On semisimplicity of Jantzen middles for the periplectic Lie superalgebra

TL;DR

This paper characterizes when blocks of the category O for the periplectic Lie superalgebra have non-semisimple Jantzen middles, linking it to the presence of atypical simple modules and implications for Kazhdan-Lusztig theory.

Contribution

It establishes a precise criterion for the semisimplicity of Jantzen middles in the category O of the periplectic Lie superalgebra, revealing the role of atypical weights.

Findings

Non-semisimple Jantzen middles occur iff the block contains an atypical simple module.

Atypical blocks do not admit Kazhdan-Lusztig theory as defined by Cline, Parshall, and Scott.

The result connects the structure of category O with the representation theory of the periplectic Lie superalgebra.

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.