On Semisimplicity of Jantzen Middles for the Periplectic Lie Superalgebra

Abstract

Abstract We prove that an integral block of the category $\mathcal O$ of the periplectic Lie superalgebra contains a non-semisimple Jantzen middle if and only if it contains a simple module of atypical highest weight. As a consequence, every atypical integral block of $\mathcal O$ does not admit a Kazhdan–Lusztig theory in the sense of Cline, Parshall, and Scott.