Translation principle for Dirac index
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 eac...
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| Format: | Article |
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Johns Hopkins University Press
2018
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| Online Access: | http://hdl.handle.net/1721.1/116090 https://orcid.org/0000-0002-9816-2395 |
| _version_ | 1811068342781345792 |
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| author | Mehdi, Salah Pandzic, Pavle Vogan, David A |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Mehdi, Salah Pandzic, Pavle Vogan, David A |
| author_sort | Mehdi, Salah |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T07:54:36Z |
| format | Article |
| id | mit-1721.1/116090 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T07:54:36Z |
| publishDate | 2018 |
| publisher | Johns Hopkins University Press |
| record_format | dspace |
| spelling | mit-1721.1/1160902022-09-23T09:33:43Z Translation principle for Dirac index Mehdi, Salah Pandzic, Pavle Vogan, David A Massachusetts Institute of Technology. Department of Mathematics Mehdi, Salah Pandzic, Pavle Vogan, David A 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. 2018-06-05T15:00:48Z 2018-06-05T15:00:48Z 2017-12 2016-04 2018-05-31T16:54:44Z Article http://purl.org/eprint/type/JournalArticle 1080-6377 0002-9327 http://hdl.handle.net/1721.1/116090 Mehdi, Salah et al. “Translation Principle for Dirac Index.” American Journal of Mathematics 139, 6 (2017): 1465–1491 © 2017 Johns Hopkins University Press https://orcid.org/0000-0002-9816-2395 http://dx.doi.org/10.1353/AJM.2017.0037 American Journal of Mathematics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Johns Hopkins University Press arXiv |
| spellingShingle | Mehdi, Salah Pandzic, Pavle Vogan, David A Translation principle for Dirac index |
| title | Translation principle for Dirac index |
| title_full | Translation principle for Dirac index |
| title_fullStr | Translation principle for Dirac index |
| title_full_unstemmed | Translation principle for Dirac index |
| title_short | Translation principle for Dirac index |
| title_sort | translation principle for dirac index |
| url | http://hdl.handle.net/1721.1/116090 https://orcid.org/0000-0002-9816-2395 |
| work_keys_str_mv | AT mehdisalah translationprinciplefordiracindex AT pandzicpavle translationprinciplefordiracindex AT vogandavida translationprinciplefordiracindex |