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The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding ofirreducible subquotients of a reducible principal series representation of$GL_n(F)$, $F$ a $p$-adic field (without specifying their multiplicities whichis done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposalof a similar kind for principal series representations of $GL_n({\mathbb R})$.Our investigation on principal series representations naturally led us toconsider the Steinberg representation for real groups, which has curiuosly notbeen paid much attention to in the subject (unlike the $p$-adic case). Ourproposal for Steinberg is the simplest possible: for a real reductive group$G$, the Steinberg of $G({\mathbb R})$ is a discrete series representation ifand 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 istypically not irreducible.

Reducible principal series representations, and Langlands parameters for real groups