Action of intertwining operators on pseudospherical $K$-types

Shiang Tang

Vol. 286 (2017), No. 1, 191–214

DOI: 10.2140/pjm.2017.286.191

Abstract

We give a concrete description of the two-fold cover of a simply connected, split real reductive group and its maximal compact subgroup as Chevalley groups. We study the representations of the maximal compact subgroups called pseudospherical representations, which appear with multiplicity one in the principal series representation. We introduce a family of canonically defined intertwining operators and compute their action on pseudospherical $K“$-types, obtaining explicit formulas of the Harish-Chandra $c$-function.

Keywords

pseudospherical representation, Shimura representation, intertwining operator, nonlinear real group, Gindikin–Karpelevich formula, Kubota cocycle, Chevalley groups, central extension

Mathematical Subject Classification 2010

Primary: 20G05, 22E50

Milestones

Received: 8 December 2015

Revised: 3 May 2016

Accepted: 28 May 2016

Published: 9 December 2016