Table Of ContentClay 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–
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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
