Geometric and homological properties of affine Deligne-Lusztig varieties

Pages 367-404 from Volume 179 (2014), Issue 1 by Xuhua He

Abstract

This paper studies affine Deligne-Lusztig varieties $X_{\tilde w}(b)$ in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of $X_{\tilde w}(b)$ for a minimal length element $\tilde w$ in the conjugacy class of an extended affine Weyl group. We then provide a reduction method that relates the structure of $X_{\tilde w}(b)$ for arbitrary elements $\tilde w$ 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 Görtz, Haines, Kottwitz and Reuman.

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Keywords
,
DOI
https://doi.org/10.4007/annals.2014.179.1.6
MR
3126571
Zbl
06260590
Milestones
Received: 29 January 2012
Revised: 13 January 2013
Accepted: 26 February 2013
Published online: 1 January 2014

Authors

Xuhua He

The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Current address:

University of Maryland, College Park, MD