Automorphic Representations of Low Rank Groups

Yuval Z. Flicker9789812568038, 981-256-803-4

The area of automorphic representations is a natural continuation of studies in number theory and modular forms. A guiding principle is a reciprocity law relating the infinite dimensional automorphic representations with finite dimensional Galois representations. Simple relations on the Galois side reflect deep relations on the automorphic side, called “liftings”. This book concentrates on two initial examples: the symmetric square lifting from SL(2) to PGL(3), reflecting the 3-dimensional representation of PGL(2) in SL(3); and basechange from the unitary group U(3, E/F) to GL(3, E), [E : F] = 2. The book develops the technique of comparison of twisted and stabilized trace formulae and considers the “Fundamental Lemma” on orbital integrals of spherical functions. Comparison of trace formulae is simplified using “regular” functions and the “lifting” is stated and proved by means of character relations. This permits an intrinsic definition of partition of the automorphic representations of SL(2) into packets, and a definition of packets for U(3), a proof of multiplicity one theorem and rigidity theorem for SL(2) and for U(3), a determination of the self-contragredient representations of PGL(3) and those on GL(3, E) fixed by transpose-inverse-bar. In particular, the multiplicity one theorem is new and recent. There are applications to construction of Galois representations by explicit decomposition of the cohomology of Shimura varieties of U(3) using Deligne’s (proven) conjecture on the fixed point formula.

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Table of contents :
CONTENTS……Page 10
PREFACE……Page 6
PART 1. ON THE SYMMETRIC SQUARE LIFTING……Page 13
INTRODUCTION……Page 15
I. FUNCTORIALITY AND NORMS……Page 28
I.1 Hecke algebra……Page 29
I.2 Norms……Page 37
I.3 Local lifting……Page 43
I.4 Orthogonality……Page 53
II.1 Fundamental lemma……Page 59
II.2 Differential forms……Page 78
II.3 Matching orbital integrals……Page 85
II.4 Germ expansion……Page 92
III. TWISTED TRACE FORMULA……Page 95
III.1 Geometric side……Page 96
III.2 Analytic side……Page 102
III.3 Trace formulae……Page 106
IV. TOTAL GLOBAL COMPARISON……Page 114
IV.1 The comparison……Page 116
IV.2 Appendix: Mathematica program……Page 133
V.1 Approximation……Page 135
V.2 Main theorems……Page 158
V.3 Characters and genericity……Page 190
VI. COMPUTATION OF A TWISTED CHARACTER……Page 198
VI.1 Proof of theorem, anisotropic case
……Page 201
VI.2 Proof of theorem, isotropic case……Page 206
PART 2. AUTOMORPHIC REPRESENTATIONS OF THE UNITARY GROUP U(3,E/F)
……Page 215
1. Functorial overview……Page 217
2. Statement of results……Page 223
I. LOCAL THEORY……Page 234
I.1 Conjugacy classes……Page 235
I.2 Orbital integrals……Page 249
I.3 Fundamental lemma……Page 256
I.4 Admissible representations……Page 286
I.5 Representations of U(2,1;C/R)……Page 295
I.6 Fundamental lemma again……Page 305
II.1 Stable trace formula……Page 323
II.2 Twisted trace formula……Page 329
II.3 Restricted comparison……Page 332
II.4 Trace identity……Page 338
II.5 The σ-endo-lifting e’
……Page 351
II.6 The quasi-endo-lifting e……Page 358
II.7 Unitary symmetric square……Page 362
III.l Local identity……Page 364
III.2 Separation……Page 372
III.3 Specific lifts……Page 381
III.4 Whittaker models and twisted characters……Page 390
III.5 Global lifting……Page 397
III.6 Concluding remarks……Page 404
PART 3. ZETA FUNCTIONS OF SHIMURA VARIETIES OF U(3)……Page 409
INTRODUCTION……Page 411
1. Statement of results……Page 414
2. The Zeta function……Page 421
I.1 The Shimura variety……Page 424
I.2 Decomposition of cohomology……Page 425
I.3 Galois representations……Page 427
II.1 Stabilization and the test function……Page 429
II.2 Functorial overview of basechange for U(3)……Page 430
II.3 Local results on basechange for U(3)……Page 436
II.4 Global results on basechange for U(3)……Page 441
II.5 Spectral side of the stable trace formula……Page 446
II.6 Proper endoscopic group……Page 447
III.1 The reflex field
……Page 448
III.2 The representation of the dual group……Page 450
III.3 Local terms at p……Page 454
III.4 The eigenvalues at p……Page 456
III.5 Terms at p for the endoscopic group……Page 458
IV.1 Representations of the real GL(2)……Page 459
IV.2 Representations of U(2,1)……Page 460
IV.3 Finite-dimensional representations……Page 471
V.1 Stable case……Page 474
V.2 Unstable case……Page 477
V.3 Nontempered case……Page 481
REFERENCES……Page 485
INDEX……Page 495