Glauberman correspondents and extensions of nilpotent block algebras

TL;DR

This paper proves Morita equivalence between extensions of nilpotent block algebras and their Glauberman correspondents under certain conditions, extending key theorems in modular representation theory.

Contribution

It provides new proofs and slight improvements of foundational results on extensions of nilpotent blocks, establishing Morita equivalence under additional group-theoretic conditions.

Findings

Proved Morita equivalence of extensions under specific conditions

Revisited and improved key theorems by Külshammer and Puig

Extended Harris-Linckelman's and Koshitani-Michler's theorems

Abstract

The main purpose of this paper is to prove that the extensions of a nilpotent block algebra and its Glauberman correspondent block algebra are Morita equivalent under an additional group-theoretic condition. In particular, Harris and Linckelman's theorem and Koshitani and Michler's theorem are covered. The ingredient to carry out our purpose is the two main results in K\"ulshammer and Puig's work "Extensions of nilpotent blocks"; we actually revisited them, giving completely new proofs of both and slightly improving the second one.