Minimal length elements of extended affine Weyl groups

Abstract

Let W be an extended affine Weyl group. We prove that the minimal length elements wO of any conjugacy class O of W satisfy some nice properties, generalizing results of Geck and Pfeiffer [On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), 79-94] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra H. We prove that TwO, where O ranges over all the conjugacy classes of W, forms a basis of the cocenter H/[H,H]. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne-Lusztig varieties He [Geometric and cohomological properties of affine Deligne-Lusztig varieties, Ann. of Math. (2) 179 (2014), 367-404].