Jeffrey Adams, Peter E. Trapa
Let G be a nonlinear double cover of the real points of a connected reductive complex algebraic group with simply laced root system. We establish a uniform character multiplicity duality theory for the category of Harish-Chandra modules for G.
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
Non-simply-laced Clusters of Finite Type via Frobenius Morphism
通过证明折叠几乎正根与折叠簇范畴的相容性，证明了在非单laced有限情形下，种子与倾斜种子之间存在一一对应关系.
The Morita equivalences of classical Brauer algebras and classical Birman-Murakami-Wenzl algebras have been well studied. Here we study the Morita equivalence problems on these two kinds of algebras of simply-laced type, especially for them with the generic parameters. We show that Brauer algebras and Birman-Murakami-Wenzl algebras of simply-laced type are Morita equivalent to the direct sums of some group algebras of Coxeter groups and some Hecke algebras of some Coxeter groups, respectively.
Small principal series and exceptional duality for two simply laced groups
作者以前开发的秩的概念被用来描述两个对应分类小酉表示分裂实形式的$E_6$和$E_7$。详细研究了小主列的情形。
Lifting of Characters for Nonlinear Simply Laced Groups
线性群的Langlands程序的一个方面是字符提升，其将组$G$上的虚拟表示与$G$的内窥镜组上的虚拟表示相关联。本文的目标是将这一理论推广到实群的非线性二重覆盖的简单花边的情况下。设$\tG$是实约化群$G$的二重覆盖.主要结果是，存在一个表示为$\Lift_G^{\tG}$的操作，将$G$的稳定虚拟字符取为0或将$\tG$的虚拟真实字符取为0，并且如果$π$是标准模块的稳定和，则可以显式地计算$\Lift_G^{\tG}(Θ_π)$。
We show the uniqueness and existence of the Euler form for a simply-laced generalized root system. This enables us to show that the Coxeter element for a simply-laced generalized root system is admissible in the sense of R.~W.~Carter. As an application, the isomorphism classes of simply-laced generalized root systems with positive definite Cartan forms are classified by Cartar's admissible diagrams associated to their Coxeter elements.
We study the decomposition problem for tensor powers of $B_2$-fundamental modules. To solve this problem singular weight technique and injection fan algorithms are applied. Properties of multiplicity coefficients are formulated in terms of multiplicity functions. These functions are constructed showing explicitly the dependence of multiplicity coefficients on the highest weight coordinates and the tensor power parameter. It is thus possible to study general properties of multiplicity coefficients for powers of the fundamental $B_2$- modules.
Let $\g$ be any simple Lie algebra over $\mathbb{C}$. Recall that there exists an embedding of $\mathfrak{sl}_2$ into $\g$, called a principal TDS, passing through a principal nilpotent element of $\g$ and uniquely determined up to conjugation. Moreover, $\wedge (\g^*)^\g$ is freely generated (in the super-graded sense) by primitive elements $ω_1, \dots, ω_\ell$, where $\ell$ is the rank of $\g$. N. Hitchin conjectured that for any primitive element $ω\in \wedge^d (\g^*)^\g$, there exists an irreducible $\mathfrak{sl}_2$-submodule $V_ω\subset \g$ of dimension $d$ such that $ω$ is non-zero on t…
Hecke-Hopf algebras were defined by A. Berenstein and D. Kazhdan. We give an explicit presentation of an Hecke-Hopf algebra when the parameter $m_{ij},$ associated to any two distinct vertices $i$ and $j$ in the presentation of a Coxeter group, equals $4,$ $5$ or $6$. As an application, we give a proof of a conjecture of Berenstein and Kazhdan when the Coxeter group is crystallographic and non-simply-laced. As another application, we show that another conjecture of Berenstein and Kazhdan holds when $m_{ij},$ associated to any two distinct vertices $i$ and $j,$ equals $4$ and that the conjectur…
The Kazhdan-Lusztig category of W-algebras of simply-laced Lie algebras at irrational levels
研究W-代数在简单李代数上非有理水平的Kazhdan-Lusztig分类，通过量子哈密顿约简与affine顶点代数建立braided张量等价。
We establish, for all simply laced types, a q,t-character formula, first conjectured by Nakajima. It relates, on one hand, the structure of the $\ell$-weight spaces of standard modules regarded as modules over the Heisenberg subalgebra of some quantum affine algebra and, on the other hand, the t-dependence of their q,t-characters, as originally defined in terms of the Poincaré polynomials of certain Lagrangian subvarieties in quiver varieties of the corresponding simply laced type. Our proof is essentially geometric and generalizes to arbitrary simply laced types earlier representation theoret…
Cluster algebras and preprojective algebras : the non simply-laced case
本文推广了Geiklit，Leclerc和Schröer关于单李群的极大幂幺子群的坐标环的簇结构的结果.这样，在非简单花边的情况下的集群结构可以被看作是在简单花边的情况下的集群结构的投影。这使我们能够证明，集群单项式是线性无关的非简单花边的情况下。
We show that every Weyl module for a current algebra has a filtration whose successive quotients are isomorphic to Demazure modules, and that the path model for a tensor product of level zero fundamental representations is isomorphic to a disjoint union of Demazure crystals. Moreover, we show that the Demazure modules appearing in these two objects coincide exactly. Though these results have been previously known in the simply laced case, they are new in the non-simply laced case.
