On modified extension graphs of a fixed atypicality

In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras gl(m|n), osp(M|2n) and q_m. We consider a simplified version of the extension graph which is produced from the Ext¹-graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor has indecomposable isotypical components. This yields semisimplicity of Duflo-Serganova functor for Fin(gl(m|n)) and for Fin(osp(M|2n)).