Abstract

Classically, the BGG category O of a complex semisimple Lie algebra is a subcategory of its category of representations which is particularly well-behaved. It contains all the highest weight modules and so in particular all the finite dimensional representations, and it has nice combinatorics (e.g. BGG reciprocity). There is a natural analogue of this category for quantum groups, which more precisely corresponds to the integral category O (i.e. the direct sum of all the integral blocks). A few years ago, Tobias Schmidt defined a category O for the Arens-Michael envelope of the enveloping algebra of a p-adic Lie algebra. This Arens-Michael envelope can be defined as a certain Fréchet completion of the enveloping algebra, and it satisfies certain properties which makes it what is called a Fréchet-Stein algebra. These algebras have a nice category of modules, called coadmissible, and Schmidt defined his category O as a certain subcategory of the category of all coadmissible modules. His main result was that his category is equivalent to the usual category O of the Lie algebra. In this talk, we will explain how to construct a quantum analogue of the Arens-Michael envelope and a category O for it. We will then see that the analogue of Schmidt’s theorem is also true for our category.