Abstract

In 1981, Beilinson and Bernstein used their celebrated localisation theorem to prove the Kazdhan-Lusztig conjecture on characters of highest weight modules. The theorem established a correspondence between representations of a complex semisimple Lie algebra and modules over certain sheaves of differential operators on the flag variety of the associated algebraic group, and it is considered as one of the starting points of geometric representation theory. Since then there have been many generalisations of this result, as well as analogues of it in different contexts. For example, Backelin and Kremnizer proved a localisation theorem for representations of quantum groups. More recently, Ardakov and Wadsley proved a localisation theorem working with certain completed enveloping algebras of p-adic Lie algebras. In this talk I will explain what these two specific theorems say and how one might attempt to combine the ideas involved in their proofs to obtain a localisation theorem for certain p-adic completions of quantum groups.