Research

Kostant-Kumar modules are cyclic submodules of the tensor product of two highest weight irreducible modules over a symmetrizable Kac-Moody Lie algebra over an algebraically closed field of characteristic 0.  A characterisation of the multiplicity of irreducibles is well known for the tensor product. We generalise this characterisation to Kostant-Kumar modules. We also give a generators and relations description of Kostant-Kumar modules in the finite dimensional and symmetric Kac-Moody algebra case. As an application of these results, we obtain Schur-positivity type results for symmetric functions corresponding to Kostant-Kumar modules. and obtain an upper bound for these refined multiplicities.

For a finite dimensional semisimple Lie algebra, the multiplicity of an irreducible  inside the tensor product of two highest weight irreducible representations can be realised as the dimension of the Kostant–Parthasarathy–Ranga Rao–Varadarajan multiplicity space. A result of Panyushev and Yakimova shows how to split the multiplicity space for into symmetric and alternating parts when the representation is self-dual.

For finite-dimensional semisimple or symmetric Kac–-Moody, we extend this to Kostant--Kumar modules when the Weyl group element is an involution and for all dominant weights. We show that the multiplicities of an irreducible in the symmetric and alternating parts of teh Kostant--Kumr module are given by the dimensions of the eigenspaces of the Tits lift of the Weyl group element on the multiplicity space corresponding to eigenvalues +1 or -1. This generalises the result for the full tensor product. In type A, we also discuss connections to combinatorial interpretation due to Carré  and Leclerc in terms of domino tableaux.