Abstract
We characterize the tempered part of the automorphic Langlands category mathfrak {D}({text {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 a smooth affine curve, the Borel–Moore homology of the indscheme {text {Maps}}(Sigma ,G) vanishes.
Highlights
The present paper is devoted to the study of the tempered condition appearing on the automorphic side of the geometric Langlands conjecture
The theory of loop group actions on differential graded (DG) categories is very convenient when dealing with such situations, and in particular when dealing with the Hecke action
For a map φ : Y → Z, we denote by Z∧Y its formal completion, which is by definition the fiber product Z ×ZdR YdR
Summary
The present paper is devoted to the study of the tempered condition appearing on the automorphic side of the geometric Langlands conjecture. The candidate is QCoh(LSG ), the DG category of quasi-coherent sheaves on the stack LSG := LSG (X ) of de Rham G -local systems on X This is the so-called spectral side of the conjecture. While this is known to be true for G abelian (see [1, Remark 11.2.7], as well as [33,34,39,40]), it is false for more general groups: two reasons for this failure are explained in [1, Section 1.1.2] and [23, Section 0.2.1]. Both reasons point at the fact that the spectral side of (1.1) is too small to match the automorphic side. This is a DG category that sits between QCoh(LSG ) and IndCoh(LSG )
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