Ben Elias, Geordie Williamson
We construct a functor from the Hecke category to a groupoid built from the underlying Coxeter group. This fixes a gap in an earlier work of the authors. This functor provides an abstract realization of the localization of the Hecke category at the field of fractions. Knowing explicit formulas for the localization is a key technical tool in software for computations with Soergel bimodules.
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The Hecke category is bigraded
本文证明了 Hecke 范畴具有双分次结构，并完整分类了其上的所有分次及保持对象的自等价。
Soergel calculus for monodromic Hecke categories
本文引入两个2-范畴，分类单映射Hecke代数。通过Abe的Soergel双模理论和Elias-Williamson图示算术扩展，证明两者等价并关联到奇性sheaves的分类。
Cyclotomic double affine Hecke algebras and affine parabolic category O, I
利用Orbifold KZ联络构造了一个从A型仿射抛物范畴O到分圆有理双仿射Hecke代数范畴O的函子。我们给出了几个关于这个函子的结果。
Gradings on the Hecke category, and categorification with unequal parameters
本文研究了Hecke范畴上的分级，并提出了一种用于分层化具有不等参数的Hecke代数的方法。
Blocks of category $\mathcal{O}$ for rational Cherednik algebras and of cyclotomic Hecke algebras of type G(r,p,n)
利用多重划分的“剩余等价性”对有理Cherednik代数和G（r，p，n）型分圆Hecke代数的$\mathcal{O}$类块进行了分类.
Let G be a reductive algebraic group with a Borel subgroup B. We define the quasi-coherent Hecke category for the pair (G,B). For any regular Noetherian G-scheme X we construct a monoidal action of the Hecke category on the derived category of B-equivariant quasi-coherent sheaves on X. Using the action we define the Demazure Descent Data on the latter category and prove that the Descent category is equivalent to the derived category of G-equivariant sheaves on X.
A note on Hecke patterns in Category O
我们研究了BGG-范畴O的导出自等价族，它们对应于Hecke代数的标准生成元的作用。考虑了非分级和分级两种情况。
We attempt to give a gentle (though ahistorical) introduction to Koszul duality phenomena for the Hecke category, focusing on the form of this duality studied in joint work of Achar, Riche, Williamson, and the author. We illustrate some key phenomena and constructions for the simplest nontrivial case of (finite) $\mathrm{SL}_2$ using Soergel bimodules, a concrete algebraic model of the Hecke category.
Centers of degenerate cyclotomic Hecke algebras and parabolic category O
证明了与B_d（l）型复反射群相关联的退化分圆Hecke代数的中心由交换生成元中的对称多项式组成.退化分圆Hecke代数的块的分类是一个简单的推论。进而证明了李代数gl_n（C）的抛物范畴O的整块的中心由它的泛包络代数的中心生成.
Quantum category O vs affine Hecke category
本文研究奇数次单位根处量子范畴O与仿射Hecke范畴的关联，通过范畴形变、构造函子并结合自主发展的Rouquier-Soergel理论，证明仿射范畴O整块与仿射Hecke范畴新t-结构心的最高权范畴等价。
We initiate the study of Gaitsgory's central sheaves using complexes of Soergel bimodules, in extended affine type A. We conjecture that the complex associated to the standard representation can be flattened to a central complex in finite type A, which is the complex sought after in a conjecture of Gorsky-Negut-Rasmussen.
We describe an algorithm for computing the $p$-canonical basis of the Hecke algebra, or one of its antispherical modules. The algorithm does not operate in the Hecke category directly, but rather uses a faithful embedding of the Hecke category inside a semisimple category to build a "model" for indecomposable objects and bases of their morphism spaces. Inside this semisimple category, objects are sequences of Coxeter group elements, and morphisms are (sparse) matrices over a fraction field, making it quite amenable to computations. This strategy works for the full Hecke category over any base …
The Hecke category is emerging as a fundamental object in representation theory. We give a motivated introduction to this category in both its geometric (via parity sheaves) and diagrammatic (generators and relations) incarnations. We also discuss the p-canonical basis and Koszul duality for the Hecke category.
