Block decomposition via the geometric Satake equivalence

TL;DR

This paper offers a new geometric proof for the block decomposition of representations of reductive algebraic groups over positive characteristic fields, using the geometric Satake equivalence and Smith-Treumann theory.

Contribution

It introduces a novel geometric approach to understanding blocks in representation categories, providing bounds on weight chains and new proofs for quantum group decompositions.

Findings

Bound for the length of minimal chains linking weights in the same block.

New proof for the block decomposition of quantum groups at roots of unity.

Application of Smith-Treumann theory in the Satake category.

Abstract

We give a new proof for the description of the blocks in the category of representations of a reductive algebraic group $\mathbf{G}$ over a field of positive characteristic $\ell$ (originally due to Donkin), by working in the Satake category of the Langlands dual group and applying Smith-Treumann theory as developed by Riche and Williamson. On the representation theoretic side, our methods enable us to give a bound for the length of a minimum chain linking two weights in the same block, and to give a new proof for the block decomposition of a quantum group at an $\ell$-th root of unity.