In this talk, we will describe a class of Lie algebra representations termed Kostant-Kumar modules, which first arose in the 1980's in the context of the celebrated Parthasarathy-Ranga Rao-Varadarajan (PRV) conjecture. Our approach is based on the theory of Lakshmibai-Seshadri paths and crystal bases, which provide a powerful combinatorial framework for understanding the structure and decomposition of these representations.
We consider affine-sl2 which is the simplest (infinite-dimensional) affine Lie algebra and describe some of its Kostant-Kumar modules. The combinatorics underlying these will now involve constructs resembling electron-hole pairs (charged partitions or semi-infinite wedge products). I will give a gentle introduction to all these ideas and finally bring them together to prove a decomposition and character formula for the Kostant-Kumar modules of interest.
This is based on joint work with Mrigendra Singh Kushwaha and KN Raghavan (arxiv:2409.09328).
