Reducible principal series representations, and Langlands parameters for real groups

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.