We construct an action of the Hecke monoid on the category of rational modules for the quantum negative Borel subgroup of the quantum general linear group. We also show that this action restricts to the category of polynomial modules for this quantum subgroup and induces an action on the category of modules for quantised Borel-Schur algebras.
Endoscopy for Modular Hecke Categories
本文推广复代数栈宇称层理论，构建扭等变宇称层体系，构造模版本的单值Hecke范畴，并给出其在Hecke代数模范畴化与内窥群范畴幺半等价中的两项应用。
Explicit $H_3$ relation in the diagrammatic Hecke category
本文通过显式线性组合计算并验证了 $H_3$ 型缺失的三色关系，完善了 Elias-Williamson 的图式 Hecke 范畴表示。
Categorifications and cyclotomic rational double affine Hecke algebras
Varagnolo和Vasserot证明了CRDAHA的范畴O与A型仿射抛物范畴O的一个子范畴之间的等价性。我们证明了这个猜想。作为应用，我们证明了Rouquier关于CRDAHA的单模维数的一个猜想和Chuang-Miyachi关于CRDAHA范畴O的Koszul对偶的一个猜想.
Modular affine Hecke category and regular unipotent centralizer
本文在温和显式假设下，用 Langlands 对偶仿射旗簇上的反常层几何地描述了约化代数群中正则幂零元中心化子的表示范畴，为后续建立仿射 Hecke 代数各几何实现之间的范畴等价奠定基础。
We compute the trace decategorification of the Hecke category for an arbitrary Coxeter group. More generally, we introduce the notion of a strictly object-adapted cellular category and calculate the trace for such categories.
A Lie-theoretic construction of representations of the degenerate affine and double affine Hecke algebras of type $BC_n$
设G=GL（N），K=GL（p）xGL（q），其中p+q=N，n为正整数.构造了一个从（G，K）上的Harish-Chandra模范畴到Bn型退化仿射Hecke代数的表示范畴的函子，以及一个从G/K上的K-单值扭D-模范畴到BC n型退化双仿射Hecke代数的表示范畴的函子，第二个函子是第一个函子的推广.这些函子是分别来自q-alg/9710037和math/0702670的A型函子的推广。
We study the diagrammatic Hecke category associated with the affine Weyl group of type $\tilde{A}_2$. More precisely we find a (surprisingly simple) basis for the Hom spaces between indecomposable objects, that we call indecomposable double leaves.
Duality between GL(n,R) and the degenerate affine Hecke algebra for gl(n)
定义了一个从$GL(n,R)$的Harish-Chandra模范畴到$gl(k)$的退化仿射Hecke代数的有限维表示范畴的正合函子$F_{n,k}$.在一定的自然假设下，我们证明了函子映射标准模到标准模（或零）和不可约到不可约（或零）。
We give a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. We offer a definition of a graded algebra $\mathcal{H}$ whose category of finite-dimensional ungraded nilpotent modules is equivalent to the category of finite-dimensional modules over an associated degenerate affine Hecke algebra. Moreover, unlike the traditional grading on degenerate affine Hecke algebras, this grading factors through central characters, and thus gives a grading to the irreducible representa…
Let $\textbf{G}$ be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block $\text{Rep}_0(\textbf{G})$ of $\text{Rep}(\textbf{G})$ by wall-crossing functors. This action was conjectured to exist by Riche-Williamson. Our method uses constructible sheaves and relies on Smith-Treumann theory.
