# On the Bonnaf\'e--Dat--Rouquier Morita equivalence

## TL;DR

This paper extends Morita equivalence results to new cases involving Deligne-Lusztig varieties, specifically for semisimple elements in type D with non-cyclic centralizer component groups, broadening previous work.

## Contribution

It proves a Morita equivalence via cohomology of Deligne-Lusztig varieties in cases not previously covered, notably for type D with non-cyclic centralizers.

## Key findings

- Established Morita equivalence for specific semisimple elements in type D
- Extended the applicability of Deligne-Lusztig cohomology in representation theory
- Provided new cases where Morita equivalence holds beyond prior known scenarios

## Abstract

We prove that the cohomology group of a Deligne-Lusztig variety defines a Morita equivalence in a case which is not covered by the argument by Bonnaf\'e, Dat and Rouquier, specifically we consider the situation for semisimple elements in type $D$ whose centralizer has non-cyclic component group.