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

Tempered D-modules and Borel-Moore homology vanishing