Computing the associated cycles of certain Harish-Chandra modules

Abstract

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 ). In [MPVZ] we proved that for any representation X of Gelfand-Kirillov dimension 1 2 dim(G R /K R ), 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 is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In this paper we compute these coefficients explicitly. H D (M ) = Ker(D)/(Im(D) ∩ Ker(D); it is a module for the spin double coverK of K R (finite-dimensional if M is admissible). This invariant was introduced in [V2], it turned out to be very interesting and also quite computable; see for example [HP1], [HP2], [HKP], [HPR], [HPP], [HPZ], [BP1], [BP2], [BPT], [MP], [MZ].

Dirac index and associated cycles of Harish-Chandra modules

“…The following Lemma now completes the proof that constants c k as in the theorem exist; the integrality of the constants holds, but this is to be established by computing the values of the constants; see sections 6 and 7 and [13].…”

“…The computations of the constants are somewhat involved and details will appear in [13]. Here we first list (in Table 1) all of the classical real groups for which the conjecture applies.…”

“…In Sections 6 and 7 we list (1) the groups G R for which the hypothesis of Theorem A holds and (2) the complex G-orbits O C corresponding to P K (λ) under the Springer correspondence. As an example we include the computation of the constants appearing in Theorem A in one classical case; the other cases will appear in [13]. A table with all constants is included for the exceptional groups; the computations were done using a computer.…”