A Poset Connected to Artin Monoids of Simply Laced Type
设W是一个类型为单缀Dynkin图的外尔群.在相互交换反射集的几个W-轨道上，描述了一个偏序集，它在相应的Artin群A的线性表示中起作用。偏序集推广了由高度给出的W的正根上的许多通常阶的性质。本文利用偏序集定义了A的正幺半群的一个线性表示。
This paper investigates a non simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-Calabi-Yau categories over arbitrary fields. It results that 2-Calabi-Yau or stably 2-Calabi-Yau categories having a cluster tilting subcategory with neither loops nor 2-cycles do have the generalized version of cluster structure. This is in particular the case of cluster categories over non-algebraically closed fields.
Let $\mathfrak{g}$ be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline{D}$ on the coordinate ring $\mathbb{C}[N]$ related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when $\mathfrak{g}$ is of type $A_n$ or $…
We provide an explicit construction for a class of commutative, non-associative algebras for each of the simple Chevalley groups of simply laced type. Moreover, we equip these algebras with an associating bilinear form, which turns them into Frobenius algebras. This class includes a 3876-dimensional algebra on which the Chevalley group of type E8 acts by automorphisms. We also prove that these algebras admit the structure of (axial) decomposition algebras.
Let $Λ$ be a simply laced root lattice and $w$ an elliptic automorphism of $Λ$ of order $d$. This paper gives a construction that begins with a central extension of the group of coinvariants $Λ_w$ and produces a semisimple Lie algebra of Dykin type $Λ$ with an automorphism of order $d$ lifting $w$. The input for this construction naturally arises when considering certain families of algebraic curves. The construction generalizes one used by Thorne to study plane quartics and one used by the author and Thorne to study a family of genus-2 curves.
2-roots for simply laced Weyl groups
本文提出并研究了Kac-Moody代数正交根对称化张量积构成的“2-根”，证明了单连通Y形Dynkin图Weyl群的反射表示对称平方子模具有由2-根组成的典范基，并揭示了其簇代数列符号相干矩阵表示及完全可约性等结构性质。
A geometric construction of crystal graphs using quiver varieties: Extension to the non-simply laced case
我们考虑了Lusztig和Nakajima的Kac簇到所有可对称化的Kac-Moody李代数的推广。为了处理非简单的花边的情况下，人们认为可容许的自同构的一个和不可约的组件的固定的品种，这一自同构。我们定义了一个晶体结构上的这些不可约的组件，并表明，所获得的晶体是同构的那些相关的晶体基地的下半部分的泛包络代数和不可约的最高权表示的非简单的花边卡茨-穆迪李代数。作为应用，我们实现了晶体的旋量表示的so_{2n+1}上的自共轭杨图，适合内的n × n盒。
We establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the (q,t)-characters of certain irreducible representations, among which fundamental representations, are obtained as quantum cluster variables. This approach gives a new algorithm to compute these (q,t)-characters. As an application, we prove that the quantum Grothendieck ring of a larger category of representations of the Borel subalgebra of the quantum affine algebra, defi…
We study the (complex) Hecke algebra $\mathcal{H}_S(\mathbf{q})$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q} \in \left( \mathbb{C} \setminus\{\text{roots of unity}\} \right)^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra $\mathcal{H}_S(\mathbf{q})$ and the algebra $\mathcal{H}_R(\mathbf{q}|_R)$ with $R\subseteq S$. …
