On the induction functor from group algebras to distribution algebras

TL;DR

This paper explores the induction functor from finite groups of Lie type to algebraic groups over finite fields, linking cohomology and representation theory to transfer data between these structures.

Contribution

It investigates the connections between a fundamental theorem and the induction functor using filtrations, truncation, and cohomological methods.

Findings

Established links between the induction functor and cohomology of $G$.

Transferred data from Frobenius kernels to finite groups of Lie type.

Connected algebraic group theory with finite group representations.

Abstract

Let $G$ be a reductive algebraic group scheme defined over ${\mathbb F}_{p}$ and $k$ be an algebraically closed field of characteristic $p$. There are two associated families of finite group schemes, the $r$-th Frobenius kernels, denoted by $G_r$, and the fixed points of the iterated Frobenius map, the finite groups of Lie type, denoted by $G(\mathbb{F}_q).$ Bendel, Nakano and Pillen initiated the investigation of the induction functor $\operatorname{ind}_{G(\mathbb{F}_q)}^G-$. Using filtrations and truncation, large amounts of data coming from the algebraic group and the Frobenius kernels can be transferred to the finite group. This paper looks at connections between a fundamental theorem of Chastkofsky and Jantzen and the induction functor via the cohomology and representation theory of $G$.