Representation theoretic harmonic spinors for coherent families

Mehdi, Salah; Parthasarathy, Raghuveer

doi:10.1007/s13226-010-0011-3

Cited by 3 publications

(1 citation statement)

References 20 publications

(24 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To fix this problem, we will replace Dirac cohomology by the Dirac index. (We note that there is a relationship between Dirac cohomology and translation functors; see [MP], [MPa1], [MPa2], [MPa3]. )…”

Section: Introductionmentioning

confidence: 99%

Translation principle for Dirac index

Mehdi¹,

Pandžić²,

Vogan³

2017

American Journal of Mathematics

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%