Tempered D-modules and Borel–Moore homology vanishing

We characterize the tempered part of automorphic Langlands category D-mod(Bun_G) using geometry big cell in affine Grassmannian. deduce that, for $G$ non-abelian, D-modules have no de Rham cohomology with compact supports. The latter fact boils down to a concrete statement, which we prove Ran space and some explicit t-structure estimates: non-abelian $\Sigma$ smooth curve, Borel-Moore homology indscheme $Maps(\Sigma,G)$ vanishes.