The first half of the talk is introductory in nature.  We will introduce the
approach developed by Ginzburg, Kazhdan and Lusztig of studying the affine
Hecke algebra via K-theory of the Steinberg stack.  In the way the Springer
theory is sheafy over the nilpotent cone, one might hope for a sheafy
realization of the affine Hecke algebra.  This question motivates our study
of derived loop spaces.