The Large Affine Hecke Category is Calabi-Yau
本文证明大仿射海克斯类别定义了一个定向的、完全扩展的拓扑场论，并探讨了诱导相干卷积类别在何种条件下能定义此类场论。
Quantum Harish-Chandra bimodules at roots of unity and affine Hecke category
本文研究量子群在q为奇阶单位根时的Harish-Chandra双模范畴，将其与仿射Soergel双模范畴及非交换Springer解联系起来。
A Hecke action on $G_1T$-modules
本研究在特征$p>0$的代数闭域连通约化群$G$的$G_1T$-模主块上定义Hecke范畴作用，通过构造带Hecke作用且等价于Andersen-Jantzen-Soergel组合范畴的新范畴实现，为模表示论中Hecke作用研究提供新构造。
Ideas from Hodge theory have found important applications in representation theory. We give a survey of joint work with Ben Elias which uncovers Hodge theoretic structure in the Hecke category ("Soergel bimodules"). We also outline similarities and differences to other combinatorial Hodge theories.
Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $\mathcal{O}$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group $W$. We establish two generalizations of this result. On the one hand to the extension of the Hecke algebra associated to the normaliser of a reflection subgroup and on the other hand to the extension of the Hecke algebra by a lattice.
Calculating double-leaf coefficients in the Hecke category
针对Abe-Bott-Samelson双模态射，提出计算双叶系数的算法，通过修正Bruhat上三角性实现前向代入求解，并在Sage中实现，用于计算H_3型Zamolodchikov三色关系并简化表达。
Linear structure on a finite Hecke category in type A
本文为GL(n)构造了Grothendieck-Springer解上的等变导出范畴作用，实现了扩展仿射到有限Hecke代数投影的部分范畴化，并计算了关键的外幂。
Heisenberg algebra and a graphical calculus
引入了一种新的涉及双伴随函子和退化仿射Hecke代数的代数学的平面图演算。微积分导致添加剂幺半范畴的Grothendieck环包含积分形式的海森堡代数在无穷多个变量。我们构造了这一范畴中生成对象的乘积之间的态射的向量空间的基。
Isotropic meta Kazhdan--Lusztig combinatorics I: Ext-quiver presentation for the Hecke category
本文基于杯帽元Kazhdan-Lusztig组合学和Temperley-Lieb图示法，为各向同性Grassmannian的反球面Hecke范畴的基本代数提供Ext-有向图和关系呈现。
Algèbres de Hecke avec paramètres et représentations d'un groupe p-adique classique: préservation du spectre tempéré
设G是一个正交或辛p-adic群（不一定是可裂的）或一般线性p-adic群的内部形式。在以前的文章中，我们证明了G的光滑表示范畴的Bernstein分支等价于某些带参数的Hecke代数上的右模范畴，或者更一般地等价于这种代数与有限群代数的半直积上的右模范畴.本文的目的是证明这种等价性保持了调和谱和离散级数表示。
The Parabolic K-motivic Hecke Category
本文定义并研究了Kac-Moody群的抛物K-动机海克曼范畴，通过奇异K理论Soergel双模给出组合描述，探讨了量子K理论导出Satake等价的猜想。
Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras
定义了分圆有理双仿射Hecke代数范畴O上的i-限制函子和i-诱导函子。这就产生了一个简单模的同构类集合上的晶体，它同构于Fock空间的晶体。
Derived Satake category and Affine Hecke category in mixed characteristics
本文通过构造连接等特征与 Witt 向量仿射 Grassmannian 的新仿射 Grassmannian，在混合特征情形下建立了导出 Satake 等价及仿射 Hecke 代数两种范畴化之间的 Bezrukavnikov 等价。
Quiver varieties, category O for rational Cherednik algebras, and Hecke algebras
我们与理性的Cherednik代数的表示与复杂的反射群G（m，1，n）的层Nakajima的变种与扩展Dynkin gaphs通过一个Z-代数的建设。当定义Cherednik代数的参数改变时，定义Cherednik代数的稳定性条件也改变。我们解释的顺序O类几何使用这种关系，我们也与几何的a-函数的Hecke代数不等参数。
The affine Hecke category is a monoidal colimit
该研究证明仿射Hecke范畴等价于标准抛物子群对应的有限型Hecke子范畴的幺半稳定∞范畴余极限，其方法可推广到环群D-模等类似问题。
Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. In this paper, we investigate the relationship between the categories $\CC^0_{A_{N-1}^{(1)}}$ and $\CC^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur-Weyl duality functors $\F^{(t)}$ from $R^{A_\infty}\gmod$ to $\CC^0_{A_{N-1}^{(t)}}$ $(t=1,2)$.
