In-Person Talk

Given a connected reductive group G and a Levi subgroup M,Braverman-Gaitsgory and Laumon constructed geometric Eisenstein functors which take Hecke eigensheaves on the moduli stack Bun_{M} of M-bundles on a curve to eigensheaves on the moduli stack Bun_{G} of G-bundles. Recently, Fargues and Scholze constructed a general candidate for the local Langlands correspondence by doing geometric Langlands on the Fargues-Fontaine curve. In this talk, we explain recent work on carrying the theory of geometric Eisenstein series over to the Fargues-Scholze setting. In particular, we explain how, given the eigensheaf S_{\chi} on Bun_{T} attached to a smooth character \chi of the maximal torus T, one can construct an eigensheaf on Bun_{G} under a certain genericity hypothesis on \chi, by applying a geometric Eisenstein functor to S_{\chi}. Assuming the Fargues-Scholze correspondence satisfies certain expected properties, we fully describe the stalks of this eigensheaf in terms of normalized parabolic inductions of the generic \chi. This explicit description of the eigensheaf has several useful applications to describing the cohomology of local and global Shimura varieties and, time permitting, we will explain this.