The coarse quotient for affine Weyl groups and pseudo-reflection groups

TL;DR

This paper classifies sheaves on the coarse quotient of affine Weyl groups acting on dual Cartan spaces, providing a pointwise descent criterion that generalizes previous results to pseudo-reflection groups.

Contribution

It introduces a pointwise criterion for descent of equivariant sheaves on affine Weyl group quotients and extends this to arbitrary finite pseudo-reflection groups.

Findings

Classified sheaves on the coarse quotient via pointwise descent criteria.

Proved the criterion for arbitrary finite groups acting on vector spaces.

Generalized a recent result of Lonergan to pseudo-reflection groups.

Abstract

We study the coarse quotient $\mathfrak{t}^*//W^{\text{aff}}$ of the affine Weyl group $W^{\text{aff}}$ acting on a dual Cartan $\mathfrak{t}^*$ for some semisimple Lie algebra. Specifically, we classify sheaves on this space via a "pointwise" criterion for descent, which says that a $W^{\text{aff}}$-equivariant sheaf on $\mathfrak{t}^*$ descends to the coarse quotient if and only if the fiber at each field-valued point descends to the associated GIT quotient. We also prove the analogous pointwise criterion for descent for an arbitrary finite group acting on a vector space. Using this, we show that an equivariant sheaf for the action of a finite pseudo-reflection group descends to the GIT quotient if and only if it descends to the associated GIT quotient for every pseudo-reflection, generalizing a recent result of Lonergan.