On some representations of degenerate affine Hecke algebras of type $BC_{n}$
设G=U（p，q），K=U（p）× U（q）.在arXiv：0801.1530中，作者构造了一个从（G，K）的Harish-Chandra模范畴到Bn型退化仿射Hecke代数的表示范畴的函子.本文研究了G的主列模在该函子下的像。我们通过构造{y_i}的一族公共特征向量来描述图像的dAHA模块结构。在秩为1的情况下，我们给出了生成元y_{1}关于U（g）的中心元的显式表达式，并利用中心特征标计算了生成元y_{1}的特征值.
We introduce a class of equivalences, which we call generalized semi-infinite Hecke equivalences, between certain categories of representations of graded associative algebras which appear in the setting of semi-infinite cohomology for associative algebras and categories of representations of related algebras of Hecke type which we call semi-infinite Hecke algebras. As an application we obtain an equivalence between a category of representations of a non-twisted affine Lie algebra $\widehat{\mathfrak g}$ of level $-2h^\vee-k$, where $h^\vee$ is the dual Coxeter number of the underlying semisimp…
We state a conjecture that relates the derived category of smooth representations of a p-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of GL_n by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and Emerton-Helm.
An orthogonal form for level two Hecke algebras with applications
这是对Khovanov弧代数与Grassmannians、一般线性超群和Walled Brauer代数的O范畴相关的一些最近结果的调查。重点讨论了二级分圆Hecke代数的Young正交型的推广。
Hecke categories, idempotents, and commuting stacks
本研究建立了仿射Hecke范畴水平迹的幂等完备化这一拓扑范畴，与半幂零交换栈的等变导出范畴这一几何范畴间的联系，完善了此前的相关研究提案。
Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras II
设$\g$是$A^{(1)}_n$ $(n \ge 1)$或$D^{(1)}_n$ $(n \ge 4)$型的无扭仿射Kac-Moody代数，$\g_0$是$\g$的有限维单李子代数.对于每个具有$\g_0$型的Dynkin群$Q$，Hernandez和Leclerc（\cite{HL11}）引入了有限维可积$\uqpg$ -模范畴的一个张量子范畴$\CC_Q$，并证明了$\CC_Q$的Grothendieck环同构于与$\g_0$相关的幂单群$N$的坐标环$\C [N]$.利用在\cite{KKK13}中引入的广义量子仿射Schur-Weyl对偶，构造了从有限维分次$R$ -模范畴到$\CC_Q$范畴的正合函子$\F$，其中$R$表示与$\g_0$相关联的对称Schur-Hecke代数.我们证明了函子$\F$诱导的同态与Hernandez和Leclerc的同态一致，并证明了函子$\F$将简单模发送到简单模。
The Diagrammatic Spherical Category
本文通过图示方法构造了球形模的范畴化，并证明其与已有的代数球形范畴等价。
Let $F$ be a local field of mixed characteristic, let $k$ be a finite extension of its residue field, let ${\mathcal H}$ be the pro-$p$-Iwahori Hecke $k$-algebra attached to ${\rm GL}_{d+1}(F)$ for some $d\ge1$. We construct an exact and fully faithful functor from the category of supersingular ${\mathcal H}$-modules to the category of ${\rm Gal}(\overline{F}/F)$-representations over $k$. More generally, for a certain $k$-algebra ${\mathcal H}^{\sharp}$ surjecting onto ${\mathcal H}$ we define the notion of $\sharp$-supersingular modules and construct an exact and fully faithful functor from t…
We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an "affine" version of the construction of [14]. Explicitly, we show that the aforementioned trace is generated by the objects $E_{\textbf{d}} = \text{Tr}(Y_1^{d_1} \dots Y_n^{d_n} T_1 \dots T_{n-1})$ as $\textbf{d} = (d_1,\dots,d_n) \in \mathbb{Z}^n$, where $Y_i$ denote the Wakimoto objects of [9] and $T_i$ denote Rouquier complexes. We compute certain categorical commutators between the $E_{\textbf{d}}$'s and show that they match the categorical commutators between the sheaves $\math…
Modular affine Hecke category and regular centralizer
本文在正特征系数下，对仿射旗簇上的倾斜反常层范畴及其自由单值变体给出组合描述，以替代特征 0 的 Soergel 理论，并推进仿射 Hecke 代数两种几何实现之间的范畴等价构造。
Idempotents, traces, and dimensions in Hecke categories
本文研究了Hecke范畴中不可分解对象对应幂等元的计算方法并提供了相关系数的闭式公式，同时通过部分迹的递归公式计算渐近Hecke范畴的范畴维数。
We categorify the Hecke algebra with parameters 1 and v using a variation of the category of Soergel bimodules.
