Closure relations of Newton strata in Iwahori double cosets

We consider the Newton stratification on Iwahori double cosets for a connected reductive group. We prove the existence of Newton strata whose closures cannot be expressed as a union of strata, and show how this is implied by the existence of non-equidimensional affine Deligne–Lusztig varieties. We also give an explicit example for a group of type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_4$$\end{document}A4.


Introduction
Let G be a connected reductive group defined over F = F q ((t)) which for the purpose of this introduction is assumed to be split. LetF = F q ((t)) and OF its ring of integers. Let σ be the morphism on G(F) induced by the Frobenius automorphism ofF over F.
For any element b of G(F) we consider its σ -conjugacy class By results of Rapoport-Richartz [16,Thm. 3.6] and Viehmann [17], the closure of a Newton stratum is given by For applications to the geometry of the special fiber of moduli spaces of shtukas, or of Shimura varieties, one wants to understand the relation between this stratification and double cosets under parahoric subgroups of G.
We begin by recalling the case of a hyperspecial maximal parahoric subgroup K of G. We fix a maximal torus T of G and a Borel subgroup B containing it. By the Cartan decomposition we have  (1.2) whenever N [b],μ is non-empty, compare [17]. Furthermore, N [b],μ is pure of codimension equal to the length of any maximal chain between [b] and [μ] in B(G).
We now describe the situation in the case we are interested in for this work. Let I be an Iwahori subgroup of G. Then the affine Bruhat decomposition gives where W denotes the extended affine Weyl group.
Again, we define Newton strata in a given double coset by They are equipped with the structure of locally closed reduced subschemes of I x I . However, it turns out to be a difficult task to describe which N [b],x are non-empty. For some results in this direction compare [4] and [9]. Let Contrary to the behaviour in the hyperspecial case, there are x and In other words the subset B(G) x ⊂ B(G) can be non-saturated. Also, the Newton strata are in general no longer of codimension equal to the length of any maximal chain between [b] and the σ -conjugacy class [b x ] corresponding to the generic point of I x I .
Using an example of a non-equidimensional affine Deligne-Lusztig variety given in [3,5] together with [14,Cor. 3.11], one sees that in general, the N [b],x ⊆ I x I are no longer pure of any fixed codimension.
In this paper we study the closures of Newton strata in a given Iwahori double coset. By (1.1) we obtain is not a union of strata. Thus in such cases, the decomposition of I x I into its intersections with the various σconjugacy classes is in general not a stratification (although we continue to call the subschemes N [b],x Newton strata). The proof of Theorem 3.1 relies on topological strong purity, which we review in Sect. 2.
In Sect. 4 we construct an explicit pair is not a union of strata. For this, we use the reduction method à la Deligne and Lusztig as proved in [6] to reduce the computation of dimensions of Newton strata in I x I to similar computations for cordial elements x which then follow from the general theory of cordial elements as in [14], compare also Sect. 2.4. Our approach is inspired by the construction of a non-equidimensional affine Deligne-Lusztig variety for [b] = [1] given by Görtz and He in [3, Sec. 5], but adapted to find a maximal class for which the corresponding Newton stratum is not equi-dimensional. For the family of elements x in the affine Weyl group that we obtain in this way, we can still give a complete description of the subset B(G) x , as well as of all codimensions of irreducible components of Newton strata.

Notation and review of previous results
In this section we fix the notation and recall the necessary theory of Newton strata and affine Deligne-Lusztig varieties.

