Scott Larson, Anna Romanov
We construct two categorifications of the Lusztig--Vogan module associated to a real reductive algebraic group. The first categorification is given by semisimple complexes in an equivariant derived category, and the second is constructed as a module category over Soergel bimodules. Our categorifications are related by taking equivariant hypercohomology.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Lusztig-Vogan categories of equal rank 2
本文研究对应等秩实群的秩2 Lusztig-Vogan范畴，分类其不可分解对象、刻画生成Soergel双模的作用，还提供适用于任意有限秩该范畴的算法，复现了对应Lusztig-Vogan模的W-图。
In this manuscript, we compute explicitly the Lusztig-Vogan bijection for local systems of some classical, special, nilpotent orbits. Using these results, we prove a conjecture of Achar and Sommers on regular functions of some covers of classical nilpotent orbits.
Let $G$ be a connected complex reductive algebraic group with Lie algebra $\mathfrak{g}$. The Lusztig--Vogan bijection relates two bases for the bounded derived category of $G$-equivariant coherent sheaves on the nilpotent cone $\mathcal{N}$ of $\mathfrak{g}$. One basis is indexed by $Λ^+$, the set of dominant weights of $G$, and the other by $Ω$, the set of pairs $(\mathcal{O}, \mathcal{E})$ consisting of a nilpotent orbit $\mathcal{O} \subset \mathcal{N}$ and an irreducible $G$-equivariant vector bundle $\mathcal{E} \rightarrow \mathcal{O}$. The existence of the Lusztig--Vogan bijection $γ\c…
Iterated Lusztig-Vogan Bijection and Distinguished Weights
本文研究特殊权，证明其在反转与取负复合下的反对称性，并揭示其分布的多项式渐近及与电话号码数的关联，同时给出 $n \leq 4$ 的显式结果。
Let G be a reductive group over an algebraically closed field k of very good characteristic. The Lusztig-Vogan bijection is a bijection between the set of dominant weights for G and the set of irreducible G-equivariant vector bundles on nilpotent orbits, conjectured by Lusztig and Vogan independently, and constructed in full generality by Bezrukavnikov. In characteristic 0, this bijection is related to the theory of 2-sided cells in the affine Weyl group, and plays a key role in the proof of the Humphreys conjecture on support varieties of tilting modules for quantum groups at a root of unity.…
Affine matrix-ball construction (abbreviated AMBC) was developed by Chmutov, Lewis, Pylyavskyy, and Yudovina as an affine generalization of Robinson-Schensted correspondence. We show that AMBC gives a simple way to compute a distinguished (or Duflo) involution in each Kazhdan-Lusztig cell of affine symmetric groups. We then use AMBC to give the first known canonical presentation for the asymptotic Hecke algebras of extended affine symmetric groups. As an application, we show that AMBC gives a conceptual way to compute Lusztig-Vogan bijection. For the latter we build upon prior works of Achar a…
Computing the Cousin-Zuckerman Resolution and the Lusztig-Vogan Bijection
本文通过结合Langlands分类与Knapp-Zuckerman分类，计算了平凡表示的Cousin-Zuckerman解析，并证明了Lusztig-Vogan双射及最低K-类型映射。
Singularities of orbit closures in loop spaces of symmetric varieties
本文研究对称簇环路空间中Iwahori轨道闭包的奇点，推广Lusztig-Vogan工作至仿射情形，证明IC-复形的纯性与奇偶消失，并应用于仿射Lusztig-Vogan模及相对Langlands对偶，回答Lusztig的问题。
Categorification of a recursive formula for Kazhdan-Lusztig polynomials
我们得到了Bott-Samelson双模的自同态代数上分次胞腔模和分次单模的分支规则。这些规则使我们能够对Kazhdan-Lusztig多项式的一个著名的递归公式进行分类。
Vogan classes and cells in the unequal parameter case
Kazhdan和Lusztig证明了Vogan类是等参数情况下细胞的联合。我们推广了这个结果在不等参数的情况下。
Categorification of the Kazhdan-Lusztig basis of the Temperley-Lieb algebra by bimodules
我们利用Soergel双模的类似物实现了Temperley-Lieb代数。关键是么半群结构不是由通常的张量积给出的，而是由稍微复杂一点的运算给出的。
In order to compute Hermitian forms on representations of real reductive groups, in the unequal rank case, it is necessary to compute twisted Kazhdan-Lusztig-Vogan polynomials. These were defined by Lusztig and Vogan (Quasisplit Hecke algebras and Symmetric Spaces, Duke, 2014) and discussed further by Adams and Vogan (Parameters for twisted representations, 2015). These notes contain the details necessary to go from what is in those papers to an explicit algorithm. This algorithm has been implemented in the Atlas of Lie Groups and Representations software.
