Geometric and homological properties of affine Deligne-Lusztig varieties

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This paper studies affine Deligne-Lusztig varieties X(b) in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of X(b) for a minimal length element in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in HL to the affine case. We then provide a reduction method that relates the structure of X(b) for arbitrary elements 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 Görtz, Haines, Kottwitz and Reuman in GHKR.