Xuhua He, Sian Nie
In this paper, we study the relation between the cocenter and the representations of an affine pro-$p$ Hecke algebra. As a consequence, we obtain a new criterion on the supersingular representation: a (virtual) representation is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter.
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We determine a basis of the cocenter (i.e., the zeroth Hochschild homology) of the degenerate affine Hecke-Clifford and spin Hecke algebras in classical types.
In this paper, we give explicit descriptions of the centers and cocenters of $0$-Hecke algebras associated to finite Coxeter groups.
In this paper, we study the relation between the cocenter and the representation theory of affine Hecke algebras. The approach is based on the interaction between the rigid cocenter, an important subspace of the cocenter, and the dual object in representation theory, the rigid quotient of the Grothendieck group of finite dimensional representations.
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).
In this paper, we study the relation between the cocenter $\overline{\tilde \ch_0}$ and the finite dimensional representations of an affine $0$-Hecke algebra $\tilde \ch_0$. As a consequence, we obtain a new criterion on the supersingular modules: a (virtual) module of $\tilde \ch_0$ is supersingular if and only if its character vanishes on the non-supersingular part of $\overline{\tilde \ch_0}$.
We study the cocenter of the cyclotomic quiver Hecke algebra $R^Λ_α$ associated to an {\it arbitrary} symmetrizable Cartan matrix $A=(a_{ij})_{i,j}\in I$, $Λ\in P^+$ and $α\in Q_n^+$. We introduce a notion called "piecewise dominant sequence" and use it to construct some explicit homogeneous elements which span the maximal degree component of the cocenter of $R^Λ_α$. We show that the minimal degree components of the cocenter of $R^Λ_α$ is spanned by the image of some KLR idempotent $e(ν)$, where each $ν\in I^α$ is piecewise dominant. As an application, we show that the weight space $L(Λ)_{Λ-α}…
On the cocenter of the cyclotomic Hecke algebra of type $G(r,1,n)$
本文基于Geck和Pfeiffer的工作，构建了类型$G(r,1,n)$的循环Hecke代数$\mathscr{H}_{n,K}$的整数基，并证明其余核和中心的维度与基域特征、Hecke参数和循环参数无关。
Trace forms on the cyclotomic Hecke algebras and cocenters of the cyclotomic Schur algebras
本文在A型分圆Hecke代数上定义了统一的迹型，并构造了双基及分圆Schur代数余中心的显式基，证明其维数与基域、Hecke参数及分圆参数无关。
In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive group over a nonarchimedean local field is contained in the rigid cocenter. As applications, we prove the trace Paley-Wiener theorem and the abstract Selberg principle for mod-$l$ representations.
Cocenters of Hecke algebras $\mathcal H$ play an important role in studying mod $\ell$ or $\mathbb C$ harmonic analysis on connected $p$-adic reductive groups. On the other hand, the depth $r$ Hecke algebra $\mathcal H_{r^+}$ is well suited to study depth $r$ smooth representations. In this paper, we study depth $r$ rigid cocenters $\overline{\mathcal H}^{\mathrm{rig}}_{r^+}$ of a connected reductive $p$-adic group over rings of characteristic zero or $\ell\neq p$. More precisely, under some mild hypotheses, we establish a Jordan decomposition of the depth $r$ rigid cocenter, hence find an exp…
