  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.
