On the Dirac Series of $U(p,q)$

Abstract

This paper computes the Dirac index of all the weakly fair Aq(λ) modules of U (p, q). Although counter-examples have been found to a conjecture of Vogan on the unitary dual of U (p, q) phrased by Trapa in 2001, we believe that any irreducible unitary representation of U (p, q) with non-zero Dirac cohomology must be a weakly fair Aq(λ) module.

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

“…A complete classification of the Dirac series has been obtained for the following Lie groups: complex F 4 [DD], complex E 6 [D1]; complex classical Lie groups [BDW]; several real exceptional Lie groups [DDH, DDY]. Results on U (p, q) have been recorded in [DW2]. For instance, Example 6.3 of [DDY] says that the Dirac series of split real F 4 go beyond elliptic representations.…”