In this paper, we study some relation between the cocenter $\bar H(G)$ of the Hecke algebra $H(G)$ of a connected reductive group $G$ over an nonarchimedean local field and the cocenter $\bar H(M)$ of its Levi subgroups $M$. Given any Newton component of $\bar H(G)$, we construct the induction map $\bar i$ from the corresponding Newton component of $\bar H(M)$ to it. We show that this map is surjective. This leads to the Bernstein-Lusztig type presentation of the cocenter $\bar H(G)$, which generalizes the work \cite{HN2} on the affine Hecke algebras. We also show that the map $\bar i$ we cons…
Affine Hecke algebras arise naturally in the study of smooth representations of reductive $p$-adic groups. Finite dimensional complex representations of affine Hecke algebras (under some restriction on the isogeny class and the parameter function) has been studied by many mathematicians, including Kazhdan-Lusztig \cite{KL}, Ginzburg \cite{CG}, Lusztig \cite{L1}, Reeder \cite{Re}, Opdam-Solleveld \cite{OS}, Kato \cite{Kat}, etc. The approaches are either geometric or analytic. In this note, we'll discuss a different route, via the so-called "cocenter-representation duality", to study finite dim…
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 $\ch$ 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 $T_w$ in the cocenter of $\ch$. In the process, we obtain a Bernstein presentation of the cocenter of $\ch$. We also obtain a comparison theorem among the class polynomials of $\ch$ and of its parabolic subalgebras, which is analogous to the Hodge-Newton decomposition theor…
Cocenter of Hecke algebras of Kac-Moody groups
本文证明了 Kac-Moody 群泛型 Hecke 代数的余中心由共轭类典范元素构成基，通过重整化轨道积分新框架解决了线性无关性开放问题。
On the structure of the affine asymptotic Hecke algebras
本文通过构造非平凡中心扩张的例子证明Lusztig猜想的较弱形式是最优的，同时证明仿射Hecke代数到仿射渐近Hecke代数的Lusztig同态诱导上中心同构，并探讨中心扩张与上中心结构的关联。
Trace Paley-Wiener theorem for Braverman-Kazhdan's asymptotic Hecke algebra
本文研究 Braverman-Kazhdan 渐近 Hecke 代数，通过分类其不可约表示证明余中心同构猜想，并在 GL_n 情形给出显式描述。
Centralizers in Hecke Algebras of Any Coxeter Group
针对任意Coxeter群Hecke代数中抛物子代数中心化子在不定情形下研究不足的问题，结合代数技术与Davis复形几何工具，给出中心化子的基、分类无限Coxeter群有限部分共轭类并定义适配的类多项式变体，推广了中心相关已知结果。
A geometric realization of the asymptotic affine Hecke algebra
本研究证明了渐近仿射Hecke代数$J$的几何描述猜想，给出其不可约表示参数化的新几何证明、谱描述的新证明及余中心描述的证明，基于对$J$的新代数描述。
Mirabolic Hecke algebras, Schur-Weyl duality and Frobenius character formulas
本文研究了偏微分Hecke代数的新的呈现方式，建立了其与量子群之间的Schur-Weyl对偶性，并推导出Frobenius特征公式及递归的Murnaghan-Nakayama规则。
Trace radicals and cocenters of free products
研究含幺结合代数\(A\)和\(B\)自由积\(A*B\)的迹根，通过结合余中心分解与有限维表示构造，证明\(A*B\)的迹根是\(A\)和\(B\)迹根的直和，得出\(A*B\)是迹RFD的充要条件是\(A\)和\(B\)都是迹RFD。
The center conjecture for the cyclotomic KLR algebras $R_β^Λ$ asserts that the center of $R_β^Λ$ consists of symmetric elements in its KLR $x$ and $e(ν)$ generators. In this paper we show that this conjecture is equivalent to the injectivity of some natural map $\barι_β^{Λ,i}$ from the cocenter of $R_β^Λ$ to the cocenter of $R_β^{Λ+Λ_i}$ for all $i\in I$ and $Λ\in P^+$. We prove that the map $\barι_β^{Λ,i}$ is given by multiplication with a center element $z(i,β)\in R_β^{Λ+Λ_i}$ and we explicitly calculate the element $z(i,β)$ in terms of the KLR $x$ and $e(ν)$ generators. We present an explic…
We prove that the affine Yokonuma-Hecke algebra defined by Chlouveraki and Poulain d'Andecy is a particular case of the pro-$p$-Iwahori-Hecke algebra defined by Vignéras.
We prove that cyclotomic Yokonuma--Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.
