Reducible principal series representations, and Langlands parameters for real groups

TL;DR

This paper extends the understanding of principal series representations from p-adic to real groups, proposing a new framework for their subquotients and analyzing the Steinberg representation's properties in real reductive groups.

Contribution

It introduces a novel approach for analyzing principal series representations of real groups and explores the properties of the Steinberg representation in this context.

Findings

Steinberg representation for real groups is a discrete series if the group admits a discrete series.

Steinberg representation forms a full L-packet of size W_G/W_K.

The paper proposes a similar classification framework for real groups as in p-adic cases.

Abstract

The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of irreducible subquotients of a reducible principal series representation of $GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities which is done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal of a similar kind for principal series representations of $GL_n({\mathbb R})$. Our investigation on principal series representations naturally led us to consider the Steinberg representation for real groups, which has curiuosly not been paid much attention to in the subject (unlike the $p$-adic case). Our proposal for Steinberg is the simplest possible: for a real reductive group $G$, the Steinberg of $G({\mathbb R})$ is a discrete series representation if and only if $G({\mathbb R})$ has a discrete series, and makes up a full $L$-packet of representations of $G({\mathbb R})$ (of size $W_G/W_K$), so is typically not irreducible.