Geometric and homological properties of affine Deligne-Lusztig varieties

Abstract

This paper studies affine Deligne-Lusztig varieties Xw(b) in the affine flag variety of a quasi-split tamely ramified group. We describe the geometric structure of Xw(b) for a minimal length elementw in the conjugacy class of an extended affine Weyl group, generalizing one of the main results in [18] to the affine case. We then provide a reduction method that relates the structure of Xw(b) for arbitrary elementsw 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 in [10].2000 Mathematics Subject Classification. 14L05, 20G25.By the definition of defect, def(b) = def(τ ) = n − ℓ(c). Therefore dim Xw(b) dw(b).By combining Theorem 12.1 and Corollary 10.4, we have that Corollary 12.2. If G is simple and δ = id. Letw ∈W ′ and b ∈ G(L) be a basic element with supp(η(w)) = S and κ(w) = κ(b). Then dim Xw(b) = dw(b).Remark. The split case was first conjectured in [10, Conjecture 1.1.3]. A weaker result for some split classical groups was proved in [8].

Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties

“…Let B(W , σ) be the set of σ-conjugacy classes ofW and B(W , σ) str be the set of straight σ-conjugacy classes ofW . Following [11], there exists a commutative diagram…”

“…In [11,Theorem 6.1], we give a criterion about the non-emptiness pattern of affine Deligne-Lusztig varieties in affine flag varieties in terms of class polynomials of affine Hecke algebras. The computation of class polynomials, however, is very hard in general.…”

“…It is then proved by Lucarelli [14] for classical split groups and then by Gashi [1] for unramified cases. The general case is proved in [11,Theorem 7.1]. Notice that if P J is a special maximal parahoric subgroup and µ is minuscule with respect toW , X(µ, b) J is in fact a single affine Deligne-Lusztig variety.…”

“…The proof is based on the reductive method in [11]à la Deligne and Lusztig, some remarkable combinatorial properties onW established in [12] and the Grothendieck conjecture for split groups proved by Viehmann in [26].…”

Cocenters and representations of pro-𝑝 Hecke algebras

“…We say that an elementw ∈W (1) is straight if (w n ) = n (w) for any n ∈ N. By [ [12], σ-conjugacy classes of connected reductive p-adic groups [8] and representations of affine Hecke algebras with non-zero parameters [4].…”