A finite presentation for extended affine Weyl groups (simply laced types)
广义仿射外尔群是广义仿射根系的外尔群。扩张仿射外尔群的有限表示仅对零值$\leq 2$是已知的，其中对于零值2只有一个已知的这样的表示。给出了单缀扩张仿射外尔群类的有限表示.如果秩为$>1$，则我们的表示是零值自由的，并且对于秩1，它是针对零值$\leq 3$给出的。所有类型的生成元和关系是统一给出的，对于给定的零点，它们可以从相应的有限Cartan矩阵和半格中读取，这些矩阵和半格涉及根系的结构。
Brauer Algebras of Simply Laced Type
图代数介绍了布劳尔，描述了中心化代数张量积的自然表示的正交群有一个介绍的发电机和关系，只依赖于图的类型安在n个节点。这里我们描述一个依赖于任意M型图的代数。研究了An、Dn、E6、E7、E8型时的结构。我们确定表示并求出维数。该代数是一般半单的，并且包含Coxeter型M的群代数作为子代数。它是这些类型的Birman-Murakami-Wenzl代数的环同态。这一事实将用于以后的工作确定结构的Birman-Murakami-Wenzl代数的简单laced球面型。
Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras
本文研究A型分裂实Kac-Moody代数极大紧子代数的有限维旋量表示，分析其可约性、半单性等性质，借助旋量扩张Weyl群得到表示矩阵由实根的部分参数化结果。
Monomials of q and q,t-characters for non simply-laced quantum affinizations
Nakajima介绍了q，t-特征标的态射用于有限维表示的简单花边量子仿射代数：它是q-特征标（无限变量单项式之和）的Frenkel-Reshetikhin态射的t-变形。在数学QA/0212257中，我们推广了非单缀量子仿射代数的q，t-特征标的构造。本文首先证明了数学QA/0212257中的一个猜想：q和q，t-特征标的单项式在非单纯形情形下相同（单纯形情形由Nakajima处理），且系数非负。特别地，这些q，t-特征标可以被认为是q-特征标的t-变形。在证明中，我们证明了对于A，B，C型量子仿射代数和A^{（1）}型量子环面代数，基本表示的l-权空间是1维的。最后，我们展示和使用的推广结果的弗伦克尔-Reshetikhin，-Mukhin和Nakajima：对于一般的量子仿射，我们证明了l-最高权最高权简单模块的l-重量低于最高的l-重量在单项式的意义。
We study the ODE/IM correspondence for ODE associated to $\hat{\mathfrak g}$-valued connections, for a simply-laced Lie algebra $\mathfrak g$. We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called $Ψ$-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.
An approach to non simply laced cluster algebras
设$Δ$是一个有向值图，它具有一组满足一定稳定性条件的容许自同构.证明了与值商图$Δ/G$相关联的（无系数）簇代数$\mathcal A(Δ/G)$是通过$G$的作用与$Δ$相关联的簇代数的商$π(\mathcal A(Δ))$的子代数.当$Δ$是Dynkin图时，证明了$\mathcal A(Δ/G)$与$π(\mathcal A(Δ))$重合.作为应用的一个例子，我们证明了仿射值图是突变有限的，给出了Seven的一个结果的另一种证明。
We construct the positive principal series representations for $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ where $\mathfrak{g}$ is of simply-laced type, parametrized by $\mathbb{R}_{\geq 0}^r$ where $r$ is the rank of $\mathfrak{g}$. We describe explicitly the actions of the generators in the positive representations as positive essentially self-adjoint operators on a Hilbert space, and prove the transcendental relations between the generators of the modular double. We define the modified quantum group $\mathbf{U}_{\mathfrak{q}\tilde{\mathfrak{q}}}(\mathfrak{g}_\mathbb{R})$ of the modular double …
We put forward the exact version of two-parameter generating functions with $τ$-invariance, which allows us to give a unified and inherent definition for the Drinfeld realization of two-parameter quantum affine algebras for all the untwisted types. As verification, we first construct their level-one vertex representations of $U_{r,s}(\widehat{\mathfrak{g}})$ for simply-laced types, which in turn well-detect the effectiveness of our definitions both for $(r,s)$-generating functions and $(r,s)$-Drinfeld realization in the framework of establishing the two-parameter vertex representation theory.
