A construction of representations of affine Weyl groups

Abstract

Let $G$ be a complex, semisimple, simply connected algebraic group with Lie algebra ${\frak g}$. We extend scalars to the power series field in one variable ${\rm C}((\pi))$, and consider the space of Iwahori subalgebras containing a fixed nil-elliptic element of ${\frak g} \otimes {\rm C}((\pi))$, i.e. fixed point varieties on the full affine flag manifold. We define representations of the affine Weyl group in the homology of these varieties, generalizing Kazhdan and Lusztig's topological construction of Springer's representations to the affine context.