Notation
Let G be a connected reductive group over the local field F ∼ = F q ((t)), where F q is the finite field with q elements. LetF ∼ = F q ((t)) be the completion of the maximal unramified extension of F, and let σ be the Frobenius automorphism of F over F, mapping all coefficients to their q th powers. We also denote the induced automorphism of G(F) by σ . Let be the absolute Galois group of F.
Fix S, a maximalF-split torus in G defined over F and containing a maximal F-split torus. Let T be the centralizer of S in G, a maximal torus. Consider the apartment A of GF associated to SF . The Frobenius σ acts on A and we fix a σ -stable alcove a. Let I ⊂ G(F) be the Iwahori subgroup corresponding to a.
For every x ∈ W we choose a representative in G(F), which we denote again by x. We consider the affine Bruhat decomposition To define a length function on the extended affine Weyl group, we consider the decomposition W ∼ = W a , where W a is the affine Weyl group of G and is the subset of elements of W which fix the chosen Iwahori subgroup. We extend the length function from W a to W by setting (ω) = 0 for ω ∈ .
Let N T be the normalizer of T in G. The (relative) Weyl group W 0 is defined to be W 0 = N T (F)/T (F), and the extended affine Weyl group is W = N T (F)/(T (F) ∩ I ). Since T and I are σ -stable, the Frobenius induces automorphisms of W 0 and W , which we also denote by σ . For a coweight μ ∈ X * (T ) we denote by t μ ∈ T (F) the image of t under μ.
In situations where G is assumed to be quasi-split, we also use the following notation. Choosing a σ -stable vertex of a determines a section W 0 → W of the natural projection map W → W 0 .
By convention, the dominant Weyl chamber is opposite to the unique chamber containing a. This choice determines a set of simple roots . Consider S, the set of simple reflections of W 0 . For any i ∈ S, we denote by α i ∈ the corresponding simple root.
For any subset J ⊂ S, let W J be the subgroup of W a generated by J . Denote by J W the set of minimal length representatives of the cosets W J \ W . Any element of the extended affine Weyl group can be written in a unique way as vt μ w with μ ∈ X * (T ), v and w in W 0 and and t μ w in S W . [10,11] and [16], the elements of B(G) are determined by two classifying invariants, the Newton point ν ([b]) and the Kottwitz point

σ -conjugacy classes and Newton strata
The second invariant is given by the image under the Kottwitz map κ G : B(G) → π 1 (G) , where the fundamental group π 1 (G) is defined as the quotient of X * (T ) by the coroot lattice. The Kottwitz map is induced by a homomorphism G(F) → π 1 (G) . Furthermore, it is trivial on I . In particular, κ G (b) = κ G (x) for every b ∈ I x I . Thus when restricting to I x I , the Newton point alone is sufficient to uniquely determine a class.
Two elements in X * (T ) Q,dom satisfy ν 1 ≤ ν 2 in the dominance (partial) order, if and only if their difference ν 2 −ν 1 is a linear combination of positive coroots with non-negative, rational coefficients. This induces a partial order on the set B(G): we For G = GL n we have an interpretation of these notions in terms of the Newton polygon. Let T ⊂ B be the subgroups of diagonal, respectively of upper triangular matrices. Then I is the subgroup of GL n (OF ), whose image modulo t is the Borel subgroup opposite to B. In this case coincides with the classical Newton point of the isocrystal (F n , bσ ). Let p ν be the polygon associated with ν, i.e. the graph of the continuous piecewise linear function [0, n] → R mapping 0 to 0 and of slope ν i on the interval [i − 1, i]. In this case, the subset of elements of X * (T ) Q actually appearing as Newton points consists of the dominant ν ∈ Q n , whose associated Newton polygon p ν has break points and end point with integral coordinates.

Newton strata in Iwahori double cosets
Our main objects of study are Newton strata in an Iwahori double coset. These are defined for x ∈ W and [b] ∈ B(G) as Recall that the loop group associated with G is the ind-group scheme representing the functor on F q -algebras R → G(R((t))). When non-empty, by [16,Thm. 3.6] a Newton stratum N [b],x is the set of F q -valued points of a locally closed subset of I x I (and thus of LG), which we equip with the structure of a reduced subscheme.
Recall that a subscheme Z of LG is bounded if it is contained in a finite union of double cosets I x I . Let I n be the kernel of the projection map I → I (OF /(t n )).
We can then define the codimension of N [b],x ⊂ I x I as the codimension of its image in I x I /I n . This is independent of the choice of n, compare [14,Rem 3.4]. Similarly, the closure N [b],x of a Newton stratum in I x I is the preimage of its closure in the quotient I x I /I n .
For a class [b], we also consider The specialization theorem of [16,Thm. 3.6] implies that this set is closed in I x I and contains N [b],x as an open subset.
Since Iwahori double cosets are irreducible, for x ∈ W we denote by [b x ] the σ -conjugacy class in the generic point of I x I , and its Newton point By [5,Sec. 2], the Newton stratification on I x I /I n satisfies topological strong purity. Considering inverse images under the projection map I x I → I x I /I n we obtain that for any ,x ⊂ Z is either empty or pure of codimension at most 1 in Z .