In this paper, we seek to prove the equality of the $q$-graded fermionic sums conjectured by Hatayama et al. in its full generality, by extending the results of Di Francesco and Kedem to the non-simply laced case. To this end, we will derive explicit expressions for the quantum $Q$-system relations, which are quantum cluster mutations that correspond to the classical $Q$-system relations, and write the identity of the $q$-graded fermionic sums as a constant term identity. As an application, we will show that these quantum $Q$-system relations are consistent with the short exact sequence of the…
BMW algebras of simply laced type
最近发现的A_n型Artin群的表示--辫群，可以通过BMW代数构造.我们引入了D_n和E_n型的相似代数，并由此得到了相应类型的Artin群的新的忠实表示.我们建立这些代数的有限维。此外，它们有理想I_1和I_2，其中I_2包含在I_1中，使得I_1的商是Hecke代数，I_1/I_2是相应的Artin群的模，推广了Lawrence-Krammer表示。最后给出了BMW代数的结构、维数和抛物子代数，以及对A_n以外的单缀球型Brauer代数的变形的推广。
For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion basis which is strong, i.e. gives the set of dimension vectors of the finitely generated indecomposable modules for the cluster-tilted algebra. This shows in particular that every companion basis of a cluster-tilted algebra of simply-laced Dynkin type is strong. Thus we give a proof of Parsons's conjecture.
We construct the positive principal series representations for $U_q(g_R)$ where $g$ is of type $B_n$, $C_n$, $F_4$ or $G_2$, parametrized by $R^r$ where $r$ is the rank of $g$. We show that under the representations, the generators of the Langlands dual group $U_{\tilde{q}}({}^Lg_R)$ are related to the generators of $U_q(g_R)$ by the transcendental relations. We define the modified quantum group of the modular double and show that the representations of both parts of the modular double commute with each other, and there is an embedding into the $q$-tori polynomials.
Let $U_q'(\mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $\mathscr{C}_{\mathfrak{g}}$ be the category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules. We write $\mathscr{C}_{\mathfrak{g}}^0$ for the monoidal subcategory of $\mathscr{C}_{\mathfrak{g}}$ introduced by Hernandez-Leclerc. In this paper, we associate a simply-laced finite type root system to each quantum affine algebra $U_q'(\mathfrak{g})$ in a natural way, and show that the block decompositions of $\mathscr{C}_{\mathfrak{g}}$ and $\mathscr{C}_{\mathfrak{g}}^0$ are parameterized by the lat…
Simply-laced Coxeter groups and groups generated by symplectic transvections
设W是任意的单缀型Coxeter群（可能无限但秩有限），u，v是W中的任意两个元素，i是Coxeter群W \times W中（u，v）对的约化字（长度为m）.我们将GL_m（Z）中的一个子群Gamma_i与i联系起来，该子群是由辛横截生成的.我们证明了对应于同一对（u，v）的不同约化字的子群在GL_m（Z）内是共轭的。我们还推广了前三位作者的计数结果（见AG/9802093），证明了在对u和v的某些假设下，F_2 ^m中的Gamma_i（F_2）-轨道数等于3 \times 2 ^s，其中s是u或v的约化分解中出现的简单反射数，F_2是二元场。
Miniscule representations, Gauss sum and modular invariance
在解释了朗兰兹对偶和极小表示的概念后，我们定义了一个类似的高斯和的任何紧凑的，单李群与简单的花边李代数。然后，我们展示了一个互惠性质时，李群交换其朗兰兹对偶。我们还探讨了与θ函数和模变换的关系。在非单laced的情况下，我们构造了Hecke群的一个酉表示，其中包含了有趣的新的相位因子。
Using vertex operators, we build representations of the Yangian of a simply laced Kac-Moody algebra and of its double. As a corollary, we prove the PBW property for simply laced affine Yangians.
