Block decompositions in the local p-adic Langlands program

Abstract: Let F be a finite extension of the field of p-adic numbers. The classical local Langlands correspondence relates complex representations of the Galois group of F (more precisely of the Weil-Deligne group of F) to so-called smooth representations of the group G = GL_n(F) on complex vector spaces. Using the concept of a center of a category, Joseph Bernstein has described the decomposition into blocks of the category of those smooth representations of G. The blocks are parametrized by "cuspidal data", which in turn consist of (equivalence classes of) Levi subgroups and supercuspidal representations of the latter. The category is thus highly disconnected, as there is no way to "interpolate" between different Levi subgroups and different supercuspidal representations. In this talk we will consider so-called locally analytic representations of G on p-adic vector spaces and see that the category of those representations decomposes into only finitely many blocks.

We will put this result into the context of a localization theory for those representations proposed by Emerton-Gee-Hellmann, and see that this block decomposition has a geometric analogue in terms of the Emerton-Gee stack of (phi,Gamma)-modules. This is joint work with Daniel Le and Zichuan Wang.

algebraic geometrynumber theory

Audience: researchers in the topic

FGC-HRI-IPM Number Theory Webinars

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