AbstractWe characterize the tempered part of the automorphic Langlands category $$\mathfrak {D}({\text {Bun}}_G)$$
D
(
Bun
G
)
using the geometry of the big cell in the affine Grassmannian. We deduce that, for G non-abelian, tempered D-modules have no de Rham cohomology with compact support. 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 $$\Sigma $$
Σ
a smooth affine curve, the Borel–Moore homology of the indscheme $${\text {Maps}}(\Sigma ,G)$$
Maps
(
Σ
,
G
)
vanishes.
Duality and de Rham cohomology for graded D-modules
