Functions on the commuting stack via Langlands duality

TL;DR

This paper computes the algebra of functions on commuting stacks of complex reductive groups using Betti Geometric Langlands, proving the reducedness of the invariant functions and providing new technical decompositions.

Contribution

It introduces new semi-orthogonal decompositions and calculations of endomorphisms in the context of Betti Geometric Langlands, advancing understanding of commuting stacks.

Findings

The ring of invariant functions on the commuting scheme is reduced.

Semi-orthogonal decomposition of the cocenter of the affine Hecke category.

Calculation of endomorphisms of a Whittaker sheaf in parabolic induction context.

Abstract

We calculate the dg algebra of global functions on commuting stacks of complex reductive groups using tools from Betti Geometric Langlands. In particular, we prove that the ring of invariant functions on the commuting scheme is reduced. Our main technical results include: a semi-orthogonal decomposition of the cocenter of the affine Hecke category; and the calculation of endomorphisms of a Whittaker sheaf in a diagram organizing parabolic induction of character sheaves.