Representation Theory And Automorphic Forms

This volume uses a unified approach to representation theory and automorphic forms. The invited papers, written by leading mathematicians, track recent progress in the ever expanding fields of representation theory and automorphic forms, and their association with number theory and differential geometry. Representation theory relates to number theory through Langlands' conjecture, which illuminates the deep properties of primes in number fields. The Langlands program is further analyzed in this work through automorphic functions and automorphic distributions. The relation between representation theory and differential geometry is explored via the Dirac cohomology of Index theory. Also discussed are the subjects of modular forms and harmonic analysis. The volume also branches off from representation theory into self-dual representations, and includes work from the non-standard geometric view of viable action on complex manifolds towards multiplicity-free representation theory. Both graduate students and researchers will find inspiration in this volume.

Preface	p. vii
Irreducibility and Cuspidality	p. 1
Preliminaries	p. 5
The first step in the proof	p. 15
The second step in the proof	p. 16
Galois representations attached to regular, selfdual cusp forms on GL(4)	p. 18
Two useful lemmas on cusp forms on GL(4)	p. 20
Finale	p. 21
References	p. 25
On Liftings of Holomorphic Modular Forms	p. 29
Basic facts	p. 29
Fourier coefficients of the Eisenstein series	p. 30
Kohnen plus space	p. 32
Lifting of cusp forms	p. 33
Outline of the proof	p. 34
Relation to the Saito-Kurokawa lifts	p. 35
Hermitian modular forms and hermitian Eisensetein series	p. 37
The case m = 2n + 1	p. 39
The case m = 2n	p. 40
L-functions	p. 40
The case m = 2	p. 41
References	p. 42
Multiplicity-free Theorems of the Restrictions of Unitary Highest Weight Modules with respect to Reductive Symmetric Pairs	p. 45
Introduction and statement of main results	p. 45
Main machinery from complex geometry	p. 56
Proof of Theorem A	p. 61
Proof of Theorem C	p. 68
Uniformly bounded multiplicities - Proof of Theorems B and D	p. 70
Counterexamples	p. 77
Finite-dimensional cases - Proof of Theorems E and F	p. 83
Generalization of the Hua-Kostant-Schmid formula	p. 89
Appendix: Associated bundles on Hermitian symmetric spaces	p. 103
References	p. 105
The Rankin-Selberg Method for Automorphic Distributions	p. 111
Introduction	p. 111
Standard L-functions for SL(2)	p. 115
Pairings of automorphic distributions	p. 121
The Rankin-Selberg L-function for GL(2)	p. 128
Exterior Square on GL(4)	p. 137
References	p. 149
Langlands Functoriality Conjecture and Number Theory	p. 151
Introduction	p. 151
Modular forms, Galois representations and Artin L-functions	p. 152
Lattice point problems and the Selberg conjecture	p. 156
Ramanujan conjecture for Maass forms	p. 158
Sato-Tate conjecture	p. 159
Functoriality for symmetric powers	p. 161
Functoriality for classical groups	p. 163
Ramanujan conjecture for classical groups	p. 164
The method	p. 166
References	p. 169
Discriminant of Certain K3 Surfaces	p. 175
Introduction - Discriminant of elliptic curves	p. 175
K3 surfaces with involution and their moduli spaces	p. 178
Automorphic forms on the moduli space	p. 180
Equivariant analytic torsion and 2-elementary K3 surfaces	p. 182
The Borcherds products	p. 184
Borcherds products for odd unimodular lattices	p. 186
K3 surfaces of Matsumoto-Sasaki-Yoshida	p. 188
Discriminant of quartic surfaces	p. 200
References	p. 209
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