Let G R be a simple real linear Lie group with maximal compact subgroup K R and assume that rank(G R ) = rank(K R ). For any representation X of Gelfand-Kirillov dimension 1 2 dim(G R /K R ), 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].
Let G be a finite cover of a closed connected transpose-stable subgroup of GL(n, R) with complexified Lie algebra g. Let K be a maximal compact subgroup of G, and assume that G and K have equal rank. We prove a translation principle for the Dirac index of virtual (g, K)-modules. As a byproduct, to each coherent family of such modules, we attach a polynomial on the dual of the compact Cartan subalgebra of g. This "index polynomial" generates an irreducible representation of the Weyl group contained in the coherent continuation representation. We show that the index polynomial is the exact analogue on the compact Cartan subgroup of King's character polynomial. The character polynomial was defined in [K1] on the maximally split Cartan subgroup, and it was shown to be equal to the Goldie rank polynomial up to a scalar multiple. In the case of representations of Gelfand-Kirillov dimension at most half the dimension of G/K, we also conjecture an explicit relationship between our index polynomial and the multiplicities of the irreducible components occuring in the associated cycle of the corresponding coherent family.2010 Mathematics Subject Classification. Primary 22E47; Secondary 22E46.
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