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].

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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.

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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].

Abstract. Under suitable hypotheses, we prove that a form of a projective homogeneous variety G/P defined over the function field of a surface over an algebraically closed field has a rational point. The method uses an algebrogeometric analogue of simple connectedness replacing the unit interval by the projective line. As a consequence, we complete the proof of Serre's Conjecture II in Galois cohomology for function fields over an algebraically closed field.

We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.Date: March 16, 2020. 1 2 X. HE, G. PAPPAS, AND M. RAPOPORTtwo examples above exhaust all possibilities (this statement has to be interpreted correctly, by considering the natural compactification of the modular curve). This comes down to a statement about the spectral decomposition under the action of the Hecke algebra of the ℓ-adic cohomology of modular curves. Unfortunately, the generalization of this statement to other Shimura varieties seems out of reach at the moment.The other possible interpretation of the question is to ask for good, resp. semistable, reduction of a specific class of p-integral models of Shimura varieties. Such a specific class has been established in recent years for Shimura varieties with level structure which is parahoric at p, the most general result being due to M. Kisin and the second author [30]. The main point of these models is that their singularities are modeled by their associated local models, cf. [40]. These are projective varieties which are defined in a certain sense by linear algebra, cf. [21,47]. More precisely, for every closed point of the reduction modulo p of the p-integral model of the Shimura variety, there is an isomorphism between the strict henselization of its local ring and the strict henselization of the local ring of a corresponding closed point in the reduction modulo p of the local model. Very often every closed point of the local model is attained in this way. In this case, the model of the Shimura variety has good, resp. semi-stable, reduction if and only if the local model has this property. Even when this attainment statement is not known, we deduce that if the local model has good, resp. semi-stable, reduction, then so does the model of the Shimura variety. Therefore, the emphasis of the present paper is on the structure of the singularities of the local models and our results determine local models which have good, resp. semi-stable reduction.Let us state now the main results of the paper, as they pertain to local models. See Section 3 for corresponding results for Shimura varieties, and Section 4 for results on Rapoport-Zink spaces. Local models are associated to local model triples. Here a LM triple over a finite extension F of Q p is a triple (G, {µ}, K) consisting of a reductive group G over F , a conjugacy class of cocharacters {µ} of G over an algebraic closure of F , and a parahoric group K of G. We sometimes write G for the affine smooth group scheme over O F corresponding to K. It is assumed that the cocharacter {µ} is minuscule (i.e., any root takes values in {0, ±1} on {µ}). The reflex field of the LM triple (G, {µ}, K) is the field of definition of the conjugacy class {µ}. One would like to associate to (G, {µ}, K) a local model M loc K (G, {µ}), a flat projective scheme over the ring...