Geometric Eisenstein Series on Fargues–Fontaine Curves

Let G be a reductive group over the p-adic numbers with P = MU a parabolic subgroup. A basic fact in smooth representation theory is that parabolic induction preserves the property of being admissible.  In this talk, we will discuss the analogue of this in the geometrization of the local Langlands program. In particular, smooth representations will be replaced by sheaves on Bun_G, the moduli stack of G-bundles on the Fargues–Fontaine curve, and parabolic induction will be replaced by a geometric Eisenstein functor carrying sheaves on Bun_M to sheaves on Bun_G. The property of being admissible translates into the rather bizarre property of being ULA over a point, which is a new phenomenon native to analytic variants of the geometric Langlands program. The main result we will discuss is that the geometric Eisenstein functor sends sheaves which are ULA over a point on Bun_M to sheaves which are ULA over a point on Bun_G. This generalizes the basic fact on admissibility mentioned at the beginning, and much more interestingly shows that various gluing functors on Bun_G send admissible representations to admissible representations.  Along the way, we hope to explain some of the similarities and differences between the usual geometric Langlands programs and the Fargues–Scholze geometric Langlands program, mostly stemming from the differences between l-adic sheaves on algebraic and p-adic analytic spaces, respectively.