Unitary representations of $\mathrm{GL}(n,K)$ distinguished by a Galois involution for a $p$-adic field $K$

Nadir Matringe

Vol. 271 (2014), No. 2, 445–460

DOI: 10.2140/pjm.2014.271.445

Abstract

Let $F$ be a $p$-adic field and $K$ a quadratic extension of $F$. Using Tadić’s classification of the unitary dual of $GL\left(n,K\right)$, we give the list of irreducible unitary representations of this group distinguished by $GL\left(n,F\right)$ in terms of distinguished discrete series. It is known that a generalised Steinberg representation $St\left(\rho ,k\right)$ is distinguished if and only if the cuspidal representation $\rho$ is ${\eta }^{k-1}$-distinguished for $\eta$, the character of $F^{\ast }$ with kernel consisting of the norms of $K^{\ast }$. This actually gives a classification of distinguished unitary representations in terms of distinguished cuspidal representations.

Keywords

distinguished representations of $p$-adic groups

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 22E35

Milestones

Received: 6 August 2013

Revised: 25 November 2013

Accepted: 14 February 2014

Published: 20 September 2014