We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie super algebra $\mathfrak{gl}_{n|m}(\mathbb{C})$ as formulated originally by the first author. We also prove for the first time that any integral block of category O for $\mathfrak{gl}_{n|m}(\mathbb{C})$ (and also all of its parabolic analogs) possesses a graded version which is Koszul. Our approach depends crucially on an application of the uniqueness of tensor product categorifications established recently by the second two authors.
Singularities of orbit closures in loop spaces of symmetric varieties II: twisted cases
本研究将轨道闭包奇点理论推广至扭曲情形，证明IC复形的逐点纯性与奇偶消失性，并应用于扭曲仿射Lusztig--Vogan模及Langlands对偶。
Statement of the Alexandru Conjecture
Vogan猜想（有时也称为Kazhdan-Lusztig猜想）是说，某个算法既适用于BGG模范畴，也适用于Harish-Chandra模范畴。亚历山德鲁猜想试图揭示这两个范畴的共同属性，这使得沃根的算法工作。
Hybrid Kazhdan-Lusztig basis and parabolic induction
本文通过构造BGG范畴主块上的新分层结构，为Hecke代数的混合Kazhdan-Lusztig基提供了范畴化。
Parameters for Twisted Representations
研究实约化群$G$上的厄米特型，在不等秩的情况下，得到一类新的Kazhdan-Lusztig-Vogan多项式.这些与$G$的外自同构$δ$相关联，并且与扩展群$<G,δ>$的表示有关。这些多项式由Lusztig和Vogan在“Quasisplit Hecke Algebras and Symmetric Spaces”，Duke Math.J.163（2014），983- 1034中几何地定义。为了使用他们的结果来计算多项式，需要明确地描述表示到扩展群的扩展。本文分析了这些推广，从而给出了计算多项式的完整算法。该算法正在李群和表示软件图谱中实现。
The degenerate analogue of Ariki's categorification theorem
作为高阶Schur-Weyl对偶和有限型A中的Kazhdan-Lusztig猜想的应用，我们解释了如何在基域C上导出Ariki范畴化定理的退化模拟.我们还讨论了一些补充性的问题，包括Young模，带符号张量，倾斜模和Ringel对偶。
We categorify the Beilinson-Lusztig-MacPherson integral form of quantum sl(2) specialized at a prime root of unity.
On involutions in symmetric groups and a conjecture of Lusztig
设$(W, S)$是一个Coxeter系统，它具有一个保持$S$的固定的$\leq 2$阶自同构$\ast$. Lusztig（和Vogan在某些特殊情况下）表明，空间所跨越的一套“扭曲”对合自然赋予了一个模块结构的赫克代数的$(W, S)$。Lusztig已经证明了这个模同构于Hecke代数的右理想（带有Hecke参数$u^2$），该右理想与由元素$X_{\emptyset}:=\sum_{w^\ast=w}u^{-\ell(w)}T_w$生成的$(W,S)$相关联。本文在$\ast=\text{id}$和$W$是$n$字母上的对称群的情况下证明了这个猜想.
