A colimit presentation of $\mathcal{D}(G(K))$ via the Bott-Samelson hypercover

TL;DR

This paper demonstrates that the derived category of D-modules on the loop group of a semisimple algebraic group can be constructed as a colimit of categories on parahoric subgroups, simplifying previous proofs and applying hyperdescent techniques.

Contribution

It provides a simpler proof of the colimit presentation of D-modules on the loop group using a combinatorial hyperdescent approach and develops a new model for path spaces in simplicial complexes.

Findings

Equivalence of D-modules on the loop group to a colimit over parahoric subgroups

Development of a combinatorial model for path spaces in simplicial complexes

Applications to triviality results for D-modules on Bruhat-Tits buildings and affine Springer resolutions

Abstract

Let $G$ be a semisimple, simply connected algebraic group over an algebraically closed field of characteristic zero. We prove that the $\infty$-category of D-modules on the loop group of $G$ is equivalent to the monoidal colimit of the $\infty$-categories of D-modules on the standard parahoric subgroups. This also follows from arXiv:2009.10998, but the present paper gives a simpler proof. The idea is to develop a combinatorial model for the path space of a simplicial complex, in which 'paths' are sequences of adjacent simplices, and to use a generalized version of hyperdescent for D-modules. We also give two more applications of this hyperdescent theorem: triviality of D-modules on the 'schematic Bruhat-Tits building,' which was first established by Varshavsky using a different method, and triviality of D-modules on the `simplicial affine Springer resolution.'