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Abstract
In this paper, we give explicit descriptions of the centers and cocenters of
0-Hecke algebras associated to finite Coxeter groups.
... 0-Hecke algebras are deformations of the group algebras of finite Coxeter groups with zero parameter. In [6], He gave a basis of the center of 0-Hecke algebras associated to finte Coxeter groups. The basis is closely related to maximal length elements in the conjugacy classes of W . ...
... In [7], Lusztig gave a basis of the center of affine Hecke algebras. In [6], He also mentioned a similar proof could be applied to give a basis of the center of affine 0-Hecke algebras. The basis is closely related to finite conjugacy classes in W aff . ...
Let W be an Iwahori Weyl group and W(1) be an extension of W by an abelian group. Vigneras gave a description of the R-algebra associated to W(1), and also gave a basis of the center of using the Bernstein presentation of . In this paper, we restrict to the case where and use the Iwahori-Matsumoto presentation to give a basis of the center of .
... Building on the work of Norton, Fayers [17] proved that H W (0) is a Frobenius algebra, and Huang [19] gave a tableau approach to the representation theory of the 0-Hecke algebras of classical Weyl groups from a combinatorial point of view. The algebra H W (0) is also closely related to the modular representation theory of finite groups of Lie type [8], and more structure information of H W (0) can be found in [7,18,29]. ...
Weak Bruhat interval modules of the 0-Hecke algebra in type A provide a uniform approach to studying modules associated with noteworthy families of quasisymmetric functions. Recently this kind of modules were generalized from type A to all Coxeter types. In this paper, we give an equivalent description, in a type-independent manner, when two left weak Bruhat intervals in a Coxeter group are descent-preserving isomorphic. As an application, we classify all left weak Bruhat interval modules of 0-Hecke algebras up to isomorphism, and thereby answer an open question and resolve in the affirmative a conjecture of Jung, Kim, Lee, and Oh. Additionally, for finite Coxeter groups we show that the set of minimum (or maximum) elements of all left weak Bruhat intervals in each descent-preserving isomorphism class forms an interval under the right weak Bruhat order.
... 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.
... The polynomials F w,C are not uniquely characterized by condition (1) in Theorem 3.3.9. This is because the cocenter of the affine Hecke algebra over Z[q] has a torsion part, cf.[He15, §5.2]. (In contrast, as we have mentioned above, the cocenter of the affine Hecke algebra over Z[q ±1 ] is free.) ...
We study the -action on the set of top-dimensional irreducible components of affine Deligne--Lusztig varieties in the affine Grassmannian. We show that the stabilizer of any such component is a parahoric subgroup of of maximal volume, verifying a conjecture of X.~Zhu. As an application, we give a description of the set of top-dimensional irreducible components in the basic locus of Shimura varieties.
... The interest in the Coxeter monoids and, mostly, in the Coxeter monoid algebra of finite monoids is evident looking at the wide literature. Besides the cited one of P. N. Norton, general results can be found in [12] , [15], [16], [24], [35]. In type A we can cite, among others, [5] and [9]. ...
For any Coxeter system we consider the algebra generated by the projections over the parabolic quotients. In the finite case it turn out that this algebra is isomorphic to the monoid algebra of the Coxeter monoid (0-Hecke algebra). In the infinite case it contains the Coxeter monoid algebra as a proper subalgebra. This construction provides a faithful integral representation of the Coxeter monoid algebra of any Coxeter system. As an application we will prove that a right-angled Artin group injects in Hecke algebra of the corresponding right-angled Coxeter group.
This paper gives an explicit formula of the dimension of affine Deligne-Lusztig varieties associated with generic Newton point in terms of Demazure product of Iwahori-Weyl groups.
In this paper, we study the relation between the cocenter H ~ ¯ \overline {{\tilde {\mathcal H}}} and the representations of an affine pro- p p Hecke algebra H ~ = H ~ ( 0 , − ) {\tilde {\mathcal H}}={\tilde {\mathcal H}}(0, -) . As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of H ~ {\tilde {\mathcal H}} is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter H ~ ¯ \overline {\tilde {\mathcal H}} .
The structure of a 0-Hecke algebra H of type (W, R) over a field is examined. H has 2n distinct irreducible representations, where n = ∣R∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2n indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W.
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).
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.
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 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].
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.
The work of Dipper and James on Iwahori-Hecke algebras associated with the finite Weyl groups of type A
n
has shown that these algebras behave in many ways like group algebras of finite groups. Moreover, there are “generic” features in the modular representation theory of these algebras which, at present, can only be verified in examples by explicit computations. This paper arose from an attempt to provide a conceptual explanation of these phenomena, in the general framework of the representation theory of (symmetric) algebras. We will study relations between the center of such algebras and properties of decomposition maps, and we will use this to obtain a general result about the “genericity” of the number of simple modules of Iwahori-Hecke algebras.
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
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
- D Ciubotaru
- X He
D. Ciubotaru and X. He, The cocenter of graded affine Hecke algebra and
the density theorem, arXiv:1208.0914.
Centers and simple modules for Iwahori-Hecke algebras, Finite reductive groups (Luminy, 1994)
- M Geck
- R Rouquier
M. Geck and R. Rouquier, Centers and simple modules for Iwahori-Hecke
algebras, Finite reductive groups (Luminy, 1994), Progr. Math., 141, pp.
251272, Birkhäuser, Boston, 1997.
- G Lusztig
G. Lusztig, Characters of reductive groups over a finite field, Annals of
Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ,
1984.