  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.
