Degenerate principal series representations for quaternionic unitary groups
type: Item
instance of: scholarly article
title: Degenerate principal series representations for quaternionic unitary groups
zbMATH Open document ID: 1279.22026
author: Shunsuke Yamana
published in: Israel Journal of Mathematics
publication date: 2012-11-19T00:00:00Z
review text: Let \(F\) be a non-Archimedean local field with uniformizer \(\pi\). Let \(\omega= |\cdot|\) be the absolute value such that \(|\pi|^{-1}\) is the order of the residue field. Let \(D\) be a quaternion \(F\)-algebra and let \(W= D^{2n}\) be a (right) \(D\)-vector space with basis \(\{e_1,\dots, e_n,\, f_1,\dots, f_n\}\). For \(\varepsilon=\pm 1\), we define an \(\varepsilon\)-skew Hermitian (sesquilinear) form \(\langle,\rangle\) on \(W\) such that \(\langle e_i,e_\rangle= \langle f_i,f_j\rangle= 0\) and \(\langle e_i, f_j\rangle= \delta_{ij}\). Let \(G= G_n\) be the isometric subgroup of \(\text{GL}_{2n}(D)\) stabilizing the Hermitian form. Then \(G_n\) is of type \(D_{2n}\) (resp. \(C_{2n}\)) if \(\varepsilon= 1\) (resp. \(\varepsilon= -1\)). Let \(P= P_n= M_n N_n\) denote the maximal parabolic subgroup of \(G\) stabilizing the span of the \(e_i\)'s. Here \(M_n= \text{GL}_n(D)\) and let det be the reduced norm on it. Let \(\chi: F^\times\to\mathbb{C}\) and we will continue to use \(\chi\) to denote the character \(\chi\circ\text{det}\) on \(P\). The main result of this paper is to study the reducibility and irreducible subquotients of the normalized induced degenerate principal series representation \(I_n(\chi)= \text{Ind}^{G_n}_{P_n}\chi\). Using a normalized intertwining operator \(M^*(\chi): I_n(\chi)\to I_n(\chi^{-1})\), the author is able to show that for \(\chi\) outside a finite set, \(I_n(\chi)\) is irreducible. In order to study the irreducible quotients of a reducible \(I_n(\chi)\), the author uses the dual pair correspondences of the Weil representation. In this review, we will only state the main results for \(\varepsilon=1\). First \(I_n(\chi)\) is reducible if and only if \(\chi^2= \overline\omega^{2j-1}\) for \(-n+ 1\leq j\leq n\). In the reducible case, \(I_n(\chi)\) is of the form \(I_n(\omega^{j-{1\over 2}})\otimes \widetilde\mu\) where \(\widetilde\mu\) is a quadratic character of \(G_n\). Let \(H\) be the isometry group of a right \(D\)-vector s…
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 6108373
reviewed by: Hung Yean Loke
MaRDI profile type: Publication
full work available at URL: s11856-011-0102-9
OpenAlex ID: W2088755544
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DOI: 10.1007/S11856-011-0102-9
description: scientific article; zbMATH DE number 6108373