Jantzen filtration of Weyl modules for general linear supergroups

Let $G=GL(m\mathrm{\mid }n)$ be a general linear supergroup over an algebraically closed field $k$ of odd characteristic $p$. In this paper we construct Jantzen filtration of Weyl modules $V(\lambda )$ of $G$ when $\lambda$ is a typical weight in the sense of Kac’s definition, and consequently obtain a sum formula for their characters. By Steinberg’s tensor product theorem, it is enough for us to study typical weights with aim to formulate irreducible characters. As an application, it turns out that an irreducible $G$-module $L(\lambda )$ can be realized as a Kac module if and only if $\lambda$ is $p$-typical.