Geometric and homological properties of affine Deligne-Lusztig varieties

Xuhua He University of Maryland

Arithmetic Geometry and Commutative Algebra mathscidoc:1803.07001

Ann. of Math., 179, 367-404, 2014
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}.
Affine Deligne-Lusztig varieties, $\s$-conjugacy classes
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  title={Geometric and homological properties of affine Deligne-Lusztig varieties},
  author={Xuhua He},
  booktitle={Ann. of Math.},
Xuhua He. Geometric and homological properties of affine Deligne-Lusztig varieties. 2014. Vol. 179. In Ann. of Math.. pp.367-404.
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