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

We study the coarse quotient ${\mathfrak{t}}^{\ast }\mathrm{/}\mathrm{/}{W}^{\text{aff}}$ of the affine Weyl group $W^{\text{aff}}$ acting on a dual Cartan ${\mathfrak{t}}^{\ast }$ 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 ${\mathfrak{t}}^{\ast }$ 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.