Local Newforms for GSp(4) : Lecture Notes in Mathematics - Brooks Roberts

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Local Newforms for GSp(4)

By: Brooks Roberts, Ralf Schmidt

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Paperback | 20 August 2007

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Local Newforms for GSp(4) describes a theory of new- and oldforms for representations of GSp(4) over a non-archimedean local field. This theory considers vectors fixed by the paramodular groups, and singles out certain vectors that encode canonical information, such as L-factors and epsilon-factors, through their Hecke and Atkin-Lehner eigenvalues. While there are analogies to the GL(2) case, this theory is novel and unanticipated by the existing framework of conjectures. An appendix includes extensive tables about the results and the representation theory of GSp(4).

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From the reviews: "This book gives an analog of Casselman's local Atkin-Lehner theorem for GSp(4). ! The local theory of the Novodvorsky construction is advanced by this work of Roberts and Schmidt, and the converse is also true: the Novodvorsky local integrals play an important role in the proof, especially in the supercuspidal case. ! proves an important theorem, and moreover is written in a useful and instructive way." (Daniel Bump, Mathematical Reviews, Issue 2008 g)

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A Summary	p. 1
Representation Theory	p. 27
Definitions	p. 27
Parabolically Induced Representations	p. 34
Dual Groups	p. 40
The Local Langlands Correspondence	p. 46
P[subscript 3]-Theory	p. 62
Zeta Integrals	p. 74
Paramodular Vectors	p. 85
Linear Independence	p. 85
The Level Raising Operators [theta], [theta]' and [eta]	p. 90
Level Lowering Operators	p. 107
Paramodular Vectors and P[subscript 3]-Theory	p. 117
Zeta Integrals	p. 123
Paramodular Vectors and Zeta Integrals	p. 123
Poles and P[subscript 3]-Theory	p. 128
The [eta] Principle	p. 135
The Existence Theorem for Generic Representations	p. 148
Non-supercuspidal Representations	p. 151
Double Coset Decompositions	p. 151
Induction from the Siegel Parabolic Subgroup	p. 156
Representations of Type IIIb and IVc	p. 161
Induction from the Klingen Parabolic Subgroup	p. 165
Saito-Kurokawa Representations	p. 168
Summary	p. 182
Atkin-Lehner Eigenvalues	p. 183
Hecke Operators	p. 187
Two Hecke Operators	p. 188
The Commutation Relation	p. 195
Hecke Operators and Level Raising	p. 205
Computation of Hecke Eigenvalues	p. 213
Some Consequences of Unitarity	p. 236
Proofs of the Main Theorems	p. 239
Zeta Integrals: The Unramified Case	p. 239
Zeta Integrals: The Level p Case	p. 243
The Operator R	p. 248
Zeta Integrals: The Higher Level Case	p. 252
Main Results	p. 262
Tables for Representations of GSp(4)	p. 269
Non-supercuspidal Representations	p. 269
Unitary Representations	p. 271
Jacquet Modules	p. 273
The P[subscript 3]-Filtration	p. 277
L-Parameters	p. 280
L- and [epsilon]-factors (degree 4)	p. 282
L- and [epsilon]-factors (degree 5)	p. 286
Paramodular Dimensions and Atkin-Lehner Eigenvalues	p. 290
Hecke Eigenvalues	p. 294
Parahori-invariant Vectors	p. 297
References	p. 301
Index	p. 305
Table of Contents provided by Ingram. All Rights Reserved.

Local Newforms for GSp(4)

At a Glance

Paperback

Industry Reviews

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How to return your order

Defective items

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