Clay Mathematics Proceedings Volume 13 On Certain L-Functions Conference in honor of Freydoon Shahidi July 23–27, 2007 Purdue University West Lafayette, Indiana James Arthur James W. Cogdell Steve Gelbart David Goldberg Dinakar Ramakrishnan American Mathematical Society Jiu-Kang Yu Clay Mathematics Institute Editors On Certain L-Functions Clay Mathematics Proceedings Volume 13 On Certain L-Functions Conference on Certain L-Functions in honor of Freydoon Shahidi July 23–27, 2007 Purdue University, West Lafayette, Indiana James Arthur James W. Cogdell Steve Gelbart David Goldberg Dinakar Ramakrishnan Jiu-Kang Yu Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 11F30, 11F70, 11F80, 11S37, 20G05, 20G25,22E35, 22E50, 22E55. Front cover graphic: Shahidi, Freydoon. On Certain L-Functions. American Journal of Mathematics 103:2 (1981), p.297. (cid:2)c1981 The Johns Hopkins University Press. Reprinted withpermission of The Johns Hopkins University Press. The back cover photograph of Freydoon Shahidi is used with the permission of Purdue University and Andrew Hancock. Library of Congress Cataloging-in-Publication Data OncertainL-functions: conferenceinhonorofFreydoonShahidioncertainL-functions,Purdue University,WestLafayette,Indiana,July23–27,2007/JamesArthur...[etal.],editors. p.cm. —(Claymathematicsproceedings;v.13) Includesbibliographicalreferences. ISBN978-0-8218-5204-0(alk.paper) 1.L-functions—Congresses. I.Shahidi,Freydoon. II.Arthur,James,1944– QA246.O53 2011 512.7(cid:2)3—dc22 2011003659 Copying and reprinting. Materialinthisbookmaybereproducedbyanymeansfor edu- cationaland scientific purposes without fee or permissionwith the exception ofreproduction by servicesthatcollectfeesfordeliveryofdocumentsandprovidedthatthecustomaryacknowledg- ment of the source is given. 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(cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ VisittheClayMathematicsInstitutehomepageathttp://www.claymath.org/ 10987654321 161514131211 Contents Preface vii Shahidi’sWork“OnCertainL-functions”:AShortHistoryofLanglands-Shahidi Theory 1 Steve Gelbart The Embedded Eigenvalue Problem for Classical Groups 19 James Arthur A Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4) 33 Mahdi Asgari and A. Raghuram Types and Explicit Plancherel Formulæ for Reductive p-adic Groups 55 Colin J. Bushnell, Guy Henniart and Philip C. Kutzko Jacquet Modules and the Asymptotic Behaviour of Matrix Coefficients 81 Bill Casselman The ABS Principle : Consequences for L2(G/H) 99 Laurent Clozel Orbital Integrals and Distributions 107 L. Clozel and J.-P. Labesse Functoriality for the Quasisplit Classical Groups 117 J.W. Cogdell, I.I. Piatetski-Shapiro and F. Shahidi Poles of L-Functions and Theta Liftings for Orthogonal Groups, II 141 David Ginzburg, Dihua Jiang and David Soudry On Dual R–groups for Classical Groups 159 David Goldberg Irreducibility of the Igusa Tower over Unitary Shimura Varieties 187 Haruzo Hida On the Gross-Prasad Conjecture for Unitary Groups 205 Herv´e Jacquet and Stephen Rallis On Local L-Functions and Normalized Intertwining Operators II; Quasi-Split Groups 265 Henry H. Kim and Wook Kim Reflexions on Receiving the Shaw Prize 297 Robert P. Langlands On Arthur’s Asymptotic Inner Product Formula of Truncated Eisenstein Series 309 Erez Lapid Multiplicit´e 1 dans les paquets d’Arthur aux places p-adiques 333 C. Mœglin v vi CONTENTS Unramified Unitary Duals for Split Classical p–adic Groups; The Topology and Isolated Representations 375 Goran Muic´ and Marko Tadic´ Parametrization of Tame Supercuspidal Representations 439 Fiona Murnaghan On the Sato-Tate Conjecture, II 471 V. Kumar Murty Icosahedral Fibres of the Symmetric Cube and Algebraicity 483 Dinakar Ramakrishnan Pseudo Eisenstein Forms and the Cohomology of Arithmetic Groups III: Residual Cohomology Classes 501 Ju¨rgen Rohlfs and Birgit Speh A Functor from Smooth o-Torsion Representations to (ϕ,Γ)-Modules 525 Peter Schneider and Marie-France Vigneras Motivic Galois Groups and L-Groups 603 Hiroyuki Yoshida Preface For overthreedecadesFreydoonShahidi hasbeenmakingsignificant contribu- tions to number theory, automorphic forms, and harmonic analysis. Shortly after receivinghisPh.D.in1975fromJohnsHopkins,underthedirectionofJosephSha- lika, Shahidi laid out a program to address several open problems, as stated by Langlands, by a novel method now known as the Langlands-Shahidi method. In particular,ShahidisoughttoexploittheFouriercoefficientsofEisensteinseriesand their local analogs to establish cases of Langlands functoriality. Key to this idea was the understanding of generic forms and their local components, from which he developed the theory of local coefficients. Shahidi believed this theory, combined with other methods (including converse theorems) could yield elusive examples of functoriality,suchasthesymmetricpowertransfersforGL andfunctorialityfrom 2 classical groups to general linear groups. It was well known that establishing such results would yield significant progress in number theory. Through the first 10 years (or so) of this pursuit, Shahidi produced several important results, and another 20 years of results of such stature would, alone, be