Multiplets of representations and Kostant's Dirac operator for equal rank loop groups
| dc.creator | Landweber, Gregory D. | |
| dc.date | 2000-05-05 | |
| dc.date | 2000-11-09 | |
| dc.date.accessioned | 2026-07-07T04:35:03Z | |
| dc.date.available | 2026-07-07T04:35:03Z | |
| dc.description | Let g be a semi-simple Lie algebra and let h be a reductive subalgebra of maximal rank in g. Given any irreducible representation of g, consider its tensor product with the spin representation associated to the orthogonal complement of h in g. Gross, Kostant, Ramond, and Sternberg recently proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of h, called a multiplet. Kostant then constructed a formal h-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of h-representations corresponding to the given representation of g. We reproduce these results in the Kac-Moody setting for the extended loop algebras Lg and Lh. We prove a homogeneous generalization of the Weyl-Kac character formula, which now yields a multiplet of irreducible positive energy representations of Lh associated to any irreducible positive energy representation of Lg. We construct a Lh-equivariant operator, analogous to Kostant's Dirac operator, on the tensor product of a representation of Lg with the spin representation associated to the complement of Lh in Lg. We then prove that the kernel of this operator gives the Lh-multiplet corresponding to the original representation of Lg. | |
| dc.description | 30 pages. Revised to clarify use of normal ordering in Section 7 and to correct various minor errors | |
| dc.identifier | https://arxiv.org/abs/math/0005057 | |
| dc.identifier | http://arxiv.org/abs/math/0005057 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59135 | |
| dc.subject | Representation Theory | |
| dc.subject | Mathematical Physics | |
| dc.subject | 17B67 (Primary) 17B35, 22E46, 81R10 (Secondary) | |
| dc.title | Multiplets of representations and Kostant's Dirac operator for equal rank loop groups | |
| dc.type | text |