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Minimal length elements in some double cosets of Coxeter groups
Abstract
We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570–600].
... In this subsection, we formulate consequences of the properties (P1), (P2) and (P3) of conjugacy classes in finite Coxeter groups described in [He07,Theorem 7.5] (see also [GP00, Theorem 3.2.7] for the untwisted case). ...
... for the untwisted case). Remark 3.6 serves as a dictionary between our notations and the ones of [He07]. ...
... Proof. In the language and notations of [He07] and [GKP00], the lemma can be reformulated as follows. Let δ ∈ Aut(W, S) and w ∈ W be such that supp δ (w) = S and w is of minimal length in its δ-conjugacy class O. Then: ...
This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let (W,S) be a Coxeter system. A cyclic shift of an element is a conjugate of w of the form sws for some simple reflection such that . The cyclic shift class of w is then the set of elements of W that can be obtained from w by a sequence of cyclic shifts. Given a subset such that is finite, we also call two elements K-conjugate if normalise and , where is the longest element of . Let be a conjugacy class in W, and let be the set of elements of minimal length in . Then is the disjoint union of finitely many cyclic shift classes . We define the structural conjugation graph associated to to be the graph with vertices , and with an edge between distinct vertices if they contain representatives and such that u,v are K-conjugate for some . In this paper, we compute explicitely the structural conjugation graph associated to any conjugacy class in W, and show in particular that it is connected (that is, any two conjugate elements of W differ only by a sequence of cyclic shifts and K-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element , as well as the existence of natural decompositions of w as a product of a "torsion part" and of a "straight part", with useful properties.
... The set [G u ] has the natural partial ordering by closure relations, denote u . On the other hand the second-named author [He07] (see also [He16a,§1.10.3]) introduced a partial order on [W e ], induced from the Bruhat order on minimal length elements of the elliptic conjugacy classes of W , which we denote W . ...
... By definition, C β W C α if and only if there exist minimal length elements w β ∈ C β and w α ∈ C α such that w β w α . In [He07], the second-named author constructed explicit minimal length representatives for any elliptic conjugacy class of Weyl groups of classical type. If α β, then we have the desired relation between those minimal length representatives with respect to the Bruhat order. ...
... The remainder of this section is concerned with the proof of Proposition 3.5. Following [He07] 1 , for 1 a, b n, we define ...
Let G be a reductive group over an algebraically closed field and let W be its Weyl group. In a series of papers, Lusztig introduced a map from the set [W] of conjugacy classes of W to the set of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes of W, is injective. In this paper, we show that Lusztig's map is order-reversing, with respect to the natural partial order on arising from combinatorics and the natural partial order on arising from geometry.
... Here d is the order of w and w i is the longest element of the parabolic subgroup of W generated by Π i . It was proved in [8], [7] and [11] that for any conjugacy class of W , there exists a good minimal length element. In [12], the second and third-named authors gave a general proof, which also provides an explicit construction of good minimal length elements. ...
... The δ-orbits 5,6,9,6,3). Note that ρ ∨ = (8,11,15,21,15,8). Then the full sequence gives (11,16,21,30,21,11)). ...
... Note that ρ ∨ = (8,11,15,21,15,8). Then the full sequence gives (11,16,21,30,21,11)). ...
Suppose G is a connected complex semisimple group and W is its Weyl group. The lifting of an element of W to G is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of W to the set of semisimple conjugacy classes of G. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.
... The proof uses the "partial conjugation method" of [16]. We first introduce some notation. ...
... We will use the following result [16,Proposition 3.4] (see also [22,Theorem 2.5]). ...
... Remark 6. 16. If x ∈ KW with x K ,σ x, then by definition, there exists an element in the (W K ) σ -orbit of x that is less than or equal to x in the Bruhat order. ...
In the paper four stratifications in the reduction modulo p of a general
Shimura variety are studied: the Newton stratification, the Kottwitz-Rapoport
stratification, the Ekedahl-Oort stratification and the
Ekedahl-Kottwitz-Oort-Rapoport stratification. We formulate a system of axioms
and show that these imply non-emptiness statements and closure relation
statements concerning these various stratifications. These axioms are satisfied
in the Siegel case.
... • The additivity of the admissible sets (Theorem 5.1), proved by Zhu's global Schubert varieties [26]. • The compatibility of admissible sets (Theorem 6.1), proved by the "partial conjugation method" in [8]. ...
