We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in \cite{GHKR2}. The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.

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

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