The Steinberg linkage class for a reductive algebraic group

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

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Received 1 September 2017

Received revised 27 December 2017

Accepted 5 January 2018

Published 24 May 2022