Character Formulas from Matrix Factorisations

Kiran Luecke

Published 2017-04-01Version 1

(With an Appendix by Constantin Teleman) In the spirit of Freed, Hopkins, and Teleman I establish an equivalence between the category of discrete series representations of a real semisimple Lie group G and a category of equivariant matrix factorisations on a subset of the dual of the Lie algebra, in analogy with the situation in [FT] which treated the case when G is compact or a loop group thereof. The equivalence is implemented by a version of the Dirac operator used in [FHT1-3], squaring to the superpotential W defining the matrix factorisations. Using the structure of the resulting matrix factorisation category as developed in [FT] I deduce the Kirillov character formula for compact Lie groups and the Rossman character formula for the discrete series of a real semi-simple Lie group. The proofs are a calculation of Chern characters and use the Dirac family constructed in [FHT1-3]. Indeed, the main theorems of [FHT3] and [FT] are a categorification of the Kirillov correspondence, and this paper establishes that this correspondence can be recovered at the level of characters.