Joint IAS/Princeton University Arithmetic Geometry Seminar

Given a connected reductive group G and a Levi subgroup M, Braverman-Gaistgory 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.