The block decomposition of the principal representation category of reductive algebraic groups with Frobenius maps

TL;DR

This paper investigates the structure of the principal representation category of reductive algebraic groups over finite fields, revealing its block decomposition based on central characters, and analyzing extensions of simple modules.

Contribution

It provides the first detailed block decomposition of the principal representation category for reductive algebraic groups over finite fields, based on central characters.

Findings

Block decomposition parameterized by central characters

Extension analysis of simple modules in the category

Structural insights into the category's blocks

Abstract

Let ${\bf G}$ be a connected reductive algebraic group defined over the finite field $\mathbb{F}_q$ with $q$ elements. Let $\Bbbk$ be a field such that $\op{char} \Bbbk \ne \op{char} \mathbb{F}_q$. In this paper, we study the extensions of simple modules (over $\Bbbk$) in the principal representation category $\mathscr{O}(\bf G)$ which is defined in \cite{D1}. In particular, we get the block decomposition of $\mathscr{O}(\bf G)$, which is parameterized by the central characters of ${\bf G}$.