... Remark 2.5. The proof is similar to the finite case [8,Lemma 4.4]. ...
... For w ∈ JW , we set I(J, w) = max{K ⊂ J; y(K) = K}. By [8,Corollary 2.6], t γ is conjugate by an element in W J to an element z = xw 1 , where w 1 ∈ JW and x ∈ W I(J,w 1 ) . Since z is conjugate to t γ , it is of the form t λ for some λ ∈ W 0 · µ. ...
In this paper, we prove a conjecture of Kottwitz and Rapoport on a union of
(generalized) affine Deligne-Lusztig varieties for any tamely
ramified group G and its parahoric subgroup K. We show that if and only if the group-theoretic version of Mazur's
inequality is satisfied. In the process, we obtain a generalization of
Grothendieck's conjecture on the closure relation of -conjugacy classes
of a twisted loop group.
... This is what we will do in this section. Another important technique is the "partial conjugation" method introduced in [5], which will be discussed in the next section. ...
... The following result is proved in [5]. Remark. ...
... Let J = {(1, 2), (2, 3), (4, 5), (6, 7), (7,8)} ⊂ S 0 andw ∈W J = Z 8 ⋊ W J with λ = [1, 1, 0, 1, 1, 1, 0, 0] and w = (3, 2, 1) (7,5,8,6,4). Then ℓ J (w) = 0. ...
This is a continuation of the sequence of papers \cite{HN2}, \cite{H99} in
the study of the cocenters and class polynomials of affine Hecke algebras
and their relation to affine Deligne-Lusztig varieties. Let w be a P-alcove
element, as introduced in \cite{GHKR} and \cite{GHN}. In this paper, we study
the image of in the cocenter of . In the process, we obtain a
Bernstein presentation of the cocenter of . We also obtain a comparison
theorem among the class polynomials of and of its parabolic subalgebras,
which is analogous to the Hodge-Newton decomposition theorem for affine
Deligne-Lusztig varieties. As a consequence, we present a new proof of
\cite{GHKR} and \cite{GHN} on the emptiness pattern of affine Deligne-Lusztig
varieties.
... It was first prove via a case-by-case analysis for untwisted case by Geck and Pfeiffer in [GP,Theorem 3.2.7] and for twisted case by the secondnamed author in [He1,Theorem 7.5]. A case-free proof for part (1) and (2) was found recently in [HN]. ...
... Let δ be the automorphism of order 2 of the Dynkin diagram of type A n−1 . The elliptic δ-twisted conjugacy classes in S n are in one-to-one correspondence with partitions of n where every part is odd, see [He1,§7.14]. Every irreducible S n -representation is δ-stable, i.e., Irr δ S n = IrrS n . ...
... In particular, R 0 (H B n,k,0 ) ∼ = R 0 (H D n,k ) ⊕ R δ 0 (H D n,k ), and therefore this case follows from the general type B n case using that the number of elliptic conjugacy classes in W (D n ) plus the number of δ-twisted elliptic conjugacy classes in W (D n ) equals the number of elliptic conjugacy classes in W (B n ) (see [He1,§7.20]). ...
We determine a basis of the (twisted) cocenter of graded affine Hecke
algebras with arbitrary parameters. In this setting, we prove that the kernel
of the (twisted) trace map is the commutator subspace (Density theorem) and
that the image is the space of good forms (trace Paley-Wiener theorem).
... Partial conjugation. To handle the general case, we use the partial conjugation method introduced in[11]. By partial conjugation, we mean conjugating by elements in the finite Weyl group W . ...
... Then Ad(x) • σ gives a length-preserving group automorphism on W I (x) . By [11, Proposition 2.4], we havẽW = x∈ ޓW W · σ (W I (x) x) = x∈ ޓW W · σ (x W σ (I (x)) ).Moreover, by[11, Proposition 3.4], we have the following:(a) For any w ∈W , there exist x ∈ ޓW and u ∈ W I (x) such that w → σ ux and all the simple reflections involved in the conjugations are in .ޓ By [12, Proposition 4.9], we have the following:(b) Let x ∈ ޓW and u ∈ W I (x) . ...
... More specifically, these algebras have a basis {T w } indexed by elements w of the corresponding finite Coxeter group W , and Geck-Pfeiffer discovered properties of conjugacy classes of W which imply that the value of a character χ is constant on these basis elements when they correspond to minimal length elements in the same conjugacy class, and moreover that those values can be used to compute χ(T w ) for arbitrary elements w in this conjugacy class [GP93,GKP00]. Furthermore, Geck-Michel noticed that every conjugacy class of W contains minimal length elements whose powers in the braid monoid are particularly simple, which was subsequently used to determine the "rationality" of these characters [GM97,GKP00,He07]. These results on conjugacy classes were later reused in other domains involving Weyl group elements (e.g. ...
