On a Conjecture of Barbasch and Pandžić

Abstract

Let G be a connected complex semisimple Lie group. Let J s be the irreducible ( , K) module with Zhelobenko parameters c /2 −s c /2 , where s ∈ W is an involution. A conjecture of Barbasch and Pandžić claims that the Dirac cohomology of any unitary J s is either zero or the trivial K-type with multiplicity 2 l 0 /2 , where l 0 is the split rank of G. We prove this conjecture for J s in the good range.