Translation principle for Dirac index

Abstract

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.

Dirac index and associated cycles of Harish-Chandra modules

“…The two invariants are typically not the same; for a given representation the harmonic polynomials associated to the two invariants often have different degrees. However, there is a family of representations, those having certain annihilators, for which the degrees do coincide and a relationship between the two invariants is conjectured in [12,Conjecture 7.2]. The main point of this article is to prove this conjecture, which is stated as Theorem A below.…”

“…The definition and numerous properties of the Dirac index of a Harish-Chandra module is contained in [12]. The Dirac index is related to the Dirac cohomology, but has some additional nice properties.…”

“…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].…”

“…([12, Prop. 5.2]) Let Λ be the weight lattice in h * and suppose that {X(λ)} λ∈λ 0 +Λ is a coherent family of virtual Harish-Chandra modules, then dim(DI v (X(λ))) is a W -harmonic polynomial in λ.…”

Dirac index of some unitary representations of $Sp(2n, \mathbb{R})$ and $SO^*(2n)$

“…It turns out that the Dirac index preserves short exact sequences and has nice behavior with respect to coherent continuation [22,23]. This idea is pursued in [12] to compute the Dirac index of all weakly fair A q (λ)-modules for G = U (p, q).…”