Distinguished tame supercuspidal representations and odd orthogonal periods
type: Item
instance of: scholarly article
title: Distinguished tame supercuspidal representations and odd orthogonal periods
zbMATH Open document ID: 1253.22010
DOI: 10.1090/S1088-4165-2012-00418-6
author: Jeffrey Hakim
author: Joshua Lansky
publication date: 2012-06-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.5114
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 6043205
zbMATH Keywords: supercuspidal representation
zbMATH Keywords: involution
zbMATH Keywords: distinguished representation
zbMATH Keywords: orthogonal group
MaRDI profile type: Publication
arXiv ID: 1108.5114
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published in: Representation Theory
review text: Let \(G\) be a reductive group over a local nonarchimedean field, and \(\theta\) an involution of \(G\). Let \(G^\theta\) be the subgroup of \(G\) fixed under \(\theta\). An irreducible admissible representation \(\pi\) of \(G\) is distinguished by \(G^\theta\) if there is a nontrivial \(G^\theta\) invariant linear form on the space of \(\pi\). A fundamental question is to classify the set of distinguished representations. In an important paper by \textit{J. Hakim} and \textit{F. Murnaghan} [IMRP, Int. Math. Res. Pap. 2008, Article ID rpn005, 166p. (2008; Zbl 1160.22008)], a general theory was developed to classify the distinguished supercuspidal representations based on the cuspidal datum associated to the representations, through the works of Howe, Moy and Prasad, Yu and others.NEWLINENEWLINEThe current paper further develops the theory of Hakim and Murnaghan, and applies it to the special case when \(G\) is a general linear group and \(G^\theta\) is an orthogonal group. Let \(\Psi\) be the cuspidal datum associated to the irreducible supercuspidal representation \(\pi\), one can construct from \(\Psi\) compact groups \(K^\circ\subset K\subset G\) and a representation \(\xi\) of \(K\). The group \(G\) acts on the set of involutions on \(G\) through NEWLINE\[NEWLINEg\cdot\theta (x)=g^{-1}\theta(gxg^{-1})g,\,\,x\in G.NEWLINE\]NEWLINE Let \(\langle \theta,\pi\rangle\) be the dimension of the space of \(G^\theta\) invariant linear forms on the space of \(\pi\). The first main result is an identity NEWLINE\[NEWLINE\langle \theta,\pi\rangle=\sum m_K(\Theta)\langle \Theta,\xi\rangle_KNEWLINE\]NEWLINE where the sum is over the \(K-\)orbits \(\Theta\) of involutions in the \(G-\)orbit of \(\theta\); \(m_K(\Theta)\) is an integer that is a power of \(2\), and \(\langle \Theta,\xi\rangle_K\) is the dimension of the space of \(K^{\theta'}\)-invariant linear forms on the space of \(\xi\) (with \(\theta'\in \Theta\)). In fact, this formula corrects a mistake in a prior formula …
reviewed by: Zhengyu Mao
description: scientific article; zbMATH DE number 6043205