Linus Hamann (Harvard)

Shimura Varieties are of fundamental importance in number theory, by virtue of the fact that their cohomology provides the only known geometric incarnations of the global Langlands correspondence over number fields. When the Shimura Variety is non-compact, it is often very desirable to work with the intersection cohomology of its minimal compactification, as this can be related to L^2-cohomology of the associated symmetric space, and the latter is computed in terms of the (g,K)-cohomology of the discrete spectrum of the space of automorphic forms via the work of Borel-Cassleman. In this talk, we investigate the relationship of intersection cohomology with a structure coming from the incarnation of the Shimura variety as a p-adic manifold. Namely, we study how the intersection cohomology interacts with Mantovan’s filtration, which generalizes the filtration on the cohomology of the modular curve coming from the ordinary and supersingular locus. We accomplish this by describing the intersection cohomology as a certain Hecke operator applied to a sheaf on Bun_{G}, the moduli stack of G-bundles of the Fargues-Fontaine curve, extending results of this form for the compactly supported and regular cohomology. The aforementioned filtration can then be understood via excision applied to the Harder-Narasimhan stratification of this sheaf on Bun_{G}. This leads to the hope that Borel-Cassleman’s description of the L^{2}-cohomology can be refined to a description of our sheaf in terms of direct sums of certain “sheared” Hecke eigensheaves on Bun_{G}, where the shearing encodes the known relationship between the (g,K)-cohomology and the Arthur SL_{2} of the A-parameters attached to the discrete spectrum. This is joint work in progress with Ana Caraiani and Mingjia Zhang.