Semisimplicity of the $DS$ functor for the orthosymplectic Lie superalgebra

Abstract

We prove that the Duflo-Serganova functor $D{S}_x$ attached to an odd nilpotent element $x$ of $\mathfrak{osp}(m|2n)$ is semisimple, i.e. sends a semisimple representation $M$ of $\mathfrak{osp}(m|2n)$ to a semisimple representation of $\mathfrak{osp}(m-2k|2n-2k)$ where $k$ is the rank of $x$. We prove a closed formula for $D{S}_x(L(\lambda ))$ in terms of the arc diagram attached to $\lambda$.