... These results on conjugacy classes were later reused in other domains involving Weyl group elements (e.g. Bruhat cells [EG04,CLT10,Lus11a], Deligne-Lusztig varieties [BR08,OR08], 0-Hecke algebras [He15] and partitions of the wonderful compactification [He07]), and in particular He-Lusztig applied them to construct cross sections in reductive groups out of elliptic Weyl group elements of minimal length [HL12]. ...
Recently, Lusztig constructed for each reductive group a partition by unions of sheets of conjugacy classes, which is indexed by a subset of the set of conjugacy classes in the associated Weyl group. Sevostyanov subsequently used certain elements in each of these Weyl group conjugacy classes to construct strictly transverse slices to the conjugacy classes in these strata, generalising the classical Steinberg slice, and similar cross sections were built out of different Weyl group elements by He-Lusztig. In this paper we observe that He-Lusztig's and Sevostyanov's Weyl group elements share a certain geometric property, which we call minimally dominant; for example, we show that this property characterises involutions of maximal length. Generalising He-Nie's work on twisted conjugacy classes in finite Coxeter groups, we explain for various geometrically defined subsets that their elements are conjugate by simple shifts, cyclic shifts and strong conjugations. We furthermore derive for a large class of elements that the principal Deligne-Garside factors of their powers in the braid monoid are maximal in some sense. This includes those that are used in He-Lusztig's and Sevostyanov's cross sections, and explains their appearance there; in particular, all minimally dominant elements in the aforementioned conjugacy classes yield strictly transverse slices. These elements are conjugate by cyclic shifts, their Artin-Tits braids are never pseudo-Anosov in the conjectural Nielsen-Thurston classification and their Bruhat cells should furnish an alternative construction of Lusztig's inverse to the Kazhdan-Lusztig map and of his partition of reductive groups.
... Proof. The uniqueness of w ′ follows from [15,Cor. 2.5]. ...
... 15) lies in the EKOR stratum attached to(1) τ if and only if pA 1 = A ♯ 1 , B 1 ⊆ B ♯ 1 , B 1 = τ (B 1 ),(2)s 1 τ if and only if B 1 ⊆ B ♯ 1 , B 1 = τ (B 1 ) (and on this stratum π A 1 = A ♯ 1 ), (3) s 2 τ if and only if π A 1 = A ♯ 1 (and on this stratum B 1 ⊆ B ♯ 1 , B 1 = τ (B 1 )), (4) s 1 s 0 τ if and only if B 1 ⊆ B ♯ 1 (and on this stratum π A 1 = A ...
We investigate qualitative properties of the underlying scheme of Rapoport–Zink formal moduli spaces of p -divisible groups (resp., shtukas). We single out those cases where the dimension of this underlying scheme is zero (resp., those where the dimension is the maximal possible). The model case for the first alternative is the Lubin–Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.
... This is is proved in [7], [6] and [9] via case-by-case analyses, and a general proof is in [10]. In fact, we may choose a good element having minimal length in the conjugacy class. ...
... Assuming we know the d i and S i explicitly, this gives a formula for σ(w) o(w) , and (at least in the elliptic case) o(σ(w)). Thus we need the explicit formulas of [7], [6] and [9]. See Section 9. ...
Suppose G is a reductive algebraic group, T is a Cartan subgroup, , and W=N/T is the Weyl group. If has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.
... For Σ, Σ ′ ∈ W δ,min / ≈ δ , we write Σ ′ Σ if Σ ′ w for some w ∈ Σ. By [7,Corollary 4.6], Σ ′ Σ if and only if Σ ′ w for any w ∈ Σ. In particular, is transitive. ...
... By Theorem 1.2, xwx −1 ≈ δ w ′ . By [7,Lemma 4.4], there exists w ′ ...
In this paper, we give explicit descriptions of the centers and cocenters of
0-Hecke algebras associated to finite Coxeter groups.
... By the proof of [H1,Lemma 4.4], we have the following observation ...
