The spectral gluing theorem revisited

We strengthen the gluing theorem occurring on the spectral side of the geometric Langlands conjecture. While the latter embeds IndCohN(LSG) into a category glued out of 'Fourier coefficients' parametrized by standard parabolics, our refinement explicitly identifies the essential image of such embedding

Tempered D-modules and Borel–Moore homology vanishing

We characterize the tempered part of the automorphic Langlands category D(BunG) 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 Σ a smooth affine curve, the Borel–Moore homology of the indscheme Maps(Σ,G) vanishes

Deligne-Lusztig duality on the stack of local systems

In the setting of the geometric Langlands conjecture, we argue that the phenomenon of divergence at infinity on BunG (that is, the difference between !-extensions and * -extensions) is controlled, Langlands-dually, by the locus of semisimple ˇG-local systems. To see this, we first rephrase the question in terms of Deligne–Lusztig duality and then study the Deligne–Lusztig functor DLspecGacting on the spectral Langlands DG category IndCohN(LSG). We prove that DLspecG is the projection IndCohN (LSG)↠QCoh (LSG), followed by the action of a coherent D-module StG∈D(LSG), which we call the SteinbergD-module. We argue that StG might be regarded as the dualizing sheaf of the locus of semisimple G-local systems. We also show that DL spec G, while far from being conservative, is fully faithful on the subcategory of compact objects