Dirac series of $GL(n,\mathbb{R})$

Abstract

The unitary dual of $GL(n,\mathbb{R})$ was classified by Vogan in the 1980s. Focusing on the irreducible unitary representations of $GL(n,\mathbb{R})$ with half-integral infinitesimal characters, we find that Speh representations and the special unipotent representations are building blocks. By looking at the $K$-types of them, and by using a Blattner-type formula, we obtain all the irreducible unitary $(\mathfrak{g},K)$-modules with non-zero Dirac cohomology of $GL(n,\mathbb{R})$, as well as a formula for (one of) their spin-lowest $K$-types. Moreover, analogous to the $GL(n,\mathbb{C})$ case given in [DW1], we count the number of the FS-scattered representations of $GL(n,\mathbb{R})$.

23 pages

Cited by in corpus (1)

• Dirac index of some unitary representations of $Sp(2n,\mathbb{R})$ and $S{O}^{\ast }(2n)$