Many interesting quantum algebras arising in representation theory are obtained via quantum Hamiltonian reduction, which mimics the physical process of reducing a gauge symmetry on a phase space. The most important examples for us are G-character varieties -- moduli spaces of G-local systems -- on closed surfaces: these are obtained by classical Hamiltonian reduction, and their quantizations which we introduced with Ben-Zvi and Brochier are computed by quantum Hamiltonian reduction. Related examples are multiplicative quiver varieties of Crawley-Boevey and Shaw and their quantizations which I have introduced.
In this talk, I'll consider a very general setup in which such a quantum algebra obtained by reduction is specialized to a root of unity. We see several related things happen: 1) The quantum algebra sprouts a large center, which in fact identifies with the functions on the classical moduli space, 2) The quantum algebra is, etale-locally, a matrix algebra (a.k.a. Azumaya algebra) over the smooth locus, and moreover is a "Poisson order", meaning that there is a connection relating the fibers of the quantum algebra at different points. The main point of the talk is that all of this rich structure is compatible with Hamiltonian reduction, so it can be established before reduction (where it is much easier), and carried through the process. In the case of character varieties, this confirms so-called Unicity Conjectures of Bonahon-Wong. This is joint work with Iordan Ganev and Pavel Safronov.