On some partitions of an affine flag variety

Xuhua He

Published 2009-10-27Version 1

In this paper, we discuss some partitions of affine flag varieties. These partitions include as special cases the partition of affine flag variety into affine Deligne-Lusztig varieties and the affine analogue of the partition of flag varieties into $\cb_w(b)$ introduced by Lusztig in \cite{L1} as part of the definition of character sheaves. Among other things, we give a formula for the dimension of affine Deligne-Lusztig varieties for classical loop groups in terms of degrees of class polynomials of extended affine Hecke algebra. We also prove that any simple $GL_n(\FF_q((\e)))$-module occurs as a subquotient of the cohomology of affine Deligne-Lusztig variety $X_w(1)$ for some $w$ in the extended affine Weyl group $\ZZ^n \rtimes S_n$ must occurs for some $w$ in the finite Weyl group $S_n$. Similar result holds for $Sp_{2n}$.