C. Bonnafé，M.盖克湖Iancu和T. Lam利用任意秩的标准多米诺表族对$B_n$型不等参数Hecke代数的Kazhdan-Lusztig左胞腔进行了参数化.利用G. Lusztig和C. Bonnafé，我们验证了猜想和描述的结构，每个细胞作为一个模块的基础外尔群。
英文摘要：
A conjecture of C. Bonnafé, M. Geck, L. Iancu, and T. Lam parameterizes Kazhdan-Lusztig left cells for unequal parameter Hecke algebras in type $B_n$ by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by G. Lusztig and the recent work of C. Bonnafé, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.
On the classification of irreducible representations of affine Hecke algebras with unequal parameters
设$R$是仿射Weyl群$W^e$的根数据，$H = H (R,q)$是仿射Hecke代数，其参数$q$为正，且可能不相等.那么$H$是群代数$\mathbb C [W^e]$的变形，所以比较$H$和$W^e$的表示理论是很自然的。我们定义了一个映射从不可约$H$ -表示$W^e$ -表示，我们表明，当扩展到Grothendieck群的有限维表示，这个映射成为同构，模扭转。映射可以被调整为从$H$的对偶空间到$W^e$的对偶空间的（非自然的）连续双射。我们用它来证明仿射Hecke代数版本的猜想的奥伯特，鲍姆和Plymen，它预测一个强大的和明确的几何相似性之间的对偶空间的$H$和$W^e$。一个重要的作用是发挥施瓦茨完成$S = S (R,q)$的$H$，代数的表示正是回火$H$ -表示。我们构造同构$ζ_ε: S (R,q^ε) \to S (R,q)$ $(ε>0)$和注入$ζ_0 : S (W^e) = S (R,q^0) \to S (R,q)$，连续依赖于$ε$。虽然$ζ_0$不是满射，但它在许多方面表现得像代数同构。$ζ_0$不仅扩展到有限维表示的Grothendieck群上的双射，它还在拓扑$K$ -理论和周期循环同调（前两个模挠）上诱导同构。这证明了Higson和Plymen的一个猜想，即仿射Hecke代数$H (R,q)$的$C^*$ -完备化的$K$ -理论不依…
We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type A III/IV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In the …
In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a $\mathbb{Z}/m\mathbb{Z}$-grading $\oplus_{i}\mathfrak{g}_{i}$ and a block of $G_{\underline{0}}$-equivariant complexes on the nilpotent cone of $\mathfrak{g}_{\underline{1}}$ as introduced in \cite{LY1}, we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot \cite{V} and Oblomkov-Yun \ci…
We establish an explicit algebra isomorphism between the affine Yokonuma-Hecke algebra $\widehat{Y}_{r,n}(q)$ and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type $A.$ As an application of this result, we show that $\widehat{Y}_{r,n}(q)$ is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter $q$ is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of $\widehat{Y}_{r,n}(q)$ previously obtained in [CW].
We first give a direct proof of a basis theorem for the cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q).$ Our approach follows Kleshchev's, which does not use the representation theory of $Y_{r,n}^{d}(q),$ and so it is very different from that of [ChP2]. We also present two applications. Then we prove that the cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ is cellular by constructing an explicit cellular basis, and show that the Jucys-Murphy elements for $Y_{r,n}^{d}(q)$ are JM-elements in the abstract sense. In the appendix, we shall develop the fusion procedure for $Y_{r,n}^{d}(q).$
On overview of Coxeter groups $W$, their Iwahori-Hecke algebras $H$, Lusztig's asymptotic algebras $J$, cellular algebras in general and cellularity of $H$ in particular, as well as an introduction to Gyoja's $W$-graph algebra and the $W$-graph decomposition conjecture.
In this note, we first prove that the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}$ is cellular by constructing an explicit cellular basis. We then develop the fusion procedure for $Y_{r,n}^{d}$, that is, a complete set of pairwise orthogonal primitive idempotents for $Y_{r,n}^{d}$ is defined by consecutive evaluations of a certain rational function.
We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y, and use it to give a concrete isomorphism between Y and Shoji's modified Ariki-Koike algebra. We give a cellular basis for Y and show that the Jucys-Murphy elements for Y are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi-Juyumaya algebra of braids and ties.
We define the Iwahori-Hecke algebra for an almost split Kac-Moody group over a local non-archimedean field. We use the hovel associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The fixer K of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define the Iwahori-Hecke algebra as the algebra of some K-bi-invariant functions on the group with support consisting of a finite union of double classes. As two chambers in the hovel are not always in a same apartment, this support has to be in some large subsemigroup o…
Let H = H (R,q) be an affine Hecke algebra with complex, possibly unequal parameters q, which are not roots of unity. We compute the Hochschild and the cyclic homology of H. It turns out that these are independent of q and that they admit an easy description in terms of the extended quotient of a torus by a Weyl group, both of which are canonically associated to the root datum R. For q positive we also prove that the representations of the family of algebras H (R,q^ε) come in families which depend analytically on the complex number ε. Analogous results are obtained for graded Hecke algebras an…
This paper gives a Schur-Weyl duality approach to the representation theory of the affine Hecke algebras of type C with unequal parameters. The first step is to realize the affine braid group of type $C_k$ as the group of braids on $k$ strands with two poles. Generalizing familiar methods from the one pole (type A) case, this provides commuting actions of the quantum group $U_q\mathfrak{g}$ and the affine braid group of type $C_k$ on a tensor space $M\otimes N \otimes V^{\otimes k}$. Special cases provide Schur-Weyl pairings between the affine Hecke algebra of type $C_k$ and the quantum group …
We establish a Schur type duality between a coideal subalgebra of the quantum group of type A and the Hecke algebra of type B with 2 parameters. We identify the $\imath$-canonical basis on the tensor product of the natural representation with Lusztig's canonical basis of the type B Hecke algebra with unequal parameters associated to a weight function.