Dimensions and cordial elements
Using the notation of the previous subsections we can now recall some of the previously obtained results on dimensions and closures of Newton strata. We assume for this subsection that G is quasi-split.
Newton strata in Iwahori double cosets are closely related to the study of affine Deligne-Lusztig varieties. These were introduced by Rapoport in [15] and are defined for x ∈ W and b ∈ G(F) as the locally closed reduced subschemes of the affine flag variety for G with By [14,Cor. 3.12], we have (1) Write x = vt μ w with v, w ∈ W 0 , μ ∈ X * (T ) and such that t μ w ∈ SW . Then η : In other words, neither of the two phenomena we are interested in arises for cordial x.

Non-equidimensional strata and closure relations
In this section we show that the existence of non-equicodimensional Newton strata implies that the Newton stratification fails to be a stratification. Let G be again any connected reductive group over F.

Theorem 3.1. Let x ∈ W and assume that there is a [b] ∈ B(G) x such that the corresponding Newton stratum N [b],x has irreducible components of different codimensions in I x I .
(1) For any maximal chain To prove (1) we assume that there is a maximal chain of classes in B(G) x From the topological strong purity of the Newton stratification (compare the end of Sect. 2.3) and the maximality of (3.3), it follows that Z i−1 has codimension at most 1 in Z i . Since the generic σ -conjugacy classes of Z i−1 and Z i are not equal, the two closed irreducible subschemes are not equal, and hence the codimension is equal to 1. Altogether, the codimension n 0 of N 0 is equal to m, which is independent of the choice of the irreducible component N 0 . Thus each irreducible component of N [b],x has the same codimension.
We now prove (2). Possibly replacing [b] by a larger class, we may assume that all ,x , and let Z 0 be its closure in I x I . By the same argument as above we see that Z 0 is an irreducible component of

An explicit example
In the previous section we related the existence of Newton strata whose closures are not a union of strata to the existence of non-equidimensional strata, and used this to show that the Newton stratification is in general not a stratification in the proper sense. In this section we work out an explicit example of a Newton stratum in some I x I whose closure is not a union of strata, generalizing the method of [3, 5.2]. Despite the failure to be a stratification, we can in this example determine which Newton strata in I x I are non-empty and compute all codimensions of irreducible components. We expect that this serves as a guiding example of typical behaviour of the Newton stratification also in other, more general cases. In this section we assume again that G is quasi-split.

