Two-sided cells in type $B$ (asymptotic case)

The lowest two-sided cell of weighted Coxeter groups of rank 3

In this paper, we give precise description for the lowest lowest two-sided cell c_{0} and the left cells in it for a weighted Coxeter group of rank 3. Then we show conjectures P1-P15 and \widetilde{P} hold for c_{0} and do some calculation for the based ring of c_{0}.

The based ring of the lowest two-sided cell of an affine Weyl group, III

We show that Lusztig's homomorphism from an affine Hecke algebra to the direct summand of its asymptotic Hecke algebra corresponding to the lowest two-sided cell is related to the homomorphism constructed by Chriss and Ginzburg using equivariant K-theory by a matrix over the representation ring of the associated algebraic group.

类型 $\tilde B_3$ 仿射 Weyl 群中双边胞腔的基环，IV

The Based Rings of Two-sided cells in an Affine Weyl group of type $\tilde B_3$, IV

本文计算了 $\tilde B_3$ 型仿射 Weyl 群中与 $Sp_6(\mathbb C)$ 的 Jordan 块 (21111) 幂幺类对应的双边胞腔基环，并验证了 Lusztig 关于基环结构的猜想。

On the lowest two-sided cell in affine Weyl groups

Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most |W_{0}| left cells where W_{0} is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.

Cells in modified $\imath$quantum groups of type AIII and related Schur algebras

We provide a combinatorial characterization of two-sided cells in modified $\imath$quantum groups of type AIII. Our approach is to lift a corresponding description of two-sided cells in $\jmath$-Schur algebras associated to Iwahori--Hecke algebras of type $B$. We further give a combinatorial description of two-sided cells in other two kinds of Schur-type algebras, namely $\imath$-Schur algebras and $\tilde{\imath}$-Schur algebras.

最低双边胞腔基础环的凝聚层化

A coherent categorification of the based ring of the lowest two-sided cell

本文给出了仿射 Weyl 群最低双边胞腔基础环 $J_0$ 的部分凝聚层化，并构造复形以提升其基元素及结构常数。

Calogero-Moser space, reduced rational Cherednik algebras, and two-sided cells

We conjecture that the "nilpotent points" of Calogero-Moser space for reflection groups are parametrised naturally by the two-sided cells of the group with unequal parameters. The nilpotent points correspond to blocks of restricted Cherednik algebras and we describe these blocks in the case G(m,1,n) and show that in type B our description produces an existing conjectural description of two-sided cells.

Cellularity of the lowest two-sided ideal of an affine Hecke algebra

In this paper we show that the lowest two-sided ideal of an affine Hecke algebra is affine cellular for all choices of parameters. We explicitely describe the cellular basis and we show that the basis elements have a nice decomposition when expressed in the Kazhdan-Lusztig basis. In type $A$ we provide a combinatorial description of this decomposition in term of number of paths.

Cherednik algebras and Calogero-Moser cells

Using the representation theory of Cherednik algebras at $t=0$ and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-s…

The Based Ring of Two-Sided Cells of Affine Weyl Groups of Type $\tilde A_{n-1}$

In this paper we prove Lusztig's conjecture on based ring for an affine Weyl group of type $\tilde A_{n-1}$.

Weyl群的双侧细胞与某些分裂Whittaker多项式

Two-sided cells of Weyl groups and certain splitting Whittaker polynomials

本文研究Weyl群中具有固定下降集的子集的双侧细胞数目，并证明某些Whittaker多项式在有理数域上分裂。

Asymptotic lowest two-sided cell

To a Coxeter system $(W,S)$ (with $S$ finite) and a weight function $L : W \to \NM$ is associated a partition of $W$ into Kazhdan-Lusztig (left, right or two-sided) $L$-cells. Let $S^\circ = \{s \in S | L(s)=0\}$, $S^+=\{s \in S | L(s) > 0\}$ and let $C$ be a Kazhdan-Lusztig (left, right or two-sided) $L$-cell. According to the semicontinuity conjecture of the first author, there should exist a positive natural number $m$ such that, for any weight function $L' : W \to \NM$ such that $L(s^+)=L'(s^+) > m L'(s^\circ)$ for all $s^+ \in S^+$ and $s^\circ \in S^\circ$, $C$ is a union of Kazhdan-Lusz…

Minimal and maximal elements in Kazhdan-Lusztig double sided cells of $S_n$ and Robinson-Schensted correspondance

In symmetric groups, a two-sided cell is the set of all permutations which are mapped by the Robinson-Schensted correspondence on a pair of tableaux of the same shape. In this article, we show that the set of permutations in a two-sided cell which have a minimal number of inversions is the set of permutations which have a maximal number of inversions in conjugated Young subgroups. We also give an interpretation of these sets with particular tableaux, called reading tableaux. As corollary, we give the set of elements in a two-sided cell which have a maximal number of inversions.