The center of the asymptotic Hecke category and unipotent character sheaves
本文证明了渐近Hecke范畴的中心为模ularity张量范畴，并研究了其在二面体群及H3、H4部分细胞中的应用。
Affine Hecke algebras and the Schubert calculus
本文用一种避开几何的组合方法研究了（广义）旗簇G/B的T-等变K-理论，即KT（G/B）的环结构。这里的数据是一个复约化代数群（或可对称化的Kac-Moody群）G，一个博雷尔子群B和一个极大环面T，K_T（G/B）是G/B上T-等变相干层的Grothendieck群。本文证明了Schubert类与齐次线丛（支配或反支配）的乘积和Schubert类与余维为1的Schubert类的乘积的“Pieri-Chevalley”公式。所有这些Pieri-Chevalley公式都是根据Littelmann路径模型的组合数学给出的。我们给出了秩为2的Schubert类的乘积的显式计算，这些数据使我们能够做出一个“正性猜想”，推广了Brion和Graham的定理，它们分别处理K（G/B）和H_T^*（G/B）的情况。
Given a reductive group G, Kostant and Kumar defined a nil Hecke algebra that may be viewed as a degenerate version of the double affine nil Hecke algebra introduced by Cherednik. In this paper, we construct an isomorphism of the spherical subalgebra of the nil Hecke algebra with a Whittaker type quantum Hamiltonian reduction of the algebra of differential operators on G. This result has an interpretation in terms of the geometric Satake and the Langlands dual group. Specifically, the isomorphism provides a bridge between very differently looking descriptions of equivariant Borel-Moore homolog…
In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke categor…
On two modular geometric realizations of an affine Hecke algebra
本文通过构造三类范畴的幺半等价，将正特征域上仿射 Hecke 代数与 Harish-Chandra 双模、Steinberg 簇等变凝聚层及仿射旗簇构造层联系起来，并据此证明了 Finkelberg-Mirković 关于主块几何实现的猜想。
We prove that the localization of the monoidal category $\mathcal{C}_w$ is rigid, and the category $\mathcal{C}_{w,v}$ admits a localization via a real commuting family of central objects. Note that the localization of $\mathcal{C}_{w,v}$ categorifies the open Richardson variety.
We explore the connection between the notion of Hopf category and the categorification of the infinite dimensional Heisenberg algebra via graphical calculus proposed by M.Khovanov. We show that the existence of a Hopf structure on a semisimple symmetric monoidal abelian category implies existence of a categorical action in the sense of Khovanov and thus leads to a strong categorification of this algebra.
We define a new category analogous to ${\bf FI}$ for the $0$-Hecke algebra $H_n(0)$ called the $0$-Hecke category, $\mathcal{H}$, indexing sequences of representations of $H_n(0)$ as $n$ varies under suitable compatibility conditions. We establish a new type of representation stability in this setting and prove it is implied by being a finitely generated $\mathcal{H}$-module. We then provide examples of $\mathcal{H}$-modules and discuss further desirable properties these modules possess.
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