Block decomposition via the geometric Satake equivalence

We give a new proof for the description of the blocks in the category of representations of a reductive algebraic group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="bold">G</mml:mi> </mml:math> over a field of positive characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> (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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℓ</mml:mi> <mml:mtext>th</mml:mtext> </mml:msup> </mml:math> root of unity.