Abstract. Let W be an extended affine Weyl group. We prove that minimal length elements w O of any conjugacy class O of W satisfy some special properties, generalizing results of Geck and Pfeiffer [8] on finite Weyl groups. We then introduce the "class polynomials" for affine Hecke algebra H and prove that T w O , where O runs over all the conjugacy classes of W , forms a basis of the cocenter H/ [H, H]. We also classify the conjugacy classes satisfying a generalization of Lusztig's conjecture [23].

The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties.We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can be phrased in combinatorial terms.Second, we study the Shimura varieties in which every non-basic σ-isogeny class is Hodge-Newton decomposable. We show that (assuming the axioms of [25]) this condition is equivalent to nice conditions on either the basic locus, or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions.While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure, and of possibly non-quasi-split underlying groups. This results in a large number of new cases of Shimura varieties where a simple description of the basic locus can be expected. As a striking consequence of the results, we obtain that this property is independent of the parahoric subgroup chosen as level structure.We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.

We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in [3]. The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.Proof. By Proposition 2.4.1, any σ-conjugacy class of M J (L) is represented by some element inW J . Let x = ǫ λ w, x ′ = ǫ λ ′ w ′ ∈W J such that x and x ′ are in the same σ-conjugacy class of G(L). By Kottwitz [8] and [9],. In other words,

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Abstract. We give a geometric proof that minimal length elements in a (twisted) conjugacy class of a finite Coxeter group W have remarkable properties with respect to conjugation, taking powers in the associated Braid group and taking centralizer in W .

Abstract. Fundamental elements are certain special elements of affine Weyl groups introduced by Görtz, Haines, Kottwitz and Reuman. They play an important role in the study of affine Deligne-Lusztig varieties. In this paper, we obtain characterizations of the fundamental elements and their natural generalizations. We also derive an inverse to a version of "Newton-Hodge decomposition" in affine flag varieties. As an application, we obtain a group-theoretic generalization of Oort's results on minimal p-divisible groups, and we show that, in certain good reduction reduction of PEL Shimura datum, each Newton stratum contains a minimal Ekedahl-Oort stratum. This generalizes a result of Viehmann and Wedhorn.

We determine the set of connected components of closed affine Deligne-Lusztig varieties for hyperspecial maximal parahoric subgroups of unramified connected reductive groups. This extends the work by Viehmann for split reductive groups, and the work by Chen-Kisin-Viehmann on minuscule affine Deligne-Lusztig varieties.M ⊇ T (resp. N) is the Levi subgroup (resp. unipotent radical) of P and µ ranges over the setThirdly, we show the image of the natural mapHere π 1 (M) σ is the set of σ-fixed points on π 1 (M). From the first three steps, we see immediately J b ′ (F ) acts on π 0 (X λ (b ′ )) transitively and Theorem 1.2 follows. To deduce Theorem 1.1 (i.e., π 0 (X λ (b ′ )) ∼ = π 1 (G) σ ), we need to show, under the assumption that (λ, b) is HN-irreducible, that ϕ µ is surjective, and moreover, any two points P,This is the last step of the whole proof.The first two steps are already established in [2]. In this paper, we develop the techniques used in [2] systematically and finish the last two steps. Based on Deligne-Lusztig reduction methods, we also provide a new conceptual proof for the first step, avoiding concrete computations in superbasic case.Compared to the original proof for the case that λ is minuscule, there are two main difficulties for the general case. The first lies in the third step. We need to construct affine lines inWhen λ is minuscule, the original construction of affine lines relies on the property that each element of I λ,M,b is conjugate to λ (under the Weyl group of T ). In general case, I λ,M,b becomes much more complicated. To overcome the difficulty, we introduce a new algorithm for the construction via defining the key set Θ(µ, µ ′ , λ) (see §6 for definition). The other difficulty is to prove the sufficiency direction of the "moreover" part in the last step. There is a gap in the original proof in [2]. However, this gap can be filled in (for minuscule λ) by Miaofen Chen via a case-by-case analysis. For general case, we figure out a conceptual proof using weakly dominant cocharacters (see §4 for definition), extending Chen's arguments.The paper is organized as follows. In §2, we collect basic notations and properties which are used frequently in the paper. In §3, we prove Theorem 1.1 for superbasic case. In §4, we show the existence of the Levi subgroup M as above such thatĪ λ,M,b ′ contains a weakly dominant cocharacter. In §5, we outline the proof of Theorem 1.1, where the surjectivity of ϕ µ in the fourth step is redundant. In §6 and §7, we introduce the set Θ(µ, µ ′ , λ) and establish the third step (see Proposition 5.1). In §8 and §9, we finish the last step (see Proposition 5.3). Lemma 2.1. Let b ∈ G(L). Then [b] is a superbasic σ-conjugacy of G(L) if and only if there exits τ ∈ [b] ∩ N T (L) such that p(τ ) ∈ Ω is σ-superbasic inW . Proof. It follows from [8, Proposition 3.5] and [2, Lemma 3.1.1]. 2.3. Let M ⊆ G be a Levi subgroup containing T . Replacing the triple T ⊆ B ⊆ G with T ⊆ M ∩ B ⊆ M, we can define, as in §2.1 and §2.2, Π M , Φ M ,W M , S a M , ≤ M and so on. Let J ⊆ S ...

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