Title of Talk: p-adic boundary values (3 lectures)
For real symmetric spaces there is an extensive theory of boundary values. For example, the classical Cauchy transform provides a link between the holomorphic functions on the upper half plane and certain distributions (=hyperfunctions) on the real projective line at its boundary.
In these lectures we will report on joint work in progress to set up an analogous theory for Drinfeld's p-adic symmetric space for the group GL_(n). Specifically, we will study boundary values for the top-dimensional global holomorphic forms on Drinfeld's space X.
This work has implications from several points of view. From a cohomological point of view, it gives an explicit analytic interpretation of Serre duality on the Stein space X, refining the earlier computation by Schneider and Stuhler of the deRham cohomology of X. From a representation theoretic point of view, it yields an embedding of the p-adic holomorphic discrete series into a locally analytic principal series of GL_(n). This principal series is a class of continuous representations of GL_n in p-adic locally convex vector spaces.
We hope that, by broadening the class of representations under consideration, we will eventually be able to explore p-adic analysis in the local Langlands theory of smooth representations.