... Theorem 2.5.[H1, Corollary 2.6] Let O be a W -conjugacy class ofW andw ∈ O. Then there existsw ′ of minimal length in O such that (a)w → S,σw ′ ; (b) There exist J ⊂ S, u ∈ W σ(J) and x ∈ SW σ(J) such that J = xσ(J)x −1 andw ′ = xu; ...
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 of PEL Shimura datum, each Newton stratum contains a minimal Ekedahl–Oort stratum. This generalizes a result of Viehmann and Wedhorn.
... This had been previously proved by X. He in [17,Corollary 4.5] (note that He's statement refers to minimal length elements, but multiplying by 0 one may obtain the result for maximal length elements, by a different twist). ...
Let GG be a simple algebraic group over an algebraically closed field and let θ be a graph‐automorphism of GG. We classify the spherical unipotent conjugacy classes in the coset Gθ. As a by‐product, we show that J.‐H. Lu's characterization in characteristic zero of spherical conjugacy classes in Gθ by the dimension formula also holds for spherical unipotent conjugacy classes in Gθ in positive characteristic. If θ has order 2, we provide an alternative description of the restriction to spherical unipotent conjugacy classes in Gθ, of Lusztig's map Ψ from the set of unipotent conjugacy classes in Gθ to the set of twisted conjugacy classes of the Weyl group of GG. We also show that a twisted conjugacy class in the Weyl group has a unique maximal length element if and only if it has maximum in the Bruhat order (a result previously proved by X. He).
... Demazure product. We follow [He07,§1]. For any x, y ∈W , the subset {xy ′ ; y ′ y} (resp. ...
The Demazure product gives a natural monoid structure on any Coxeter group. Such structure occurs naturally in many different areas in Lie Theory. This paper studies the Demazure product of an extended affine Weyl group . The main discovery is a close connection between the Demazure product of an extended affine Weyl group and the quantum Bruhat graph of the finite Weyl group. As applications, we obtain explicit formulas on the generic Newton points and the Demazure products of elements in the lowest two-sided cell/shrunken Weyl chambers of , and obtain an explicit formula on the Lusztig-Vogan map from the coweight lattice to the set of dominant coweights.
... [Mar14a] or [CH15]). Similar statements to (1) and (2) above were further obtained for an arbitrary Coxeter group W , but when O is replaced by some "partial" conjugacy class O = {v −1 wv | v ∈ W I }, for some finite standard parabolic subgroup W I ⊆ W (see [He07] and [Nie13]). Finally, we showed in [Mar14b, Theorem A] that for a certain class of Coxeter groups that includes the right-angled ones, (1) and (2) hold using only cyclic shifts. ...
... These minimal length double coset representatives can be used as identifiers for the double coset. See [57] for more on this. The focus of [13] is understanding W S · ω · W T with ω fixed as S and T vary over subsets of the generating reflections. ...
Let H and K be subgroups of a finite group G. Pick uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) the Borel subgroup in . This leads to new theorems for Mallows measure on permutations and new insights into the LU matrix factorization. 2) The double cosets of the hyperoctahedral group inside , which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if H and K are parabolic subgroups of , the double cosets are `contingency tables', studied by statisticians for the past 100 years.
... The partial order J,σ on J W is introduced in [He07a, Proposition 3.8] (see also [He07b,4.7]). For w, w ∈ J W , we write 1. ...
We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.
... We have a partial order ≤ K on Adm({µ}) K , induced from the Bruhat order on the associated affine Weyl group. On the finite set K Adm({µ}), we have also a partial order ≤ K,σ , introduced by He in [21] section 4. See 1.2 for more details on these sets Adm({µ}), Adm({µ}) K and K Adm({µ}). 1 The existence of the local model diagram is in fact one of the He-Rapoport axioms, and of course, one hopes that finally all the He-Rapoport axioms should be verified for the Kisin-Pappas integral models. ...
In this paper we study the geometry of reduction modulo p of the Kisin-Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR strata on these Shimura varieties, using the theories of G-zips and mixed characteristic local -Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties.
... Closure relations of fine affine Deligne-Lusztig varieties. We recall from [He07,§4] the partial order on KW . Let w, w ′ ∈ KW . ...
We investigate qualitative properties of the underlying scheme of Rapoport-Zink formal moduli spaces of p-divisible groups, resp. Shtukas. We single out those cases when the dimension of this underlying scheme is zero, resp. those where the dimension is maximal possible. The model case for the first alternative is the Lubin-Tate moduli space, and the model case for the second alternative is the Drinfeld moduli space. We exhibit a complete list in both cases.
