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

TL;DR

This paper classifies irreducible unitary representations of $GL(n, ext{R})$ with non-zero Dirac cohomology, identifying key building blocks and providing formulas for their $K$-types and counts of specific representations.

Contribution

It explicitly determines all irreducible unitary $(rak{g},K)$-modules with non-zero Dirac cohomology for $GL(n, ext{R})$, extending previous classifications.

Findings

Identified Speh and unipotent representations as building blocks.

Derived formulas for spin-lowest $K$-types.

Counted FS-scattered representations for $GL(n, ext{R})$.

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 $GL(n, \mathbb{R})$.