A categorification of the integral form of the Cartan subalgebra for quantum sl(2)
该研究通过有限维射影空间上的等变凝聚层范畴的余极限，提出量子sl(2)的Lusztig积分形式的范畴化，其非等变版本可给出整值多项式环的非半单阿贝尔范畴化。
A quantum Frobenius map a la Lusztig for $\mathfrak{sl}_2$ is categorified at a prime root of unity.
Upper semicontinuity of KLV polynomials for certain blocks of Harish-Chandra modules
我们证明了Kazhdan-Lusztig-Vogan多项式的系数附加到某些块的Harish-Chandra模满足单调性性质相对于封闭秩序的K-轨道的标志品种。
Two-parameter quantum algebras, canonical bases and categorifications
与单参数情况下的正则基理论并行地发展了双参数量子代数的正则基理论。在Lusztig工作的基础上，利用混合反常层和Deligne的权理论，给出了双参数量子代数的负部分的尺规作图法.给出了双参数量子代数的负部分的一个分类。一个双参数量子代数被证明是一个双上循环变形，只依赖于第二个参数，其单参数类似。
We study a presentation of Khovanov - Lauda - Rouquier's candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of $2$-cells in this $2$-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this $2$-category is a categorification of Lusztig's idempotent and integral quantum group $\…
Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone
在数学.AG/0005152中，在简单复代数群的零锥上的等变相干层的导出范畴上引入了某种$t$ -结构（所谓的反常$t$ -结构对应于中间反常）。在本说明中，我们表明，相同的$t$ -结构可以从一个自然的准例外集生成这个导出范畴。因此，我们得到了一个双射之间的集的主导权和对组成的幂零轨道，和一个不可约表示的中心化这个元素，由Lusztig和Vogan（并获得了其他手段在数学。RT/0010089）。
A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors
我们用基本（二维）sl（2）-模的n次张量幂幂来确定sl（n）的某个最高权范畴的奇异块直和的Grothendieck群. U（sl（2））的作用由投射函子给出，Temperley-Lieb代数的交换作用由Zuckerman函子给出.不可分解的投射函子对应于U（sl（2））中的Lusztig标准基。在对偶实现中，基本表示的n阶张量幂被确定为具有最高权类别的抛物块的直和。跨墙平移函子充当Temperley-Lieb代数的生成元，而Zuckerman函子充当U（sl（2））的生成元。
Local Systems on Nilpotent Orbits and Weighted Dynkin Diagrams
我们研究了局部系统的Lusztig-Vogan双射。我们计算的双射显式在A型的本地系统，然后表明，不同的本地系统在同一轨道上获得的主导权重相关的方式作出精确的文件。我们还给出了一个猜想（对所有群都成立），详细说明了对偶李代数中幂零轨道的加权Dynkin图在双射下如何产生。
Monoidal categorification on open Richardson varieties
本文证明了在反转冻结簇变量后，quiver Hecke代数模子范畴$\mathscr{C}_{w,v}$构成任意Dynkin型开Richardson簇坐标环的幺半范畴化。
Cell 2-representations of finitary 2-categories
研究了有限维代数上模范畴上具有对合和对合的有限2-范畴的函子2-表示。特别是，我们定义，构造和详细描述（右）细胞2-表示的启发Kazhdan-Lusztig细胞模块Hecke代数。在一些自然的假设下，我们证明了细胞2表示是强简单的，不依赖于一个正确的细胞内的双边细胞的选择。本文改进和推广了\cite{MS2}中关于$A$型Hecke代数的Kazhdan-Lusztig胞腔模范畴化的唯一性结果.
