Coefficients of Šapovalov elements for simple Lie algebras and contragredient Lie superalgebras

Abstract

We provide upper bounds on the degrees of the coefficients of Šapovalov elements for a simple Lie algebra. If $\fg$ is a contragredient Lie superalgebra and $\gc$ is a positive isotropic root of $\fg,$ we prove the existence and uniqueness of the Šapovalov element for $\gc$ and we obtain upper bounds on the degrees of their coefficients. For type A Lie superalgebras we give a closed formula for Šapovalov elements. Often the coefficients of Šapovalov elements are products of linear factors, and we provide some reasons for this coming from representation theory. We also explore the relationships between Šapovalov elements coming from different roots, and their behavior when the Borel subalgebra is changed.