Verma modules over \(p\)-adic Arens-Michael envelopes of reductive Lie algebras (Q2437946)

Let \(p\) be a prime number. The author constructs and investigates an analogue of the classical (parabolic) BGG category \(\mathcal{O}\) for the Arens-Michael envelope of a split reductive Lie algebra of a \(p\)-adic Lie group defined over a locally compact \(p\)-adic field. The construction is based on the weight theory for topological Fréchet modules over commutative Fréchet algebras. The main result established an equivalence for certain blocks of \(\mathcal{O}\) over the Arens-Michael envelope and over the original universal enveloping algebra. As a consequence, this clarifies the structure of Verma modules over the Arens-Michael envelope.