... into locally closed sub-varieties. The partial order J,σ on J W is introduced in [He07a, Proposition 3.8] (see also [He07b,4.7]). For ...
We prove a character formula for some closed fine Deligne-Lusztig varieties. We apply it to compute fixed points for fine Deligne-Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport-Zink spaces arising from the arithmetic Gan-Gross-Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.
... [Mar14a] or [CH15]). Similar statements to (1) and (2) above were further obtained for an arbitrary Coxeter group W , but when O is replaced by some "partial" conjugacy class O = {v −1 wv | v ∈ W I }, for some finite standard parabolic subgroup W I ⊆ W (see [He07] and [Nie13]). Finally, we showed in [Mar14b, Theorem A] that for a certain class of Coxeter groups that includes the right-angled ones, (1) and (2) hold using only cyclic shifts. ...
Let W be a Coxeter group. We provide a precise description of the conjugacy classes in W, yielding an analogue of Matsumoto's theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993.
... In [51], Lusztig introduced G-stable pieces for reductive groups over algebraically closed fields. The closure relation between G-stable pieces was determined in [23] and a more systematic approach using the " partial conjugation action" technique was given later in [24]. The notion and the closure relation of G-stable pieces also found application in arithmetic geometry, e.g. in the work of Pink, Wedhorn and Ziegler on algebraic zip data [58]. ...
The study of affine Deligne-Lusztig varieties originally arose from arithmetic geometry, but many problems on affine Deligne-Lusztig varieties are purely Lie-theoretic in nature. This survey deals with recent progress on several important problems on affine Deligne-Lusztig varieties. The emphasis is on the Lie-theoretic aspect, while some connections and applications to arithmetic geometry will also be mentioned.
... We also have the following relation between the minimal length elements of C and of O. This is proved in [He07b,§3]. ...
This survey article, is written as an extended note and supplement of my
lectures in the current developments in mathematics conference in 2015. We
discuss some recent developments on the conjugacy classes of affine Weyl groups
and p-adic groups, and some applications to Shimura varieties and to
representations of affine Hecke algebras.
... Given w ∈ W we denote by [15,14,17] (see also [18] for a case-free proof). min such that x ′ ≤ x for the Bruhat order. ...
We study the decomposition matrices for the unipotent -blocks of finite
special unitary groups SU for unitary primes larger than n. Up
to some unknow entries, we give a complete solution for . We
formulate a strong positivity conjecture on characters of certain intersection
cohomology complexes, which then predicts most of the remaining missing
entries. We also prove a general result for two-column partitions when
divides q+1. This is achieved using projective modules coming from the
-adic cohomology of Deligne--Lusztig varieties.
... Here the lower bound is obtained by constructing an explicit sequence from w to a minimal length element w ′ in some conjugacy class ofW (in most cases, w and w ′ are not in the same conjugacy class). The upper bound is obtained by combining the "partial conjugation" method in [10] and the dimension formula for affine Deligne-Lusztig varieties in affine Grassmannian in [6, Theorem 2.15.1] and [20, Theorem 1.1]. If w lies in the shrunken Weyl chamber, then the lower bound and the upper bound coincide and the theorem is proved. ...
This note is based on my talk at ICCM 2013, Taipei. We give an exposition of
the group-theoretic method and recent results on the questions of non-emptiness
and dimension of affine Deligne-Lusztig varieties in affine flag varieties.
... The following lemma is well-known, see for example [13]. ...
We provide a direct connection between Springer theory, via Green
polynomials, the irreducible representations of the pin cover \wti W, a
certain double cover of the Weyl group W, and an extended Dirac operator for
graded Hecke algebras. Our approach leads to a new and uniform construction of
the irreducible genuine \wti W-characters. In the process, we give a
construction of the action by an outer automorphism of the Dynkin diagram on
the cohomology groups of Springer theory, and we also introduce a q-elliptic
pairing for W with respect to the reflection representation V. These
constructions are of independent interest. The q-elliptic pairing is a
generalization of the elliptic pairing of W introduced by Reeder, and it is
also related to S. Kato's notion of (graded) Kostka systems for the semidirect
product A_W=\bC[W]\ltimes S(V).
... 3.2.10] for the case where ⋄ is the identity and He [8,Lemma 3.6] for the general case.) If we take w = w I , then one easily sees that the set of roots Φ w I is just the parabolic subsystem Φ I ⊆ Φ corresponding to I. Let Π I ⊆ Φ + I be the set of simple roots. ...
