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.
