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 $c_{\lambda \mu }^{\nu }(w)$ refine ordinary tensor product multiplicities. We study them module-theoretically. We show that $c_{\lambda \mu }^{\nu }(w)$ 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 $c_{\lambda \mu }^{\nu }(w)$ and to study Schur positivity.