fittingofa60thbirthdayconferenceandaccompanyingvolume. However,thepower and stature of his results grew significantly and continues through (and beyond) the publication of this volume. Simply put, in the last 15 to 20 years, Shahidi and the Langlands-Shahidi method have helped produce a series of significant results. Rather than a list, we will say that this record speaks for itself. That a number of events within and outside of the Langlands-Shahidi method transpired simulta- neously has, no doubt, changed the face of the Langlands program in significant ways. That Shahidi’s approach is crucial to so much recent progress is a testament to his persistence and perseverance. The influence of Shahidi’s work continues to grow, and the breadth of the applications by Shahidi, his collaborators, and others is undeniable. One of Shahidi’s contributions that should not be overlooked is his service in mentoring young mathematicians within his chosen field. To date, Shahidi has produced eight Ph.D.’s and has at least six more in progress. In addition, he has sponsored roughly 15 postdoctoral appointments at Purdue. Further, several outstanding figures have noted his more informal, but just as crucial, role as a mentor. The list of these mathematicians includes several major contributors to the field, some of whom have contributed articles for this volume. Freydoon Shahidi reached the age of 60 on June 19, 2007, and a conference was held to commemorate this occasion July 29-August 3, 2007, at Purdue Uni- versity, West Lafayette, Indiana. As Shahidi has been a member of the Mathe- matics Department at Purdue since 1977 and has been designated by Purdue as a Distinguished Professor of Mathematics, this seemed a fitting location for such a vii viii PREFACE conference. FundingforthisconferencewasprovidedbyPurdueUniversity’sMath- ematics Department and College of Science, the National Science Foundation, The Clay Mathematics Institute, The Institute for Mathematics and its Applications, and the Number Theory Foundation. Over 100 mathematicians attended, and there were 23 one-hour lectures. The conference focused on several aspects of the Langlands program, including some exposition of Shahidi’s work, recent progress, and future avenues of investigation. Far from being a retrospective, the conference emphasized the vast array of significant problems ahead. All lecturers were invited to contribute material for this volume. In addition, some important figures who were unable to attend or deferred on speaking at the conference were invited to submit articles as well. We hope this resulting volume will serve as a modest trib- ute toShahidi’s legacytodate, but should not be considered the final word on this subject. The editors wish to thank all of the authors for their willingness to contribute manuscripts of such high quality in honor of our colleague. We also wish to thank the anonymous referees for their conscientious reading of these manuscripts and their helpful comments to authors which have improved the contents. We wish to express our deep gratitude to Purdue University’s Mathematics Department and College of Science, the National Science Foundation, The Clay Mathematics Institute, The Institute for Mathematics and its Applications, and the Number Theory Foundation, for their sponsorship of the conference. We also wish tothank all of the conferees who made the conference so successful, and the staff of the Mathematics Department at Purdue University, particularly Julie Morris, for their help with organizational matters. We thank the Clay Mathematics Institute and theAmericanMathematicalSocietyforagreeingtopublishthiswork,andaspecial thanks goes to Vida Salahi for all her efforts in helping us through this process. ClayMathematicsProceedings Volume13,2011 Shahidi’s Work “On Certain L-functions”: A Short History of Langlands-Shahidi Theory Steve Gelbart For FreydoonShahidi, on his 60th birthday Some History The precursor of Langlands-Shahidi theory is Selberg’s earlier relation be- tween real analytic Eisenstein series and the Riemann zeta function. More precisely, let (cid:2) 1 ys E(z,s)= , (c,d)=1, 2 |cz+d|2s where (c,d) means the greatest common divisor of c and d. This is the simplest real analytic Eisenstein series for SL(2,R). Its zeroth Fourier coefficient is (cid:3) 1 a (y,s)= E(z,s)dx=ys+M(s)y1−s 0 0 where √ πΓ(s− 1)ζ(2s−1) (1) M(s)= 2 ; Γ(s)ζ(2s) its first Fourier coefficient is (cid:3) (2) a (y,s)= 1E(z,s)exp(−2πix)dx= 2πsy12Ks−12(2πy) 1 Γ(s)ζ(2s) 0 whichimplieswhatwewant(seep. 46of[Kub]). Thatis,themeromorphiccontin- uation of M(s) already gives the (in this case known) meromorphic continuability ofζ(s),andthefunctionalequationofE(z,s)givesthefunctionalequationofζ(s). The generalization of these ideas would have to wait many more years, when R.P.Langlands began to publish his miraculous conjectures. Langlands (1967 Whittemore Lectures) In January 1967, Langlands wrote his famous letter to Weil. It contains precursors of most of the startling conjectures that today make up most of the broad “Langlands Program”. In April 1967, Langlands delivered the Yale 2010 MathematicsSubjectClassification. Primary11F66;Secondary11F70,22E55. (cid:2)c 2011SteveGelbart 1