − 2)-factor and cuspidal π0 on the PGL(2)-factor. The analysis of the formula is intricate our main achievement is rigorously establishing the convergence of the formula. New in this case is that none of the representations of PGL(n) involved are cuspidal, they rather occur in the continuous spectrum, but discretely in our formula. ii To Lore Thaler iii ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor Professor Yuval Flicker for allowing me to work on this thesis, and for all of his time, encouragement, concern, and patience with me. I have appreciated the many conversations and discussions we have had, both mathematical and otherwise. I would also like to thank all my committee members, my advisor Professor Yuval Flicker, Professor James Cogdell, and Professor Wenzhi Luo, for agreeing to serve on my committee. I would like to thank the Department of Mathematics for their support expressed through several teaching and research fellowships that I enjoyed. iv VITA 2002-Present . Ph.D. Candidate, The Ohio State Univer- sity 2002 . M.Sc. Mathematics, Sofia University 2001-2002 . Researcher, Bulgarian Academy of Sci- ences, Institute of Mathematics and In- formatics, 2000-2001 . University of Electro Communications, Tokyo Japan 1997-2002 . Sofia University “St. Kliment Ohirdski” FIELDS OF STUDY Major Field: Mathematics Specialization: Number Theory v TABLE OF CONTENTS Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... v CHAPTER PAGE 1 Introduction . 1 2 Cusp Forms on GSp(2) with SO(4)-Periods . 10 2.1 Definitions and Notations . 10 2.2 The group . 20 2.3 Geometric side . 25 2.4 Asymptotes . 32 2.5 Matching . 39 2.6 Corresponding . 45 2.7 Spectral analysis . 51 2.8 Computing the integrals of truncated Eisenstein series . 57 2.9 The contribution from the maximal parabolics . 64 2.10 The contribution from the minimal parabolic . 70 2.11 Comparison . 79 2.12 Converse . 89 2.13 Split cycles . 93 2.14 Invariance of Fourier coefficients of cyclic automorphic forms 96 3 A Trace Formula for the Symmetric Space PGL(n)/ GL(n − 1) . 109 3.1 Notations . 109 3.2 Truncation . 115 vi 3.3 The Main Distribution . 126 3.4 Regularized Periods of Eisenstein series . 128 3.5 Periods of Truncated Eisenstein Series . 136 3.6 Limit Formula for J T (f) .................. 137 3.7 An application . 146 Bibliography . 154 vii CHAPTER 1 INTRODUCTION The relative trace formula is a tool to study period integrals of automorphic forms and their relation to the principle of functoriality. For a cusp form φ on the group G(A) of ad`elepoints on a connected reductive group G over a number field F , these integrals are of the form Z φ(h)dh. S(F )\S(A) Here S is the group of fixed points of an involution of G over F . A representation π of G(A) is called S-cyclic, or just cyclic if S is understood, if there is a φ in its representation space whose period integral is nonzero. It is expected that the cyclic representations are precisely those that are the functorial transfer from another group G0. For more on the relative trace formula see [Lap1]. The present work consists of the study of to cases of the relative trace formula and their applications. Namely, the case first case studies the pair G = GSp(2) and S = SO(4), the second deals with G = GL(n) and S = GL(n − 1). Chapter 2 concerns the determination of cusp forms on an ad´elegroup G(A) whose period − namely integral − over a closed subspace (“cycle”) arising from a subgroup H(A), is nonzero. Such forms contribute to the cohomology of the symmetric space G/H, and play a role in lifting automorphic forms to G(A) from another group 1 G0(A). Many advances in these studies so far have been made by means of the theory of the Weil representation [We1]; see Waldspurger [Wa1], Howe and Piatetski-Shapiro [HPS], [PS1], Kudla-Rallis [KR], Oda [O]. This technique has the advantage − in addition to early maturity − of constructing cusp forms on G(A) directly from such forms on G0(A). Miraculously, the cusp forms on G(A) so obtained happen to have nonzero H(A)-periods. Our approach is based on a more naive and direct method, focusing more on the representation and its properties rather than on its particular realization. Thus we integrate both the spectral and the geometric expressions for the kernel Kf (x, y) of the convolution operator on the space of cusp form on G(A), over the cycle associated with H(A). If both variables x and y are integrated over the cycle, one obtains a bi-period summation formula, involving the periods of the automorphic forms over the cycles (in Jacquet [J1], and later in [FH], this is named a “relative trace” formula, although there are no traces in that formula). The case where G(A) is G0(A) × G0(A) and H(A) is G0(A) embedded diagonally, coincides with that of the usual trace formula on G0(A); this case is also referred to as the “group case”. If the second variable is integrated over a unipotent radical of a parabolic subgroup against an additive character, the Fourier summation formula − involving Fourier co- efficients of the automorphic forms − is obtained (see Jacquet [J2], where the formula is again named “relative trace” formula, and [F5], [F6] and [F4]). Only the cusp forms on G(A) with nonzero H(A)-periods survive on the spectral side. The geometric side is compared with the geometric side of an analogous summation formula on G0(A), for matching test functions on G(A) and G0(A). The resulting identity of spectral 2 sides can be used to establish lifting from G0(A) to G(A). In summary, both the bi-period and the Fourier summation formulae are special instances of the “relative trace” formula. The derivation of the Fourier summation formula, and the characterization of the relevant orbital integrals, lead to deep chapters in global, and local, harmonic analy- sis, especially of symmetric spaces; cf., e.g., [OM], [BS]. The analytic problems thus raised might even be considered to be of greater importance than the motivating final applications in representation theory. Conversely, these applications justify some of the work which has been done on symmetric spaces. One expects to derive identi- ties of (bi-period or) Fourier-Period distributions intrinsically related to the (local) representation in question. These distributions are analogous to Harish-Chandra’s characters, which play a key role in studies of automorphic forms by means of the Selberg trace formula. To fully harvest the (bi-period, or) Fourier-Period summation formulae, one would need an analogue of the orthogonality relations of characters, due to Harish-Chandra and Kazhdan [K], for these local distributions. The summation formula has been slow to evolve possibly since its application is based on panoply of techniques, substantially different from each other. Yet it could be a source of inspiration in various branches of contemporary harmonic analysis. In chapter 2 we study the case of automorphic forms on G = GSp(2) and the cycle H = SO(4, E/F ) associated to a quadratic separable extension E of the global base field F . More precisely, G is the algebraic group of g ∈ GL(4) with gJ tg = λJ, λ = 0 w 0 1 × λ(g) ∈ GL(1),J = ( −w 0 ) , w = ( 1 0 ); A is the ring of ad´eles of F . We fix θ ∈ F which is not a square in F , put θ = ( 0 1 ) and Θ = θ1 0 , and let H be the centralizer 1 θ 0 0 θ1 3 of Θ in G. Denote by Z(F ),Z(A) the center of G(F ) and G(A) respectively. Also IX x y consider the unipotent radical N = {n = ( 0 I ); X = ( z x )} of the Siegel parabolic subgroup P of type (2,2) of G, a complex valued nontrivial character ψ of the additive 0 1 group A/F , and the character ψθ(n) = ψ(trTX) = ψ(z − θy),T = ( −θ 0 ).
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