\vS{a}povalov elements and the Jantzen sum formula for contragredient Lie superalgebras

Abstract

If $\mathfrak{g}$ is a contragredient Lie superalgebra and $\gamma$ is a root of $\mathfrak{g},$ we prove the existence and uniqueness of \v{S}apovalov elements for $\gamma$ and give upper bounds on the degrees of their coefficients. Then we use \v{S}apovalov elements to define some new highest weight modules. If $X$ is a set of orthogonal isotropic roots and $\lambda \in \mathfrak{h}^*$ is such that $\lambda +\rho$ is orthogonal to all roots in $X$, we construct a highest weight module $M^X(\lambda)$ with character $\epsilon^\lambda {p}_X$. Here $p_X$ is a function that counts partitions not involving roots in $X$. Examples of such modules can be constructed via parabolic induction provided $X$ is contained in the set of simple roots of some Borel subalgebra. However our construction works without this condition and provides a highest weight module for the distinguished Borel subalgebra. The main results are analogs of the \v{S}apovalov determinant and the Jantzen sum formula for $M^X(\lambda)$ when $\mathfrak{g}$ has type A.For the proof it is enough to study the behavior for certain "relatively general" highest weights. Using an equivalence of categories due to Cheng, Mazorchuk and Wang, the information we require is deduced from the behavior of the modules $M^X(\lambda)$ when $\mathfrak{g}=\mathfrak{gl}(2,1)$ or $\mathfrak{gl}(2,2)$. These low dimensional cases are studied in detail in an appendix.