Verma modules over p-adic Arens–Michael envelopes of reductive Lie algebras

Abstract

Let K be a locally compact p-adic field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We investigate the category Cg of coadmissible modules over the p-adic Arens–Michael envelope Uˆ(g) of U(g). Let p⊆g be a parabolic subalgebra. The main result gives a canonical equivalence between the classical parabolic BGG category of g relative to p and a certain explicitly given highest weight subcategory of Cg. This completely clarifies the “Verma module theory” over Uˆ(g).