Given a quiver automorphism with nice properties, we give a presentation of the fixed subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r,1,n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In the meanwhile, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike al…
This article introduces the duplex Hecke algebra, which is an infinite dimensional algebra generated by two Hecke algebras. This concept originates from the degenerate duplex Hecke algebra in the theory of Schur-Weyl duality related to enhanced reductive algebraic groups. We will study the finite dimensional natural representation of the duplex Hecke algebra on tensor space and prove that the duplex Hecke algebra forms a duality with the Levi type quantum group.
We define an imbedding of the Hecke algebra module carried by the involutions in a Weyl group W (defined by the author and Vogan) into a completion of the Hecke algebra. An analogous result is proved for any Coxeter group. A variant of the construction gives an imbedding of the biregular representation of a Hecke algebra of W in a completion of the Hecke algebra of W^2.
In the context of Hecke algebras of complex reflection groups, we prove that the generalized Hecke algebras of normalizers of parabolic subgroups are semidirect products, under suitable conditions on the parameters involved in their definition.
Tensor product degenerate spin affine Hecke superalgebras
本文定义了高阶退化自旋仿射Hecke超代数，建立其与高阶退化仿射Hecke-Clifford超代数的同构关系，推广了退化情形下两类超代数的对应关系，丰富了超代数结构理论研究。
DG-Enhanced Hecke and KLR Algebras
本文构造了退化与非退化仿射Hecke代数的DG-增强版本，扩展了相关同构，证明其完备化后与KLR代数同构，并推导出其同调集中于零次且同构于分圆Hecke代数。
We propose a categorification of the cyclotomic Hecke algebra in terms of the equivariant K-theory of the framed matrix factorizations. The construction generalizes the earlier construction of the authors for a categorification of the finite Hecke algebra of type A. We also explain why our construction provides a faithful realization of the Hecke algebras and discuss a geometric realization of the Jucys-Murphy subalgebra.
We prove a strong induction theorem for graded Hecke algebras and we classify the tempered and square integrable representations of such algebras using methods of equivariant homology.
We state four open problems on Iwahori-Hecke algebras. The first one states a relation between some algebras appearing in Solleveld's work and some explicit Hecke algebras appearing in the study of unipotent representations. The second one is a boundedness conjecture. The third one is a conjecture of well-definedness of an asymptotic Hecke algebra. The fourth one concerns positive conjugacy classes in a Weyl group.
We study the structure of parabolic inductions of a pro-$p$-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo $p$ representations of pro-$p$-Iwahori Hecke algebra in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vigneras, it completes the classification of irreducible modulo $p$ representations.
We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type $A$ (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic $p$. Secondly, the modular branching rules for these alg…
In this short note we give a new proof of the quantum generalization of Regev's theorems by applying the Murnaghan-Nakayama formula for skew characters of the generic Iwahori-Hecke algebra.
We prove a conjecture of Kashiwara and Miemietz on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case.
In this paper we prove that the linear Koszul duality equivalence constructed in a previous paper provides a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras.
In this paper we prove a quantum generalization of Regev's theorems in (Israel. J. Math. 195 (2013), 31--35) by applying the Schur-Weyl duality between the quantum superalgebra and Iwahori-Hecke algebra. We also present an alternative proof of the quantized generalizations using the skew character theory of Iwahori-Hecke algebras.
We classify all the decomposition matrices of the generic Hecke algebras on 3 strands in characteristic 0. These are the generic Hecke algebras associated to the exceptional complex reflection groups $G_4$, $G_8$ and $G_{16}$. We prove that for every choice of the parameters that define these algebras, all ordinary representations are obtained as modular reductions of irreducible representations.
We explicitly calculate a projective bimodule resolution for a special biserial algebra giving rise to the Hecke algebra H_q(S_4) when q=-1. We then determine the dimensions of the Hochschild cohomology groups.
This text provides an introduction and complements to some basic constructions and results in 2-representation theory of Kac-Moody algebras.
The purpose of this paper is to study induction and restriction of two-variable Hecke algebras. First, we provide the explicit form of the Mackey decomposition formula. And then, we elucidate how (anti-)involutions interact with induction product and restriction.
