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.

中文翻译：

---

淬火的D模块和Borel-Moore同源性消失

我们使用仿射Grassmannian中大单元格的几何形状来表征自守的Langlands类别\（\ mathfrak {D}（{\ text {Bun}} _ G）\）的回火部分。我们推论，对于G非阿贝尔族人，回火的D-模块在紧凑的支撑下没有de Rham的同调性。后一个事实可以归结为一个具体的陈述，我们可以使用Ran空间和一些显式的t结构估计来证明这一点：对于G非阿贝尔和\（\ Sigma \）平滑仿射曲线，indscheme的Borel-Moore同源\（{\ text {Maps}}（\ Sigma，G）\）消失了。