Working Seminar on Representation Theory

It is an old question in representation theory whether any finitely-generated smooth representation of a $p$-adic group has a resolution by representations induced from (finite-dimensional representations of) compact subgroups. We construct such a resolution in a non-unique way, with non-uniqueness coming from a choice of lifting of the representation to the "compactified category" of Bezrukavnikov and Kazhdan. The machinery for constructing the resolution starting from an object of the compactified category involves techniques coming from mirror symmetry.