Graded Hecke algebras, constructible sheaves and the p-adic Kazhdan--Lusztig conjecture
本文利用幂零锥上的等变可构造层研究分次 Hecke 代数，证明重数对应的 Kazhdan–Lusztig 猜想，并在多类群与表示上验证 Vogan 的 p-adic 版本。
Let $(W,S)$ be a Coxeter system and $\ast$ an automorphism of $W$ with order $\leq 2$ and $S^{\ast}=S$. Lusztig and Vogan ([11], [14]) have introduced a $u$-deformed version $M_u$ of Kottwitz's involution module over the Iwahori-Hecke algebra $\mathscr{H}_{u}(W)$ with Hecke parameter $u^2$, where $u$ is an indeterminate. Lusztig has proved that $M_u$ is isomorphic to the left $\mathscr{H}_{u}(W)$-submodule of ${\hat{\mathscr{H}}}_u$ generated by $X_{\emptyset}:=\sum_{w^*=w\in W}{u^{-\ell(w)}T_w}$, where ${\hat{\mathscr{H}}}_u$ is the vector space consisting of all formal (possibly infinite) su…
Special unipotent representations and the coadjoint orbit method
本文针对线性实约化群或元辛群，基于相关循环对特殊幂幺表示进行几何分类，通过变形量子化给出其哈里什 - 钱德拉模的统一构造，证明相关模有不可约循环且酉可化，突出卢茨蒂格猜想应用。
Parabolic Category O in Complex Rank via Fock Space Tensor Product Categorifications
本文研究一般线性李代数中复秩类比的抛物范畴O，通过Deligne的插值范畴定义。利用sl_Z范畴化理论证明猜想，得到恒定沿面的结构，并通过稳定性极限得到简单对象的特征公式，回答了Etingof的问题。
Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category ${\mathcal{O}}$. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to be exact. In the Koszul dual picture, exactness of BGG complexes is expressed as a certain condition on a generalized Verma flag of an indecomposable projective object in the corresponding block of parabolic category ${\mathcal{O}}$. In the second part of the paper, we construct BGG complexes in a more general setting of balanced quasi-hereditary algebras a…
In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished $p$-characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category $\mathcal{O}$. For this, w…
We study a diagrammatic categorification (the "anti-spherical category") of the anti-spherical module for any Coxeter group. We deduce that Deodhar's (sign) parabolic Kazhdan-Lusztig polynomials have non-negative coefficients, and that a monotonicity conjecture of Brenti's holds. The main technical observation is a localisation procedure for the anti-spherical category, from which we construct a "light leaves" basis of morphisms. Our techniques may be used to calculate many new elements of the $p$-canonical basis in the anti-spherical module.
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。
Construction of PBW and canonical bases in the folding extension algebras for symmetrizable types
研究源于Lusztig构造的扩张代数，扩展Varagnolo和Vasserot方法构造斜群代数，证明相关过渡矩阵性质，为对称型情形下PBW基与典范基及其关系提供几何范畴化。
Some homological properties of the category $\mathcal{O}$, II
我们表明，在充分的一般性，Lusztig的$\mathbf{a}$ -功能描述的投射维数的不可分解倾斜模块和不可分解的内射模在经常块的BGG类别$\mathcal{O}$，证明了一个猜想从第一篇论文。同时证明了投射函子下的单模的象在导出范畴中可用倾斜模的线性复形表示。这些复形又可以被解释为范畴$\mathcal{O}$的Koszul对偶中的投射函子下的单模的像。最后，我们描述了占主导地位的投射模，也投射-内射模的$\mathcal{O}$的一些子范畴，并显示如何可以使用分类分解的正则表示的Weyl群成一个直接和的细胞模块，扩展的结果已知的对称群（类型$A$）。
Deligne's category $\underline{\rm Rep}(S_t)$ is a tensor category depending on a parameter $t$ "interpolating" the categories of representations of the symmetric groups $S_n$. We construct a family of categories $\mathcal{C}_λ$ (depending on a vector of variables $λ= (λ_1, λ_2, \ldots, λ_l)$, that may be specialised to values in the ground ring) which are module categories over $\underline{\rm Rep}(S_t)$. The categories $\mathcal{C}_λ$ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to $S_n$-mod which are …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。