We present here two detailed examples of additive categorifications of the cluster algebra structure of a coordinate ring of a maximal unipotent subgroup of a simple Lie group. The first one is of simply-laced type ($A_3$) and relies on an article by Geiß, Leclerc and Schröer. The second is of non simply-laced type ($C_2$) and relies on an article by the author of this note. This is aimed to be accessible, specially for people who are not familiar with this subject.
Lattice structure in cluster algebra of finite type and non-simply-laced Ingalls-Thomas bijection
该研究通过显式刻画有限型丛代数的交换箭群，证明其丛集合的格结构与GLS路代数的挠格、$c$-寒武纪格反同构，建立了非单箭袋型的Ingalls-Thomas双射对应。
In this paper we analyze Fourier coefficients of automorphic forms on a finite cover $G$ of an adelic split simply-laced group. Let $π$ be a minimal or next-to-minimal automorphic representation of $G$. We prove that any $η\in π$ is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro--Shalika formula for cusp forms on $GL_n$. We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient in terms of these Whitt…
It is known that there exists an order isomorphism between the Weyl group orbit through a minuscule weight of a simply-laced finite-dimensional simple Lie algebra and the set of all order filters in a self-dual connected d-complete poset. In this paper, we try to extend this fact to the case of multiply-laced finite-dimensional simple Lie algebras by using the "folding" technique with respect to a Dynkin diagram automorphism.
The minimal degeneration singularities in the affine Grassmannians
简单单缀代数群的仿射Grassmannian中的极小退化奇点被确定为A型Kleinian奇点或幂零锥中极小轨道的闭包。利用交上同调和等变Chow群方法研究了非单系型的奇点。
We present a way to associate an algebra $B_G (Υ) $ with every pseudo reflection group $G$. When $G$ is a Coxeter group of simply-laced type we show $B_G (Υ)$ is isomorphic to the generalized Brauer algebra of simply-laced type introduced by Cohen,Gijsbers and Wales[10]. We prove $B_G (Υ)$ has a cellular structure and be semisimple for generic parameters when $G$ is a rank 2 Coxeter group. In the process of construction we introduce a Cherednik type connection for BMW algebras and a generalization of Lawrence-Krammer representation to complex braid groups associated with all pseudo reflection …
Homogeneous Representations of Khovanov-Lauda Algebras
本文构造了单缀Khovanov-Lauda代数的不可约分次表示。基本的组合学的斜形状和标准tableaux对应于任意简单的花边类型已经开发了彼得森，普罗克特和Stembridge。特别地，彼得森-普洛克托钩子公式给出了对应于直形的齐次不可约模的维数。
Characteristic cycles of IH sheaves of simply laced minuscule Schubert varieties are irreducible
本文证明了 E6、E7 型复余减旗流形中 Schubert 簇的交截同调复形的特征循环不可约，从而将 Boe 和 Fu 的结果推广到例外 Lie 型，并附带得到 Schubert 胞腔的 Mather 类与 CSM 类的强正性。
We describe a combinatorial realization of the crystals $B(\infty)$ and $B(λ)$ using rigged configurations in all symmetrizable Kac-Moody types up to certain conditions. This includes all simply-laced types and all non-simply-laced finite and affine types.
Let G be a semisimple simply connected group of simply laced type over C. We show that the flag manifold of G has a version defined over the "tropical" semifield Z on which the monoid G(Z) associated to G and the semifield Z acts.