The based ring the lowest generalized two-sided cell of an extended affine Weyl group

Let $\mathbf{c}_0$ be the lowest generalized two-sided cell of an extended affine Weyl group W. We determine the structure of the based ring of $\mathbf{c}_0$. For this we show that certain conjectures of Lusztig on generalized cells (called P1-P15) hold for $\mathbf{c}_0$. As an application, we use the structure of the based ring to study certain simple modules of Hecke algebras of $ W $ with unequal parameters, namely those attached to $\mathbf{c}_0$. Also we give a set of prime ideals $\mathfrak{p}$ of the center $\mathcal{Z}$ of the generic affine Hecke algebra $\mathcal{H}$ such that the …

Cells and representations of right-angled Coxeter groups

We study Kazhdan-Lusztig cells and the corresponding representations of right-angled Coxeter groups and Hecke algebras associated to them. In case of the infinite groups generated by reflections in the hyperbolic plane about the sides of right-angled polygons we obtain an explicit description of the left and two-sided cells. In particular, we prove that there are infinitely many left cells but they all form only three two-sided cells.

Kazhdan-Lusztig coefficients for the lowest two-sided cell of type $\tilde{G_{2}}$

This paper studies the Kazhdan-Lusztig coefficients $μ(u,w)$ of the Kazhdan-Lusztig polynomials $P_{u,w}$ for the lowest cell ${c_{0}}$ of an affine Weyl group of type $\widetilde{G_{2}}$ and gives an estimation $μ(u,w)\leqslant 3$ for $u,w\in c_{0}$.

基于最低双侧细胞的环中的正性性质

A positivity property in the based ring of the lowest two-sided cell

本文研究了最低双侧细胞中Hecke代数的正性性质，通过广义指数和假设条件推导了系数公式，并定义了新的正基，特别针对GL_n给出了部分其他细胞的结果。

Conjugacy classes of involutions and Kazhdan-Lusztig cells

According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz' conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types; combined with previous work, thi…

Cellularity of the p-Canonical Basis for Symmetric Groups

For symmetric groups we show that the p-canonical basis can be extended to a cell datum for the Iwahori-Hecke algebra H and that the two-sided p-cell preorder coincides with the Kazhdan-Lusztig two-sided cell preorder. Moreover, we show that left (or right) p-cells inside the same two-sided p-cell for Hecke algebras of finite crystallographic Coxeter systems are incomparable (Property A).

Cells in quantum affine sl_n

We study Lusztig's theory of cells for quantum affine $\mathfrak{sl}_n$. Using the geometric construction of the quantum group due to Lusztig and Ginzburg--Vasserot, we describe explicitly the two-sided cells, the number of left cells in a two--sided cell, and the asymptotic algebra, verifying conjectures of Lusztig.

Cellules de Calogero-Moser

Using the representation theory of Cherednik algebras at t=0 and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-sid…

Calogero-Moser versus Kazhdan-Lusztig cells

In 1979, Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter groups. They defined in particular partitions of the group in left and two-sided cells. In 1983, Lusztig generalized this theory to Hecke algebras of Coxeter groups with unequal parameters. We propose a definition of left cells and two-sided cells for complex reflection groups, based on ramification theory for Calogero-Moser spaces. These spaces have been defined via rational Cherednik algebras by Etingof and Ginzburg. We conjecture that these coincide with Kazhdan-Lusztig cells, for real reflection groups. C…

a(2)-有限 Coxeter 系统中 a 值为 2 的 Kazhdan–Lusztig 胞腔

Kazhdan--Lusztig cells of $\mathbf{a}$-value 2 in $\mathbf{a}(2)$-finite Coxeter systems

本文针对不可约 a(2)-有限 Coxeter 群，利用新引入的 stub 元素、星运算和弱 Bruhat 序，显式刻画 a 值为 2 的 Kazhdan–Lusztig 单边与双边胞腔，并计算其基数。

Tensor categories attached to exceptional cells in Weyl groups

Using truncated convolution of perverse sheaves on a flag variety Lusztig associated a monoidal category to a two sided cell in the Weyl group. We identify this category in the case which was not decided previously.

