Affine Springer fibers and small quantum groups

“I will describe a work in progress (with Michael McBreen) where we
identify a regular block in the category of small quantum group modules
with a category that can be described as the category of microlocal
sheaves on a certain affine Springer fiber. Here the specific technical
incarnation of the “microlocal sheaves”
concept is based on presenting the affine Springer fiber as a
Hamiltonian reduction of a cotangent bundle. This is heuristically
(though at
present not technically) related to a conjecture of Shan-Vasserot and
myself where this Springer fiber is used to describe the
center of the small quantum group; the conjecture was partly motivated
by a work Lachowska and Qi on a problem by I. Frenkel.
Time permitting I will also describe some ideas from a joint project
with Pablo Boixeda Alvarez, Peng Shan and Eric Vasserot aimed
at proving that conjecture.