Šapovalov elements and the Jantzen filtration for contragredient Lie superalgebras: a survey

Abstract

This is a survey of some recent results on Šapovalov elements and the Jantzen filtration for contragredient Lie superalgebras. The topics covered include the existence and uniqueness of the Šapovalov elements, bounds on the degrees of their coefficients and the behavior of Šapovalov elements when the Borel subalgebra is changed. There is always a unique term whose coefficient has larger degree than any other term. This allows us to define some new highest weight modules. If X is a set of orthogonal isotropic roots and λ ∈ h* is such that λ + ρ is orthogonal to all roots in X, we construct highest weight modules with character ελpx. Here pX is a partition function that counts partitions not involving roots in X. When |X| = 1, these modules are used to give a Jantzen sum formula for Verma modules in which all terms are characters of modules in the category O with positive coefficients.