# Tempered D-modules and Borel-Moore homology vanishing

## TL;DR

This paper characterizes the tempered automorphic D-modules via affine Grassmannian geometry and proves their de Rham cohomology with compact supports vanishes for non-abelian groups, using Ran space techniques.

## Contribution

It introduces a geometric characterization of tempered D-modules and establishes a vanishing result for Borel-Moore homology of mapping indschemes for non-abelian groups.

## Key findings

- Tempered D-modules have no de Rham cohomology with compact supports for non-abelian G.
- Borel-Moore homology of mapping indschemes vanishes for non-abelian G and affine curves.
- The characterization uses the geometry of the big cell in the affine Grassmannian.

## 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.