Elliptic representations for Sp(2n) and SO(n)

Rebecca A. Herb

Vol. 161 (1993), No. 2, 347–358

DOI: 10.2140/pjm.1993.161.347

Abstract

Let G be a connected, reductive p-adic group and let G^e denote the set of regular elliptic elements of G. Let π be an irreducible, tempered representation of G with character Θ_π, and write Θ_π^e for the restriction of Θ_π to G^e. We say π is elliptic if Θ_π^e is non-zero. In this paper we will characterize the elliptic representations for the p-adic groups Sp(2n) and SO(n). We will show for Sp(2n) and SO(2n + 1) that every irreducible, tempered representation is either elliptic or can be irreducibly induced from an elliptic representation. We will then show that this fails for the groups SO(2n). In this case there are irreducible tempered representations which cannot be irreducibly induced and are not elliptic.

Mathematical Subject Classification 2000

Primary: 22E50

Secondary: 22D30, 22E35

Milestones

Received: 1 December 1991

Revised: 15 April 1992

Published: 1 December 1993