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

TL;DR

This paper investigates the structure of coadmissible modules over the p-adic Arens-Michael envelope of a reductive Lie algebra, establishing a connection with the classical parabolic BGG category.

Contribution

It identifies a highest weight category within the nonarchimedean setting that corresponds to the classical parabolic BGG category, expanding the understanding of p-adic Lie algebra representations.

Findings

Established an explicit equivalence with the classical parabolic BGG category

Expanded the theory of modules over p-adic Arens-Michael envelopes

Replaced previous preprint with a more comprehensive final version

Abstract

Let K be a locally compact nonarchimedean field, g a split reductive Lie algebra over K and U(g) its universal enveloping algebra. We study the category C_g of coadmissible modules over the nonarchimedean Arens-Michael envelope of U(g). Let p be a parabolic subalgebra of g. The main result identifies a certain explicitly given highest weight category inside C_g with the classical parabolic BGG category of g relative to p. This paper is in final form, replaces and expands the former preprint 'BGG reciprocity for p-adic Arens-Michael envelopes of semisimple Lie algebras' and appears in Journal of Algebra.