BGG-decomposition for de Rham Banach sheaves

TL;DR

This paper extends the BGG method to complexes of Banach-modules with integrable connections on Shimura varieties, enabling new computations of p-adic automorphic cohomology and promising broader applications.

Contribution

It generalizes the BGG technique to Banach-module complexes on adic spaces, facilitating advanced cohomological computations in p-adic automorphic theory.

Findings

Extended BGG method to Banach-modules on Shimura varieties.

Provided a new computation method for p-adic automorphic cohomology.

Potential applications to Hilbert and Siegel cases.

Abstract

In this article we refer to "the BGG method" as a method in which by using Lie-algebra techniques one produces a complex of coherent sheaves on a Shimura variety, which is quasi-isomorphic to the de Rham complex of an automorphic vector bundle with integrable connection, on that Shimura variety. These BGG-complexes are in many ways "smaller" than the de Rham complex and have better cohomological properties. These BGG complexes contribute to the understanding of automorphic representations. In this article, we extend the BGG technique and apply it to complexes of Banach-modules with integrable connection on relevant open subspaces of Shimura varieties, seen as adic analytic spaces. Furthermore, we explain how this method can be used to obtain a new computation of the de Rham cohomology of certain p-adic automorphic representations in the GL_2 case. In future work, the authors intend to apply the techniques developed here to the Hilbert and Siegel cases.