A family of pairs (x, s) in W a × S
Our goal is to find an x and an N [b],x having irreducible components of different codimensions. For this we generalize the method of [3,5], where an x ∈ W a is constructed such that the variety X x (1) is not equidimensional. However, we consider larger [b], and rather want to minimize the length of maximal chains between [b] and [b x ], to make the closure relations easier to handle. Also, we do not restrict ourselves to split groups as in loc. cit. Lemma 4.1. Let x = vt μ w ∈W with v, w ∈ W 0 and t μ w ∈ S W , and let s ∈ S satisfying the following conditions: (1) sxσ (s) lies in the shrunken Weyl chamber (2) (sv) < (v) and (wσ (s)) > (w) for any u ∈ W 0 .
Proof. Condition (2) implies that (sxσ (s)) = (x) − 2, and this together with (1) shows that sx and x are also in the shrunken Weyl chamber.
By the Deligne-Lusztig reduction method (compare [6,Prop. 4.2]), X x (b) can be written as a disjoint union X x (b) = X 1 X 2 of a closed subscheme X 1 and an open subscheme X 2 , where X 1 is of relative dimension one over X sxσ (s) (b) and X 2 is of relative dimension one over X sx (b). In particular, this proves (i) and (ii).
Let b 0 be the unique basic element with κ G (b 0 ) = κ G (x). Using that x, sx, sxσ (s) are shrunken and Condition (4), [6,Cor. 12.2] implies that X x (b 0 ), X sx (b 0 ) and X sxσ (s) (b 0 ) are non-empty and that their dimensions agree with the respective virtual dimensions d x (b 0 ), d sx (b 0 ) and d sxσ (s) (b 0 ). Note that loc. cit. only considers the case where σ acts trivially on W 0 . For the generalization to the present case compare the remarks following [8,Thm. 5.3]. Furthermore, (3) and [3,Lem. 3.2.2] imply that d x (b 0 ) > d sx (b 0 ) + 1. Again, loc. cit. only considers the split case. However, the proof carries over to our situation if one makes the obvious adaptations due to the action of σ on W 0 . From these observations and the reduction method we can deduce By (i), this also shows that d sxσ (s) which is a constant independent of the class [b], and thus is positive.
We now focus on a case where all dimensions of irreducible components of X sx (b) and X sxσ (s) (b) can be computed. In the next section we show that these conditions can indeed be satisfied.
Proof. By the reduction method of [6,Prop. 4.2], Since sx and sxσ (s) are cordial, all non-empty affine Deligne-Lusztig varieties for sx or sxσ (s) are equidimensional of the respective virtual dimension. Thus by The statement on the dimension of X x (b) directly follows from Lemma 4.1 (i) and (iii) and cordiality of sx and sxσ (s).

An explicit example
In order to find a pair (x, s) as in Proposition 4.2, we use the mathematics software system. SageMath Here, to compute generic Newton points and to check cordiality we use the results of [13,Thm. 3.2] and [14,Prop. 4.2], respectively. The results of [13] offer a very useful description of the generic Newton point for split groups G and any x ∈ W of the form x = vt μ w where α, μ > M for all simple roots α where M is a constant depending on G, v and w. Under the same assumption, [14] gives a criterion to check if x is cordial in terms of some paths in the quantum Bruhat graph. We include the SageMath code in the appendix.
For G of type A 3 , an exhaustive computer search (on the finitely many possible triples (v, w, s)) shows that there are no pairs (x, s) as in Proposition 4.2. However, for type A 4 there are many such pairs. The following example is a particularly suitable one in the sense that the two classes [b sx ] and [b sxs ] (which correspond to the maximal non-empty and the maximal non-equicodimensional Newton stratum in I x I ) are close to each other in B(G). where s i denote the simple reflections in S. In this example, the constant M for the regularity assumption on μ equals 74 and is thus satisfied. Then the pair (x, s) satisfies the requirements of Proposition 4.2 and the superregularity hypothesis of [13] and [14].
By the description in (4.4) of the set of classes whose Newton strata are not equicodimensional, the next step is to find the generic Newton points associated to sx and sxs. By where ν x , ν sx and ν sxs denote the generic Newton points of I x I , I sx I and I sxs I , respectively. Observe that ν x = ν sx > ν sxs in the dominance order, and where α ∨ i denotes the coroot e i − e i+1 . From ν sx > ν sxs and Proposition 4.2 we obtain that  (2) (ν(z)) = pr (2) (ν x ) (i.e., the sum of the first two entries of ν(z) is 224), and let S 2 be its complement. Then S 2 contains Z and N [b i ],x /I n for i = 3. By the Purity Theorem, S 2 is pure of codimension 1 in I x I /I n . Up to considering an irreducible component containing Z , we can assume that S 2 is irreducible. Since Z has codimension at least 1 in S 2 , its generic Newton class [b sxs ] is then smaller than that of S 2 . By definition of S 2 , it follows that the generic class of S 2 can only be [b 3 ], and therefore, Z is contained in the closure of N b 3 . With a similar argument, we find that any irreducible component of Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/ licenses/by/4.0/.
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