Kottwitz' conjecture is concerned with the intersections of Kazhdan--Lusztig
cells with conjugacy classes of involutions in finite Coxeter groups. In joint
work with Bonnaf\'e, we have recently found a way to prove this conjecture for
groups of type and . The argument for type relies on two
ingredients which were used there without proof: (1) a strengthened version of
the "branching rule" and (2) the consideration of "-twisted"
involutions where is a graph automorphism. In this paper we deal
with (1), (2) and complete the argument for type ; moreover, we establish
Kottwitz' conjecture for -twisted involutions in all cases where
is non-trivial.
This paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne–Lusztig varieties (ADLV). The primary objective is to investigate the non-emptiness pattern, dimension, and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a full mathematical proof of a newly identified problem concerning a certain lower bound of dimension. Furthermore, we extend an open invitation to the readers by providing the source code for computing ADLV and the ML models, promoting further explorations. This paper concludes by sharing valuable experiences and highlighting lessons learned from this collaboration.
We determine which bipartite graphs embedded in a torus are move-reduced. In addition, we classify equivalence classes of such move-reduced graphs under square/spider moves. This extends the class of minimal graphs on a torus studied by Goncharov–Kenyon, and gives a toric analog of Postnikov’s and Thurston’s results on a disk.
Motivated by Lusztig’s G-stable pieces, we consider the combinatorial pieces: the pairs (w, K) for elements w in the Weyl group and subsets K of simple reflections that are normalized by w. We generalize the notion of cyclic shift classes on the Weyl groups to the set of combinatorial pieces. We show that the partial cyclic shift classes of combinatorial pieces associated with minimal-length elements have nice representatives. As applications, we prove the left-right symmetry and the compatibility of the induction functors of the parabolic character sheaves.
For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.
We prove that the perfect loop functor LX of a quasi-projective scheme X over a local non-archimedean field k satisfies arc-descent, strengthening a result of Drinfeld. Then we prove that for an unramified reductive group G , the map L G → L ( G / B ) {LG\rightarrow L(G/B)} is a v -surjection. This gives a mixed characteristic version (for v -topology) of an equal characteristic result (in étale topology) of Bouthier–Česnavičius. In the second part of the article, we use the above results to introduce a well-behaved notion of Deligne–Lusztig spaces X w ( b ) {X_{w}(b)} attached to unramified p -adic reductive groups. We show that in various cases these sheaves are ind-representable, thus partially solving a question of Boyarchenko. Finally, we show that the natural covering spaces X ˙ w ˙ ( b ) {\dot{X}_{\dot{w}}(b)} are pro-étale torsors over clopen subsets of X w ( b ) {X_{w}(b)} , and analyze some examples.
For unramified reductive groups, we determine the connected components of affine Deligne-Lusztig varieties in the partial affine flag varieties. Based on the work of Hamacher-Kim and Zhou, this result allows us to verify, in the unramified group case, the He-Rapoport axioms, the ``almost product structure" of Newton strata, and the precise description of mod p isogeny classes predicted by the Langlands-Rapoport conjecture, for the Kisin-Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.
We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig in Lfromto, (2011) and that the Kazhdan–Lusztig map in Kazhdan and Lusztig, (1998) is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the “skeleta” of affine Springer fibers.
Let H and K be subgroups of a finite group G. Pick g∈G uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) H=K= the Borel subgroup in GLn(Fq). This leads to new theorems for Mallows measure on permutations and new insights into the LU matrix factorization. 2) The double cosets of the hyperoctahedral group inside S2n, which leads to new applications of the Ewens's sampling formula of mathematical genetics. 3) Finally, if H and K are parabolic subgroups of Sn, the double cosets are ‘contingency tables’, studied by statisticians for the past 100 years.
Let G be a reductive group over an algebraically closed field and let W be its Weyl group. In a series of papers, Lusztig introduced a map from the set [W] of conjugacy classes of W to the set [Gu] of unipotent classes of G. This map, when restricted to the set of elliptic conjugacy classes [We] of W, is injective. In this paper, we show that Lusztig's map [We]→[Gu] is order-reversing, with respect to the natural partial order on [We] arising from combinatorics and the natural partial order on [Gu] arising from geometry.
The study of certain union X(μ,b) of affine Deligne–Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit dimension formula for X(μ,b) associated to sufficiently large dominant coweight μ.
