On the Duflo-Serganova functor for the queer Lie superalgebra

Abstract

We study the Duflo-Serganova functor ${\mathrm{DS}}_x$ for the queer Lie superalgebra ${\mathfrak{q}}_n$ and for all odd $x$ with $[x,x]$ semisimple. For the case when the rank of $x$ is $1$ we give a formula for multiplicities in terms of the arc diagram attached to $\lambda$. Further, we prove that ${\mathrm{DS}}_x(L)$ is semisimple if $L$ is a simple finite-dimensional module and $x$ is of rank $1$ satisfying $x^2=0$.

33 pages; for $D{S}_x$ with $x$ is of rank 1 and $x^2=0$: computed grading, and proved it is semisimple functor; other minor changes made