Endotrivial Modules for Finite Group Schemes

TL;DR

This paper extends the study of endotrivial modules from finite groups to arbitrary finite group schemes, providing new classifications and computational methods for various classes of algebraic groups.

Contribution

It investigates endotrivial modules over finite group schemes, offering classifications for simple, induced, Weyl, and tilting modules over Frobenius kernels.

Findings

Finitely generated endotrivial groups for finite group schemes.

Classification of endotrivial modules for Frobenius kernels of reductive groups.

Applicability to infinitesimal group schemes like Frobenius kernels and unipotent radicals.

Abstract

It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we investigate endotrivial modules over arbitrary finite group schemes. Our results can be applied to computing the endotrivial group for several classes of infinitesimal group schemes which include the Frobenius kernels of parabolic subgroups, and their unipotent radicals(for reductive algebraic groups). For G reductive, we also present a classification of simple, induced/Weyl and tilting modules (G-modules) which are endotrivial over the Frobenius kernel G_r of G.