Dirac cohomology and translation functors

Mehdi, Salah; Pandžić, Pavle

doi:10.1016/j.jalgebra.2012.11.022

Cited by 3 publications

(1 citation statement)

References 18 publications

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“…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,DH].…”

Section: Introductionmentioning

confidence: 99%

Computing the associated cycles of certain Harish-Chandra modules

Mehdi¹,

Pandžić²,

Vogan³

et al. 2018

Glas. Mat. Ser. III

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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].

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Section: Introductionmentioning

confidence: 99%

Computing the associated cycles of certain Harish-Chandra modules

Mehdi¹,

Pandžić²,

Vogan³

et al. 2018

Glas. Mat. Ser. III

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Corrigendum to “Dirac cohomology and translation functors” [J. Algebra 375 (2013) 328–336]

Mehdi

Pandžić

2016

Journal of Algebra

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Translation principle for Dirac index

Mehdi¹,

Pandžić²,

Vogan³

2017

American Journal of Mathematics

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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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Dirac cohomology and translation functors

Abstract: International audienc

Cited by 3 publications

References 18 publications

Computing the associated cycles of certain Harish-Chandra modules

Computing the associated cycles of certain Harish-Chandra modules

Corrigendum to “Dirac cohomology and translation functors” [J. Algebra 375 (2013) 328–336]

Translation principle for Dirac index

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