2026/07/31 by Ian M. Musson
Mathematics · #math.RT
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arxiv created 2026/07/31 · arxiv updated 2026/08/03
We study Shapovalov elements for symmetrizable, integrable Kac-Moody algebras \mathfrakg. Let γ be a positive root of \mathfrakg and m a positive integer, with suitable conditions on the pair (γ, m). The Shapovalov element θγ,m∈ U(\mathfrakb-) has the important property that if λ lies on a certain hyperplane, then θγ, m vλ is a highest weight vector of weight λ-mγ in M(λ). We give a closed fomula for all coefficients that arise in the induction step of the proof. The commutative version of this formula relates the leading terms using maps graded algebras that we call of Shapovalov algebras. This suggests a geometric setting for the study of Shapovalov elements.