A vanishing theorem for Dirac cohomology of standard modules

TL;DR

This paper establishes a criterion for the vanishing of Dirac cohomology in standard modules of graded Hecke algebras, linking it to twisted-elliptic temperedness, and applies this to analyze ladder representations in type A_n.

Contribution

It proves a vanishing theorem for Dirac cohomology of standard modules and computes it for ladder representations, connecting to Springer correspondence and Weyl group representations.

Findings

Dirac cohomology vanishes iff module is not twisted-elliptic tempered.

Non-zero Dirac cohomology modules appear with multiplicity one.

Provides explicit Dirac cohomology computations for ladder representations.

Abstract

This paper studies the Dirac cohomology of standard modules in the setting of graded Hecke algebras with geometric parameters. We prove that the Dirac cohomology of a standard module vanishes if and only if the module is not twisted-elliptic tempered. The proof makes use of two deep results. One is some structural information from the generalized Springer correspondence obtained by S. Kato and Lusztig. Another one is a computation of the Dirac cohomology of tempered modules by Barbasch-Ciubotaru-Trapa and Ciubotaru. We apply our result to compute the Dirac cohomology of ladder representations for type $A_n$. For each of such representations with non-zero Dirac cohomology, we associate to a canonical Weyl group representation. We use the Dirac cohomology to conclude that such representations appear with multiplicity one.