Kostant--Kumar modules: presentation and multiplicities

Abstract

Kostant--Kumar modules $K(λ,w,μ)$ are submodules of a tensor product $V(λ)\otimes V(μ)$ of irreducible highest weight modules over a symmetrizable Kac--Moody algebra, indexed by Weyl group elements $w$; their decomposition numbers $c^ν_{λμ}(w)$ refine ordinary tensor product multiplicities. We study them module-theoretically. We show that $c^ν_{λμ}(w)$ 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 $K(λ,w,μ)$ 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 $c^ν_{λμ}(w)$ and to study Schur positivity.

Disclosure

“ion 4.2 and proved through a sequence of reductions culminating in Section 4.10. Tensor envelopes, canonical morphisms and the Schur-positivity results of Part III are developed in Sections 5–6. Tool and computational resource disclosure: LLMs were used during the writing process to polish sentences, find typos and check references. The mathematics in this paper was entirely human-produced, with no involvement of AI tools at any stage. Sagemath [Sag25] was employed to compute ex”

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2026-08-23T05:15:00.258756Z

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