Smallness problem for quantum affine algebras and quiver varieties
射影态射的几何小性质（Borho-MacPherson）暗示了用交同调来描述它们的奇点。在本文中，我们解决了Nakajima（数学QA/0105173）提出的小性问题的某些决议的numerous品种（类似的Springer决议）：对于Kirillov-Reshetikhin模的simply-laced量子仿射代数，我们明确地刻画了Drinfeld多项式对应的小决议。我们使用一个消除定理的单项式的Frenkel-Reshetikhin q-字符，我们建立了不一定简单的花边量子仿射代数。我们还改进了（数学QA/0501202）的结果，并将主要结果推广到一般的简单花边量子仿射，特别是量子环面代数（双仿射量子代数）。
Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras
本文引入简单连通量子仿射代数模范畴的子范畴，证明其量子Grothendieck环具有量子簇代数结构，并据此分类了实单模，提出了相关猜想。
A cluster algebra is a commutative algebra whose structure is decided by a skew-symmetrizable matrix or a quiver. When a skew-symmetrizable matrix is invariant under an action of a finite group and this action is admissible, the folded cluster algebra is obtained from the original one. Any cluster algebra of non-simply-laced affine type can be obtained by folding a cluster algebra of simply-laced affine type with a specific $G$-action. In this paper, we study the combinatorial properties of quivers in the cluster algebra of affine type. We prove that for any quiver of simply-laced affine type,…
Affine Kazhdan-Lusztig polynomials on the subregular cell in non simply-laced Lie algebras: with an application to character formulae (with an appendix by Roman Bezrukavnikov, Vasily Krylov, and Kenta Suzuki)
本文将仿射Kazhdan-Lusztig多项式技术推广到非单连通情形，计算次正则幂零轨道上的显式特殊值，从而得到仿射李代数不可约表示及顶点代数的特征标公式，并应用于味对称性为$G_2$和$B_3$的Argyres-Douglas理论的Schur指数计算。
On Lusztig's canonical bases of simple Lie algebras
本文提出通过特定的2-余积构造简单李代数的标准基，并探讨了非单连通根系下的结构常数描述问题。
Quantum affine Cartan matrices, Poincare series of binary polyhedral groups, and reflection representations
本文首先利用McKay对应和量子仿射Cartan矩阵，用代数的方法快速地回顾了SU（2）的有限子群的一些不变理论结果。顺便说一下，我们发现，通常与这样一个群相关联的一些参数（a，b，h;p，q，r），因此与一个简单的Coxeter-Dynkin图相关联，对于非简单的图也有一个有意义的定义，作为一个副产品，我们将Saito的Cartan矩阵行列式公式扩展到所有情况。回到不变量理论，我们表明，对于每个不可约表示i的二元四面体，八面体，或二十面体组可以找到一个同态到一个有限的复杂的反射群，其定义的反射表示限制到i。
Classification of Zamolodchikov periodic cluster algebras
本文通过折叠和转置操作，将齐莫洛季基茨周期簇代数与成对的交换有限型卡坦矩阵建立双射关系，并分类了所有此类对。
Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case
本文构造与 ADE 型根系相关的无限秩簇代数，证明其完备化同构于移位量子仿射代数范畴 O_Z 的 Grothendieck 环，并用双 Bruhat 胞腔解释 QQ-系统关系。
We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb{Z}[q^{\pm 1}]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply-laced type and equivariant scalars $e^λ$, where $λ$ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines …
We study the principal series representations of central extensions of a split reductive algebraic group by a cyclic group of order $n$. We compute the Plancherel measure of the representation using Eisenstein series and a comparison method. In addition, we construct genuine central characters of the metaplectic torus in the simply-laced case.
Let g be a semi-simple Lie algebra. In this paper we study the spaces of based quasi-maps from the projective line P^1 to the flag variety of g (it is well-known that their singularities are supposed to model the singularities of the so called semi-infinite Schubert varieties which are hard to define directly). In the first part of the paper we show that the above spaces are normal and in the case when g is simply laced they are also Gorenstein and have rational singularities. In the second part of the paper we compute the character of the ring of functions on the above spaces; in view of the …
We investigate Fourier coefficients of automorphic forms on split simply-laced Lie groups G. We show that for automorphic representations of small Gelfand-Kirillov dimension the Fourier coefficients are completely determined by certain degenerate Whittaker vectors on G. Although we expect our results to hold for arbitrary simply-laced groups, we give complete proofs only for G=SL(3) and G=SL(4). This is based on a method of Ginzburg that associates Fourier coefficients of automorphic forms with nilpotent orbits of G. Our results complement and extend recent results of Miller and Sahi. We also …
Let $C$ be a simply laced generalized Cartan matrix. Given an element $b$ of the generalized braid semigroup related to $C$, we construct a collection of mutation-equivalent quivers with potentials. A quiver with potential in such a collection corresponds to an expression of $b$ in terms of the standard generators. For two expressions that differ by a braid relation, the corresponding quivers with potentials are related by a mutation. The main application of this result is a construction of a family of $CY_3$ $A_\infty$-categories associated to elements of the braid semigroup related to $C$. I…
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。