Dirac operators for algebraic families

TL;DR

This paper introduces algebraic families of Dirac operators for real reductive Lie groups and proves Vogan's conjecture relating infinitesimal characters and Dirac cohomology.

Contribution

It develops a framework for algebraic families of Dirac operators and establishes Vogan's conjecture within this context.

Findings

Proved Vogan's conjecture for algebraic families of Dirac operators.

Established a deformation family connecting reductive groups and Cartan motion groups.

Extended the theory of Dirac operators to algebraic family settings.

Abstract

We introduce algebraic families of Dirac operators for the deformation family (and other related families) associated with a real reductive Lie group that interpolates the reductive group and the corresponding Cartan motion group. We prove Vogan's conjecture in this setting, relating the infinitesimal character of an algebraic family of Harish-Chandra modules and its Dirac cohomology.