Tempered D-modules and Borel-Moore homology vanishing

Abstract

We characterize the tempered part of the automorphic Langlands category D-mod(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 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 of the indscheme $Maps(\Sigma ,G)$ vanishes.

References in corpus (2)

• Sheaves of categories with local actions of Hochschild cochains
• The spectral gluing theorem revisited

Cited by in corpus (1)

• Deligne-Lusztig duality on the stack of local systems