Even-primitive vectors in induced supermodules for general linear supergroups and in costandard supermodules for Schur superalgebras

Let $G=GL(m|n)$ be the general linear supergroup over an algebraically closed field $K$ of characteristic zero and let $G_{ev}=GL(m)\times GL(n)$ be its even subsupergroup. The induced supermodule $H^0_G(\lambda)$, corresponding to a dominant weight $\lambda$ of $G$, can be represented as $H^0_{G_{ev}}(\lambda)\otimes \Lambda(Y)$, where $Y=V_m^*\otimes V_n$ is a tensor product of the dual of the natural $GL(m)$-module $V_m$ and the natural $GL(n)$-module $V_n$, and $\Lambda(Y)$ is the exterior algebra of $Y$. For a dominant weight $\lambda$ of $G$, we construct explicit $G_{ev}$-primitive vectors in $H^0_G(\lambda)$. Related to this, we give explicit formulas for $G_{ev}$-primitive vectors of the supermodules $H^0_{G_{ev}}(\lambda)\otimes \otimes^k Y$. Finally, we describe a basis of $G_{ev}$-primitive vectors in the largest polynomial subsupermodule $\nabla(\lambda)$ of $H^0_G(\lambda)$ (and therefore in the costandard supermodule of the corresponding Schur superalgebra $S(m|n)$). This yields a description of a basis of $G_{ev}$-primitive vectors in arbitrary induced supermodule $H^0_G(\lambda)$.