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Ann Arbor, MI 48106 L-function for the Standard Tensor Product Representation of GSp{2) X GSp{2) dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Dihua Jiang, B.S., M.S. ***** The Ohio State University 1994 Dissertation Committee: Prof. Stephen Rallis Approved by Prof. Avner Ash Prof. David Ginzburg Adviser Department of Mathematics to my wife Ying and my son Fan n A cknowledgements I wish to express my gratitude and indebtedness to my advisor and my teacher, Professor Stephen Rallis, for his bringing me into the mathematics area of Auto- morphic Representations and L-functions, for his suggestion and formulation of the research project for my Ph. D. Dissertation, and for sharing his ideas and insights during my implementation of the project. His guidance, advice, trust, patience, and supports are greatly appreciated. I am grateful to Prof. D. Ginzburg for helpful conversations during my working with the project and for his careful reading this dissertation. I would like to thank Prof. A. Ash, Prof. C. Rader, and Prof. R. Stanton for their frequent help. My thanks also go to Prof. A. Ash, Prof. .1. Hsia, Prof. M. Rosen (Brown University), Prof. K. Rubin, Prof. A. Silverberg, and Prof. R. Stanton for their wonderful topic courses on Number Theory, Algebraic Geometry, and Representation Theory. Finally, I would like to thank my family for their love, patience, and understanding throughout this time. m V i t a Oct 17, 1958 ................ Born in Wenzhou, P. R. China. 1982 ...............................B.Sc., Zhejiang Normal University, P. R. China. 1987 ...............................M.Sc., Department of Mathematics, East China Normal University, Shanghai, P. R. China. 1989-1994 .................... Graduate Teaching Associate, Department of Mathematics, The Ohio State University, Columbus, Ohio. R epresentation T h e o r y a n d N u m b e r T h e o r y Major Field: Mathematics IV T a b l e o f C o n t e n t s ACKNOWLEDGEMENTS ...................................................................................... iii VITA ............................................................................................................................. iv CHAPTER PAGE I Introduction .......................................................................................................... 1 II L-function for the Standard Tensor Product Representation of GSp{2) x GSp{2) ................................................................................................................ 9 2.1 P relim in a ries ......................................................................................... 9 2.1.1 The Group of Symplectic Similitudes ........................................ 9 2.1.2 Root Systems and Representations ........................................... 11 2.1.3 The Negligibility of O rbits ........................................................... 12 2.2 Rankin-Selberg Global Zeta Integral ......................................... 14 2.2.1 The Whittaker functions on GSp{2) ...................................... 15 2.2.2 The global zeta-integral .............................................................. 19 2.3 Unramified Local Zeta Integrals ........................ 25 2.3.1 Reduction ....................................................................................... 26 2.3.2 Computation of 29 2.3.3 The Unramified Local Zeta Integral ........................................... 37 2.4 Local Langlands Factor of Degree 16 for GSp{2) x GSp{2) . 40 2.4.1 Kostant-Rallis A lgorithm ........................................................... 41 2.4.2 Symmetric Space (S'p(4),5p(2) x Sp{2))................................. 43 2.4.3 Branching formula for Sp{2) 4- SL{2) x S L { 2 )......................... 47 2.4.4 Multiplicity Z((mi,m 2 ; ni,n 2 )) .................................................... 49 2.4.5 Orbit Decomposition on {B x B)\K e....................................... 52 2.4.6 Homogeneity di((m i,m 2 ;ni, 7 1 2 )) .............................................. 61 2.4.7 Application of Kostant-Rallis’ Algorithm ................................ 85 2.4.8 The Proof of Theorem 2 .3 .1 4 ................................................... 89 2.5 The Fundamental Id e n tity ............................................................... 90 2.6 Notations Used in Chapter II ......................................................... 92 III The Local Theory of Rankin-Selberg Convolution ...................................... 93 3.1 Som e E s t im a t e s .................................................................................. 94 3.1.1 Estimates of Whittaker Functions: ......................................... 94 3.1.2 Estimates of Some Integrals ...................................................... 99 3.1.3 Estimate of the Partial ‘Fourier’ TVansform.......................... 105 3.2 Nonvanishing of Local Zeta I n te g r a ls ........................................ 112 3.2.1 Nonarchimedean C a se s: ..................................................... 112 3.2.2 Archimedean C a se s: .................................................................. 120 3.3 Absolutely Convergence of Local Zeta Integrals .................. 126 3.3.1 Nonarchimedean C a se s: ............................................................ 127 3.3.2 Archimedean C a se s: ................................................................... 132 3.4 Meromorphic Continuation of Local Zeta integrals 136 3.4.1 Nonarchimedean Case: 136 3.5 Notations Used in Chapter I II ..................................................... 146 IV The Poles of Eisenstein Series Attached to Sp{n) ...................................... 147 4.1 P^-Orbital Decomposition on \ S p {n )............................... 150 4.2 The P"-constant term of Eisenstein Series ............................... 155 4.3 Local Analyses of Intertwining Operators ............................... 162 4.3.1 Intertwining Operator ^(g): general sections ................. 163 4.3.2 Intertwining Operator general sections ................... 166 4.3.3 Intertwining Operator ^(s): general sections .................... 174 4.3.4 Intertwining Operators U^^(s), % (s), and U^^{s): spherical section ............................................................................................ 182 4.4 Poles of Eisenstein Series: n < 3 .................................................. 188 4.4.1 Proof of Theorem 4.4.4 for the case of s ....................... 194 4.4.2 Proof of Theorem 4.4.4 for the case of s = |: ....................... 196 4.5 Poles of Eisenstein S e r i e s .............................................................. 207 4.6 P roof o f Two L e m m a s .................................................................... 216 4.7 Notations Used in Chapter I V ..................................................... 227 VI V Degenerate Principal Series Representations of Sp{n) over a p-adic Field 228 5.1 Basic F a c t s ........................................................................................... 229 5.2 Computations of Hecke Operators ............................................... 236 5.2.1 Hecke Operator A ...................................................................... 236 5.2.2 Hecke Operator W ...................................................................... 255 5.3 P roof of th e M ain T h eo rem ........................................................... 258 5.3.1 The Dimension of ifomsp( 4 )(/ 3 (s),/ 3 (—s)): ......................... 258 5.3.2 The Proof of the Main Theorem : ............................................. 261 5.4 N otations U sed in Chapter V ........................................................ 269 BIBLIOGRAPHY.....................................................................................................