Geometric and homological properties of affine Deligne-Lusztig varieties

This paper studies affine Deligne-Lusztig varieties Ka, (b) in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of K(sic) (b) for a minimal length element 71; in the conjugacy class of an extended affine Weyl group. We then provide a reduction method that relates the structure of X,(sic) (b) for arbitrary elements 71; in the extended affine Weyl group to those associated with minimal length elements. Based on this reduction, we establish a connection between the dimension of affine Deligne-Lusztig varieties and the degree of the class polynomial of affine Hecke algebras. As a consequence, we prove a conjecture of Gortz, Haines, Kottwitz and Reuman.