Affine Springer fibers and representations

Relating representation categories of interest, such of modules over the quantum group, to topology of loop spaces has been an important theme in representation theory for some decades. I will describe a result of this sort involving a geometric object that has not appeared in this context until now, an affine Springer fiber, and mention its applications. Based on a joint work with Pablo Boixeda Alvarez, Michael McBreen and Zhiwei Yun.