Asymptotic Semigroups and Two-sided Weak Orders

Various partial orders related to the structures of dual canonical monoids are investigated. It is shown that the nilpotent variety of a dual canonical monoid is equidimensional; its dimension is found. It is shown in type A that certain intervals of the Putcha poset of a dual canonical monoid are isomorphic to the Renner monoids of matrices. The notion of a two-sided weak order on a normal reductive monoid is introduced. A criterion, in terms of type maps, for the covering relations in a two-sided weak order to have degree 2 is found. It is shown that, for the unique equivariant divisor of a …

RSK bases and Kazhdan-Lusztig cells

From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, 'RSK bases' are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory of the symmetric group are discussed.

胞腔2-表示的端态射

Endomorphisms of cell 2-representations

本文在自然假设下确定了与强正则双边胞腔相关联的fiat 2-范畴的胞腔2-表示的端态射范畴，并完全刻画了仅含一个非恒等双边胞腔的J-单fiat 2-范畴，同时证明在正分次情形下额外限制可去除。

Generalized induction of Kazhdan-Lusztig cells

Following Lusztig, we consider a Coxeter group $W$ together with a weight function. Geck showed that the Kazhdan-Lusztig cells of $W$ are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of $W$ which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of a certain finite parabolic subgroup of $W$ are cells in the whole group, and we decompose the affine Weyl group $\tilde{G}_{2}$ into left and two-sided cells for a whole class of weight functions.

Kazhdan-Lusztig cells in planar hyperbolic Coxeter groups and automata

Let C be a one- or two-sided Kazhdan--Lusztig cell in a Coxeter group (W,S), and let Reduced(C) denote the set of reduced expressions of all w in C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(C) is regular. In this paper we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.

Representations of Coxeter groups and homology of Coxeter graphs

We classify a class of complex representations of an arbitrary Coxeter group via characters of the integral homology of certain graphs. Such representations can be viewed as a generalization of the geometric representation and correspond to the second-highest 2-sided cell in the sense of Kazhdan-Lusztig. We also give a description of the cell representation provided by this 2-sided cell, and find out all its simple quotients for simply laced Coxeter system with no more than one circuit in the Coxeter graph.

A parametrization of unipotent representations

We define a map from the unipotent representations of a split semisimple group over a finite field to (essentially) the set of pairs of left cells representations of the Weyl group in the same two-sided cell. We use this map to parametrize the unipotent representations.

Character sheaves on disconnected groups, IX

We associate a two-sided cell to any (parabolic) character sheaf. We study the interaction of the duality operator for character sheaves and the operation of "twisted induction".

Simple transitive 2-representations of small quotients of Soergel bimodules

In all finite Coxeter types but $I_2(12)$, $I_2(18)$ and $I_2(30)$, we classify simple transitive $2$-rep\-re\-sen\-ta\-ti\-ons for the quotient of the $2$-category of Soergel bimodules over the coinvariant algebra which is associated to the two-sided cell that is the closest one to the two-sided cell containing the identity element. It turns out that, in most of the cases, simple transitive $2$-representations are exhausted by cell $2$-representations. However, in Coxeter types $I_2(2k)$, where $k\geq 3$, there exist simple transitive $2$-representations which are not equivalent to cell $2$-r…

An asymptotic cell category for cyclic groups

In his theory of unipotent characters of finite groups of Lie type, Lusztig constructed modular categories from two-sided cells in Weyl groups. Broué,Malle and Michel have extended parts of Lusztig's theory to complex reflection groups. This includes generalizations of the corresponding fusion algebras, although the presence of negative structure constants prevents them from arising from modular categories. We give here the first construction of braided pivotal monoidal categories associated with non-real reflection groups (later reinterpreted by Lacabanne as super modular categories). They ar…

Kazhdan-Lusztig Basis and A Geometric Filtration of an affine Hecke Algebra, II

An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the corresponding Lie algebra. In this paper we will show that the two-sided ideals are in fact the two-sided ideals of the affine Hecke algebra defined through two-sided cells of the corresponding affine Weyl group after the two-sided ideals are tensored by rational numbers field. This proves a weak form of a conjecture of Ginzburg proposed in 1987.

