Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras: A Survey

This is a survey of some recent results on Sapovalov elements and the Jantzen filtration for contragredient Lie superalgebras. The topics covered include the existence and uniqueness of the Sapovalov elements, bounds on the degrees of their coefficients and the behavior of Sapovalov 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 $\lambda \in h^*$ is such that $\lambda +\rho$ is orthogonal to all roots in X, we construct highest weight modules with character $\epsilon^\lambda p_X$. Here $p_X$ 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.