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Invariant representations of $\mathrm {GSp}(2)$ under tensor product with a quadratic character

About this Title

Ping-Shun Chan, Department of Mathematics, The Ohio State University, Columbus, Ohio 43210

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 204, Number 957
ISBNs: 978-0-8218-4822-7 (print); 978-1-4704-0571-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-09-00599-7
Published electronically: October 16, 2009
Keywords: $\mathrm {GSp}(2)$, automorphic representations, $p$-adic harmonic analysis
MSC: Primary 11F70, 11F72, 11F85

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Table of Contents

Chapters

  • 1. Introduction
  • 2. $\varepsilon$-Endoscopy for $\mathrm {GSp}(2)$
  • 3. The Trace Formula
  • 4. Global Lifting
  • 5. The Local Picture
  • A. Summary of Global Lifting
  • B. Fundamental Lemma

Abstract

Let $F$ be a number field or a $p$-adic field. We introduce in Chapter 2 of this work two reductive rank one $F$-groups, $\mathbf {H_1}$, $\mathbf {H_2}$, which are twisted endoscopic groups of $\mathrm {GSp}(2)$ with respect to a fixed quadratic character $\varepsilon$ of the idèle class group of $F$ if $F$ is global, $F^\times$ if $F$ is local. When $F$ is global, Langlands functoriality predicts that there exists a canonical lifting of the automorphic representations of $\mathbf {H_1}$, $\mathbf {H_2}$ to those of $\mathrm {GSp}(2)$. In Chapter 4, we establish this lifting in terms of the Satake parameters which parameterize the automorphic representations. By means of this lifting we provide a classification of the discrete spectrum automorphic representations of $\mathrm {GSp}(2)$ which are invariant under tensor product with $\varepsilon$.

The techniques through which we arrive at our results are inspired by those of Kazhdan (1984), which introduced the trace formula twisted by a character. In particular, our techniques involve comparing the spectral sides of the trace formulas for the groups under consideration. We make use of the twisted extension of Arthur’s trace formula, and Kottwitz-Shelstad’s stabilization of the elliptic part of the geometric side of the twisted trace formula.

When $F$ is local, in Chapter 5 we provide a classification of the irreducible admissible representations of $\mathrm {GSp}(2, F)$ which are invariant under tensor product with the quadratic character $\varepsilon$ of $F^\times$. More precisely, we use the global results from Chapter 4 to express the twisted characters of these invariant representations in terms of the characters of the admissible representations of $\mathbf {H}_i(F)$ ($i = 1, 2$). These (twisted) character identities provide candidates for the liftings predicted by the conjectural local Langlands functoriality. The proofs are inspired by Kazhdan and rely on Sally-Tadić’s classification of the irreducible admissible representations of $\mathrm {GSp}(2, F)$, results on endoscopic lifting to $\mathrm {GSp}(2)$ by Weissauer, and Flicker’s results on the lifting from $\mathrm {PGSp}(2)$ to $\mathrm {PGL}(4)$.

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