We introduce the notion of minimal reduction type of an affine Springer fiber, and use it to define a map from the set of conjugacy classes in the Weyl group to the set of nilpotent orbits. We show that this map is the same as the one defined by Lusztig, and that the Kazhdan-Lusztig map is a section of our map. This settles several conjectures in the literature. For classical groups, we prove more refined results by introducing and studying the ``skeleta'' of affine Springer fibers.
The study of certain union of affine Deligne-Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit formula of for sufficiently large dominant coweight .
Let F be a p-adic field and let E/F be a finite unramified extension. For the Weil restriction ResE/FGLn of the general linear group GLn, we determine the connected components of closed affine Deligne-Lusztig varieties of arbitrary parahoric level.
We show that the totally nonnegative part of a partial flag variety G/P (in the sense of Lusztig) is a regular CW complex, confirming a conjecture of Williams. In particular, the closure of each positroid cell inside the totally nonnegative Grassmannian is homeomorphic to a ball, confirming a conjecture of Postnikov.
In this paper, we study the -ordinary locus of a Shimura variety with parahoric level structure. Under the Axioms in \cite{HR}, we show that -ordinary locus is a union of some maximal Ekedahl-Kottwitz-Oort-Rapoport strata introduced in \cite{HR} and we give criteria on the density of the -ordinary locus.
Let denote the affine Kac-Moody group associated to and the associated affine Grassmanian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of . In the case of ordinary (non-equivariant) K-theory we find an explicit closed form for the structure constants. We also determine an inductive formula for the structure constants in the torus-equivariant cohomology ring, and use this formula to find closed forms for some of the structure constants.
Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y in exactly one point. In this paper this is generalized to the case where w is replaced by any elliptic element in the Weyl group of minimal length d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to an affine space of dimension d and Y is replaced by a unipotent class Y' of codimension d in such a way that the intersection of V' and Y' is finite.
Let \W\ be an extended affine Weyl group. We prove that the minimal length elements \w_{{\mathcal{O}}}\ of any conjugacy class \{\mathcal{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 \T_{w_{{\mathcal{O}}}}\, where \{\mathcal{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].
This paper is a contribution to the general problem of giving an explicit
description of the basic locus in the reduction modulo p of Shimura
varieties. Motivated by \cite{Vollaard-Wedhorn} and
\cite{Rapoport-Terstiege-Wilson}, we classify the cases where the basic locus
is (in a natural way) the union of classical Deligne-Lusztig sets associated to
Coxeter elements. We show that if this is satisfied, then the Newton strata and
Ekedahl-Oort strata have many nice properties.
Let (W,I) be a finite Coxeter group. In the case where W is a Weyl group, Berenstein and Kazhdan in [A. Berenstein, D. Kazhdan, Geometric and unipotent crystals. II. From unipotent bicrystals to crystal bases, in: Quantum Groups, in: Contemp. Math., vol. 433, Amer. Math. Soc., Providence, RI, 2007, pp. 13–88] constructed a monoid structure on the set of all subsets of I using unipotent χ-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.
We prove that the Deligne–Lusztig varieties associated to elements of the Weyl group which are of minimal length in their twisted class are affine. Our proof differs from the proof of He and Orlik–Rapoport and it is inspired by the case of regular elements, which correspond to the varieties involved in Broué's conjectures.
Let (W,S) be a finite Coxeter system and B = B(W) the corresponding Artin-Tits braid group. The natural map B → W has a canonical section r:W → B defined by the condition that if w ∈ W is written as a reduced expression in the generators in S then r(w)is the corresponding product taken in B. The main result of the present paper is as follows. Let C be a conjugacy class in W whose elements have order d say. Then there exists an element w ∈ C of minimal length in C such that r(w)d is a product of terms of the form where the following hold: dI is a non-negative even integer, I runs over a sequence of subsets of S which decreases (which implies that the terms commute), and wI is the longest element in the corresponding parabolic subgroup of W. Such an element will be called a ‘good’ element in C. The result is proved case by case, using the classification of irreducible finite Coxeter groups and the knowledge of representatives
of minimal length from the article by Geck and Pfeiffer in Advances in Math. 102 (1993) 79–94. The main application of this result concerns the problem of calculating character values of Iwahori-Hecke
algebras. The generic Iwahori-Hecke algebra H associated with (W,S) is a quotient of the group algebra of B by the ideal generated by quadratic relations of the form (s-q)(s+1) where s ∈ S and q is an indeterminate. Thus, H is an algebra over a suitable field of rational functions in the variable ∼q. The above result implies that if w is a good element in the class C of W, then the eigen values of the standard basis element Tw of H in an irreducible representation of H are roots of unity times fractional powers of ∼q, and the fractional powers occurring can be explicitly determined from the ordinary character table of W. This result is used to compute the character table of the Iwahori-Hecke algebra of type E8. To determine the roots of unity, we use additional relations coming from the modular representation theory of ∼H. This completes the program of determining the character tables of Iwahori-Hecke algebras. 1991 Mathematics Subject Classification: 20C20, 20F36.
