# The Steinberg linkage class for a reductive algebraic group

## TL;DR

This paper studies a specific subcategory of rational modules for a reductive algebraic group in positive characteristic, establishing an explicit equivalence with the entire module category and linking key functors.

## Contribution

It introduces the Steinberg component and provides an explicit equivalence with the category of rational modules, connecting Frobenius contraction and projection functors.

## Key findings

- Established an explicit equivalence between the Steinberg component and the category of rational G-modules.
- Connected the Frobenius contracting functor to the projection functor onto the Steinberg component.
- Derived consequences for the structure and representation theory of G-modules.

## Abstract

Let G be a reductive algebraic group over a field of positive characteristic and denote by C(G) the category of rational G-modules. In this note we investigate the subcategory of C(G) consisting of those modules whose composition factors all have highest weights linked to the Steinberg weight. This subcategory is denoted ST and called the Steinberg component. We give an explicit equivalence between ST and C(G) and we derive some consequences. In particular, our result allows us to relate the Frobenius contracting functor to the projection functor from C(G) onto ST .