Geometry and coefficients of \v{S}apovalov elements for KM Lie superalgebras

Abstract

We study Shapovalov elements for symmetrizable, integrable Kac-Moody algebras $\mathfrak{g}$. Let $\gamma$ be a positive root of $\mathfrak{g}$ and $m$ a positive integer, with suitable conditions on the pair $(\gamma, m).$ The Shapovalov element $\theta_{\gamma,m}\in U(\mathfrak{b}^-) $ has the important property that if $\lambda$ lies on a certain hyperplane, then $\theta_{\gamma, m} v_\lambda$ is a highest weight vector of weight $\lambda -m\gamma$ in $M(\lambda)$. 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.