Local theta correspondence of depth zero representations and theta dichotomy.
type: Item
instance of: scholarly article
title: Local theta correspondence of depth zero representations and theta dichotomy.
zbMATH Open document ID: 1039.22012
author: Shu-Yen Pan
published in: Journal of the Mathematical Society of Japan
publication date: 2003-06-30T00:00:00Z
review text: Let \(F\) be a \(p\)-adic field with odd residual characteristic, with ring of integers \(\mathbb Z_F\) and let \(D\) be a central division algebra over \(F\) with an involution \(\tau\) and ring of integers \(\mathbb Z_D\). Let \(\varpi_F\) (resp. \(\varpi_D\)) be a uniformizer of \(F\) (resp. \(D\)), and let \(\psi\) be a continuous character of \(F\). Let \(\varepsilon\), \(\varepsilon'\) in \(\{-1,1\}\) with \(\varepsilon\varepsilon'=-1\), and let \((\mathcal V,\langle\;,\;\rangle)\) (resp. \((\mathcal V',\langle\;,\;\rangle')\)) denote a finite-dimensional vector space over \(D\), equipped with a nondegenerate \(\varepsilon\)-Hermitian form. One can define a skew-symmetric \(F\)-bilinear form \(\langle\!\langle\;,\;\rangle\!\rangle\) on the space \(\mathcal W=\mathcal V\otimes \mathcal V'\) by \(\langle\!\langle\;,\;\rangle\!\rangle={\text{Trd}}_{D| F}(\langle\;,\;\rangle\otimes \tau\circ\langle\;,\;\rangle')\) where \({\text{Trd}}_{D| F}\) denotes the reduced trace from \(D\) to \(F\), and the pair \((U(\mathcal V),U(\mathcal V'))\) forms a reductive dual pair over \(F\) in the symplectic group \({\text{Sp}}(\mathcal W)\). Let \(\widetilde{\text{Sp}} (\mathcal W)\) be the metaplectic cover of \({\text{Sp}}(\mathcal W)\), and let \(\widetilde U(\mathcal V)\) (resp. \(\widetilde U(\mathcal V'))\) denote the inverse image of \(U(\mathcal V)\) (resp. \(U(\mathcal V')\)) in \(\widetilde {\text{Sp}}(\mathcal W)\). Since \(\widetilde U(\mathcal V)\) and \(\widetilde U(\mathcal V')\) commute with each other in \(\widetilde {\text{Sp}}(\mathcal W)\), their product \(\widetilde U(\mathcal V)\cdot \widetilde U(\mathcal V')\) is a subgroup of \(\widetilde {\text{Sp}}(\mathcal W)\), and, by restricting the Weil representation \(\omega_\psi\) of \(\widetilde{\text{Sp}}(\mathcal W)\) to that subgroup, one obtains a one-to-one correspondence (the Howe correspondence or local theta correspondence) between irreducible smooth representations of \(\widetilde U(\mathcal V)\) and i…
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 1943286
zbMATH Keywords: Howe correspondence
zbMATH Keywords: dual pair
zbMATH Keywords: \(p\)-adic classical group
zbMATH Keywords: level zero representation
zbMATH Keywords: Weil representation
reviewed by: Anne-Marie Aubert
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/1191591993
OpenAlex ID: W2070342499
DOI: 10.2969/JMSJ/1191591993
description: scientific article; zbMATH DE number 1943286