Translation principle for Dirac index
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1504.08307.pdf
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372.56 KB
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Author(s)
Mehdi, Salah
Pandzic, Pavle
Vogan, David A
Date Issued
December 2017
Journal
American Journal of Mathematics
Publisher
Johns Hopkins University Press
Citation
Mehdi, Salah et al. “Translation Principle for Dirac Index.” American Journal of Mathematics 139, 6 (2017): 1465–1491 © 2017 Johns Hopkins University Press
Version
Original manuscript
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 suchmodules, 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 by King 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 occurring in the associated cycle of the corresponding coherent family.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1353/AJM.2017.0037