Truncated convolution of character sheaves

Let G be a reductive connected group over an algebraic closure of a finite field. I define a tensor structure on the category of perverse sheaves on G which are direct sums of unipotent character sheaves in a fixed two-sided cell, in accordance with a conjecture I have made in 2004. I also show that that the resulting monoidal category is equivalent to the centre of a monoidal category which I defined in 1997 (a categorical version of the J-ring attached to the same two-sided cell), thus verifying a conjecture of Bezrukavnikov, Finkelberg, Ostrik. A possible interpretation of unipotent charact…

渐近Plancherel公式与Lusztig的渐近代数对于$\tilde{\mathsf{A}}_n$

The asymptotic Plancherel formula and Lusztig's asymptotic algebra for $\tilde{\mathsf{A}}_n$

本文通过构造平衡的单元模系统，证明渐近Plancherel定理，并发展每个双侧Kazhdan-Lusztig单元的相对Satake同构，给出Lusztig渐近代数的新显式构造。

Kazhdan-Lusztig 映射与 Langlands 对偶

Kazhdan-Lusztig map and Langlands duality

本文利用仿射旗簇几何定义从双边胞腔到 Weyl 群共轭类的 Kazhdan-Lusztig 映射，作为经典理论的仿射类比，并通过仿射 Springer 纤维上同调中的 Springer 表示证明，为 Lusztig 猜想提供证据。

On the Kazhdan--Lusztig cells in type $E_8$

In 1979, Kazhdan and Lusztig introduced the notion of "cells" (left, right and two-sided) for a Coxeter group $W$, a concept with numerous applications in Lie theory and around. Here, we address algorithmic aspects of this theory for finite $W$ which are important in applications, e.g., run explicitly through all left cells, determine the values of Lusztig's $\ba$-function, identify the characters of left cell representations. The aim is to show how type $E_8$ (the largest group of exceptional type) can be handled systematically and efficiently, too. This allows us, for the first time, to solv…

渐近 Schur 代数与 q-Schur 代数的胞腔性

Asymptotic Schur algebras and cellularity of q-Schur algebras

本文证明了有限型 q-Schur 代数的胞腔性，推广了 Geck 的定理，并研究了渐近 Schur 代数的特殊模与 Schur 代数的左胞腔表示，推广了 Lusztig 的相关工作。

Automata and cells in affine Weyl groups

Let W~ be an affine Weyl group, and let C be a left, right, or two-sided Kazhdan--Lusztig cell in W~. Let Reduced (C) be the set of all reduced expressions of elements of C, regarded as a formal language in the sense of the theory of computation. We show that Reduced (C) is a regular language. Hence the reduced expressions of the elements in any Kazhdan--Lusztig cell can be enumerated by a finite state automaton.

On the affine Schützenberger involution

We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at -1.

On tensor categories attached to cells in affine Weyl groups

We prove a conjecture by Lusztig, which describes the tensor categories of perverse sheaves on affine flag manifolds, with tensor structure provided by truncated convolution, in terms of the Langlands dual group. We also give a geometric (categorical) description of Lusztig's bijection between two-sided cells in an affine Weyl group, and unipotent conjugacy classes in the Langlands dual group. The main tool is the sheaf-theoretic construction of the center of the affine Hecke algebra due to Gaitsgory (based on ideas of Beilinson and Kottwitz), see math.AG/9912074.

Sign insertion and Kazhdan-Lusztig cells of affine symmetric groups

Combinatorics of Kazhdan-Lusztig cells in affine type $A$ was originally developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy, and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC) which gives an analog of Robinson-Schensted correspondence for affine symmetric groups. An alternative approach to Kazhdan-Lusztig theory in affine type $A$ was developed by Blasiak in his work on catabolism. He introduced sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of sign insertion process determines uniquel…