Completed cohomology of Hilbert modular varieties below middle degree

Abstract: We study the locally analytic vectors of completed cohomology of Hilbert modular varieties below middle degree. Our main result says that the universal enveloping algebra action has to factor through certain quotients. Application includes bounds on the Gelfand-Kirillov dimension and a density result. The method is a continuation of our joint work with Kai-Wen Lan.