We define and study a family of partitions of the wonderful compactification \bar{G} of a semi-simple algebraic group G of adjoint type. The partitions are obtained from subgroups of G \times G associated to triples (A_1, A_2, a), where A_1 and A_2 are subgraphs of the Dynkin graph \Gamma of G and a : A_1 \to A_2 is an isomorphism. The partitions of \bar{G} of Springer and Lusztig correspond respectively to the triples (\emptyset, \emptyset, \id) and (\Gamma, \Gamma, \id).
Let G be a connected, simple algebraic group over an algebraically closed field. There is a partition of the wonderful compactification of G into finite many G-stable pieces, which were introduced by Lusztig. In this paper, we will investigate the closure of any G-stable piece in . We will show that the closure is a disjoint union of some G-stable pieces, which was first conjectured by Lusztig. We will also prove the existence of cellular decomposition if the closure contains finitely many G-orbits.
In this paper we give an elementary method for classifying conjugacy classes of involutions in a Coxeter group (W, S). The classification is in terms of (W-equivalence classes of certain subsets of S).
Let W be a finite Coxeter group and let F be an automorphism of W that leaves the set of generators of W invariant. We establish certain properties of elements of minimal length in the F-conjugacy classes of W that allow us to define character tables for the corresponding twisted Iwahori–Hecke algebras. These results are extensions of results obtained by Geck and Pfeiffer in the case where F is trivial.
We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in [G. Lusztig, Parabolic character sheaves, II, Mosc. Math. J. 4 (4) (2004) 869–896]. We also prove some properties of the character sheaves on the group compactification.
If G is a group and
an automorphism of G, one has the twisted conjugation action of G on itself
This paper collects a number of results — more or less well known — for the case that G is a simply connected semisimple
group.
This note is a sequel to part I [Trans. Am. Math. Soc. 353, No. 7, 2725–2739 (2001; Zbl 0996.20003)]. We establish the minimal basis theory for the centralizers of parabolic subalgebras of Iwahori-Hecke algebras associated to finite Coxeter groups of any type, generalizing the approach introduced in [J. Algebra 221, No. 1, 1–28 (1999; Zbl 0940.20011)] from centers to the centralizer case. As a pre-requisite, we prove a reducibility property in the twisted J-conjugacy classes in finite Coxeter groups, which is a generalization of results of M. Geck and R. Rouquier [Prog. Math. 141, 251–272 (1997; Zbl 0868.20013] and part I [loc. cit.].
this paper to prove a similar statement and to describe a similar algorithm for Weyl groups and their Hecke algebras of any given type. Our approach which is completely elementary can be described entirely within the Weyl group itself, as follows
We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.
We define the notion of character sheaf on a possibly disconnected reductive group. We show that the restriction functor carries a character sheaf to a direct sum of character sheaves.
The main theme of this paper is establishing the "generalized Springer correspondence" in complete generality that is, for not necessarily connected reductive algebraic groups.
These are notes for the Aisenstadt lectures given in may/june 2002 at CRM, Montreal. The main object is the study of Iwahori-Hecke algebras arising from reductive groups over finite or p-adic fields. We try to extend various results known in the equal parameter case to the case of not necessarily equal parameters.ai
We define and study convolution of parabolic character sheaves. As an application we attach to any parabolic character sheaf the orbit of a tame local system on the maximal torus under a subgroup of the Weyl group.
We continue the study of character sheaves on a not necessarily connected reductive group. We prove orthogonality formulas for certain characteristic functions.
We study a class of perverse sheaves on the variety of pairs (P,gU_P) where P runs through a conjugacy class of parabolics in a connected reductive group G and gU_P runs through G/U_P. This is a generalization of the theory of character sheaves.
Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy
- A A Beilinson
- J Bernstein
- P Deligne
A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Soc. Math. France, Paris, 1982, pp. 5–171.