Dirac Series of GL(n, ℝ)

Abstract

Abstract The unitary dual of $GL(n, {\mathbb {R}})$ was classified by Vogan in the 1980s. In particular, the Speh representations and the special unipotent representations are the building blocks of the unitary dual with half-integral infinitesimal characters. In this paper, we classify all irreducible unitary $({\mathfrak {g}}, K)$-modules with non-zero Dirac cohomology for $GL(n, {\mathbb {R}})$, as well as a formula for (one of) their spin-lowest $K$-types. Moreover, analogous to the $GL(n,{\mathbb {C}})$ case given in [12], we count the number of the FS-scattered representations of $GL(n, {\mathbb {R}})$.