Kostant--Kumar modules: presentation and multiplicities
Kostant--Kumar modules $K(\lambda,w,\mu)$ are submodules of a tensor product $V(\lambda)\otimes V(\mu)$ of irreducible highest weight modules over a symmetrizable Kac--Moody algebra, indexed by Weyl group elements $w$; their decomposition numbers $c^\nu_{\lambda\mu}(w)$ refine ordinary tensor product multiplicities. We study them module-theoretically. We show that $c^\nu_{\lambda\mu}(w)$ is computed by a natural quotient of the Kostant--Parthasarathy--Ranga Rao--Varadarajan multiplicity space, via orthogonal projection onto a Demazure module. For $\mathfrak{g}$ finite-dimensional semisimple or symmetric Kac--Moody, we present $K(\lambda,w,\mu)$ by generators and relations, extending the presentation of Demazure modules due to Joseph, Polo and Mathieu. We apply the presentation to obtain upper bounds on $c^\nu_{\lambda\mu}(w)$ and to study Schur positivity.