Dirac index and associated cycles of Harish-Chandra modules

Abstract

Let GR be a simple real linear Lie group with maximal compact subgroup KR and assume that rank(GR)=rank(KR). For any representation X of Gelfand-Kirillov dimension 12dim⁡(GR/KR), we consider the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing X. Under a technical condition involving the Springer correspondence, we establish an explicit relationship between this polynomial and the multiplicities of the irreducible components occurring in the associated cycle of X. This relationship was conjectured in [12].