We first present an Iwahori-Matsumoto presentation of affine Yokonuma-Hecke algebras $\widehat{Y}_{r,n}(q)$ to give a new proof of the fact, which was previously proved by Chlouveraki and Sécherre, that $\widehat{Y}_{r,n}(q)$ is a particular case of the pro-$p$-Iwahori-Hecke algebras defined by Vignéras, meanwhile, we give one application. Using the new presentation, we then give a third presentation of $\widehat{Y}_{r,n}(q),$ from which we immediately get an unexpected result, that is, the extended affine Hecke algebra of type $A$ is a subalgebra of the affine Yokonuma-Hecke algebra.
We define an analogue of the Dirac operator for the degenerate affine Hecke-Clifford algebra. A main result is to relate the central characters of the degenerate affine Hecke-Clifford algebra with the central characters of the Sergeev algebra via Dirac cohomology. The action of the Dirac operator on certain modules is also computed. Results in this paper could be viewed as a projective version of the Dirac cohomology of the degenerate affine Hecke algebra.
We prove a series of conjectures of Enomoto and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type B. The main ingredient of the proof is a new graded Ext-algebra associated with quiver with involutions that we compute explicitly.
Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields
本文定义了局部域上 split Kac-Moody 群的 Iwahori-Hecke 代数完备化，利用 masure 工具确定其中心并证明同构于球面 Hecke 代数，同时将 Hecke 代数构造推广至特殊球面面。
In this paper, we prove a decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2 with positive weight function. Then we discuss some applications of this kind of decomposition to Lusztig's conjectures P1-P15.
L'Algèbre de Hecke chirale nilpotente
该研究将手性Hecke代数推广至幂零轨道，通过等型分量分析，在非容许整数水平上构造了简单仿射顶点代数的新模及融合方式。
We study a twisted Euler-Poincaré pairing for graded affine Hecke algebras, and give a precise connection to the twisted elliptic pairing of Weyl groups defined by Ciubotaru-He. The Ext-groups for an interesting class of parabolically induced modules are also studied in a connection with the twisted Euler-Poincaré pairing. We also study a certain space of graded Hecke algebra modules which equips with the twisted Euler-Poincaré pairing as an inner product.
This paper proves a periodic property of Jucys-Murphy elements of the degenerate and non-degenerate cy- clotomic Hecke algebras of type A. We do this by first giving a new closed formula for the KLR idempotents e(i) which, it tuns out, is very efficient computationally.
I withdraw this article because one of the references has changed in the meantime.
The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra isomorphism relating tDaHCa to tsDaHa is obtained.
Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.
We generalize graded Hecke algebras to include a twisting two-cocycle for the associated finite group. We give examples where the parameter spaces of the resulting twisted graded Hecke algebras are larger than that of the graded Hecke algebras. We prove that twisted graded Hecke algebras are particular types of deformations of a crossed product.
We introduce the spin Hecke algebra, which is a q-deformation of the spin symmetric group algebra, and its affine generalization. We establish an algebra isomorphism which relates our spin (affine) Hecke algebras to the (affine) Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the spin (affine) Hecke algebra and a nonstandard presentation of the usual (affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke algebra is introduced to provide a link between these algebras. Various algebraic structures for the spin (affine) Hecke algebra are establishe…
The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.
This is an extended and corrected version of the author's Diplomarbeit. A class of algebras called generic pro-$p$ Hecke algebras is introduced, enlarging the class of generic Hecke algebras by considering certain extensions of (extended) Coxeter groups. Examples of generic pro-$p$ Hecke algebras are given by pro-$p$-Iwahori Hecke algebras and Yokonuma-Hecke algebras. The notion of an orientation of a Coxeter group is introduced and used to define `Bernstein maps' intimately related to Bernstein's presentation and to Cherednik's cocycle. It is shown that certain relations in the Hecke algebra …
We use Bernstein's presentation of the Iwahori-Matsumoto Hecke algebra to obtain a simple proof of the Satake isomorphism and, in the same stroke, compute the center of the Iwahori-Matsumoto Hecke algebra.
Based on a recent result of Mathas and the author, we prove that Uno's conjecture on representation types of Hecke algebras is true for all Hecke algebras of classical type.
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
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