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 in the finite dimensional case. We first extend this to symmetrizable Kac-Moody Lie algebras using Kashiwara's Global Basis. Then, 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.