Dean Alvis
This manuscript contains tables giving the multiplicities with which irreducible characters of exceptional Weyl groups appear in characters induced from certain reflection subgroups containing maximal parabolic subgroups.
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We examine various consequences of the existence of exceptional representations of an irreducible Weyl group. (These are notes from a talk in the MIT Lie groups seminar.)
On the new bases attached to Weyl groups of exceptional type
该研究针对例外型外尔群的幂么表示格罗滕迪克群，定义新基以精确并修正此前相关结果。
Using truncated convolution of perverse sheaves on a flag variety Lusztig associated a monoidal category to a two sided cell in the Weyl group. We identify this category in the case which was not decided previously.
Coxeter Graphs for Super Weyl Groups of Exceptional Classical Lie Superalgebras
本文研究例外经典李超代数的基本系统，确定对应超韦尔群的Coxeter图，验证了CLS中关于D(2,1,α)、F(4)和G(3)超韦尔群Coxeter图的猜想。
This note discusses the bijection between the exceptional subcategories of representations of quivers and generalized non-crossing partitions of Weyl groups. We give a new proof of the Ingalls-Thomas-Igusa-Schiffler bijection by using the exchange property of the Weyl groups of the Kac-Moody Lie algebras.
We compute all the spin fake degrees for the exceptional Weyl groups, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in its spin coinvariant algebra. The spin fake degrees are all shown to be palindromic polynomials.
Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S(g) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.
Let $\mathfrak{g}$ be the exceptional complex simple Lie algebra of type $G_2$. We provide a concrete cyclicity condition for the tensor product of fundamental representations of the Yangian $Y(\mathfrak{g})$. Using this condition, we show that every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of $Y(\mathfrak{g})$.
On the new bases attached to families of Weyl groups
本文研究有限域上分裂约化群幂幺表示的复化格罗滕迪克群的新基，对例外型采用与早期不同的定义，属数学领域的相关研究。
Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion that ensure that the restriction of invariant polynomials to subspaces is surjective. In another paper we will apply our criterion to problems in Fourier analysis on projective/injective limits, specifically to theorems of Paley--Wiener type.
The purpose of this paper is to answer the question whether it is possible to realize simultaneously the relations $N_{α,β}=-N_{-α,-β}$, $N_{α,β}=N_{β,-α-β}=N_{-α-β,α}$ and $N_{α,β}N_{-α,-β}=-\frac{1}{2}q(p+1)\langleα,H_α\rangle$ by the structure constants of the Lie algebra ${\mathfrak g}_2$. We show that if the structure constants obey the first relation, the three last ones are violated, and vice versa. Contrary to the second case, the first one uses the Cartan matrix elements to derive the structure constants in the form of $\langleβ,H_α\rangle$. The commutation relations corresponding to …
Let $(G,G^Γ)$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^Γ)$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module $π_K$ that is discretely decomposable as a $(\mathfrak{g}^Γ,K^Γ)$-module. In this article, three assumptions will be made. Firstly, $G$ is an exceptional Lie group of Hermitian type, i.e., $G=\mathrm{E}_{6(-14)}$ or $\mathrm{E}_{7(-25)}$. Secondly, $G^Γ$ is noncompact. Thirdly, there exists an element $σ\inΓ$ corresponding to a symmetric pair of anti-holomorphic type such that $π_K$ i…
Elliptic Weyl group elements and unipotent isometries with p=2
该研究针对特征2代数闭域上的典型群，证明了与Weyl群椭圆共轭类关联的幺幂共轭类Φ(C)可通过幺幂类闭包关系刻画，补充了此前奇特征和例外群情形的相关结果。
Characteristic polynomials for classical Lie algebras
该研究利用 Weyl 群轨道与不变多项式理论，计算经典复李代数及 G2 型李代数有限维表示的特征多项式，并证明其可分解为 Weyl 群不变的不可约轨道因子。
Action of the restricted Weyl group on the $L$-invariant vectors of a representation
本文是一项研究简报，综述了针对实李群$G$，确定限制Weyl群最长元$w_0$在$L$-不变向量子空间上作用非平凡的表示集合的相关近期工作。
We study in this paper the restricted roots for a class of spherical homogeneous spaces of semisimple groups which includes simply connected symmetric spaces. For these spaces we give a detailed description (case by case) of the set of roots of the group associated with each restricted root of the space (the nest of the restricted root). As an application, we obtain a refinement of the Weyl dimension formula in the case of spherical representations, expressing the dimension as a product over the set of indivisible positive restricted roots.
Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let ${\mathfrak g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional ${\mathfrak g}$-module over K. For any point $v\inV$, the {\it isotropy subalgebra} of $v$ in $\mathfrak g$ is ${\mathfrak g}_v=\{x\in{\mathfrak g}/x\cdot v=0\}$. A restricted ${\mathfrak g}$-module V is called exceptional if for each $v\in V$ the isotropy subalgebra ${\mathfrak g}_v$ contains a non-central element (that is, ${\mathfrak g}_v\not\subseteq {\mathfrak z(\ma…
Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and let $W$ be a finite-dimensional $K$-space of dimension greater than or equal to $5.$ In this paper, we give the structure of certain Weyl modules for $G=\mbox{Spin}(W)$ in the case where $p\neq 2,$ as well as the dimension of the corresponding irreducible, finite-dimensional, rational $KG$-modules. In addition, we determine the composition factors of the restriction of certain irreducible, finite-dimensional, rational $K\mbox{SL}(W)$-modules to $\mbox{SO}(W).$
We initiate the study of an algebra of symmetries for the 3D Dirac-Dunkl operator associated with the Weyl group of the exceptional root system $G_2$. For this symmetry algebra, we give both an abstract definition and an explicit realisation. We then construct ladder operators, using an intermediate result we prove for the Dirac-Dunkl symmetry algebra associated with arbitrary finite reflection group acting on a three-dimensional space.
Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, sta…
We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence.
For Weyl groups of classical types, we present formulas to calculate the restriction of Springer representations to a maximal parabolic subgroup of the same type. As a result, we give recursive formulas for Euler characteristics of Springer fibers for classical types. We also give tables of those for exceptional types.
We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We a…
For a symmetrizable Kac-Moody algebra the category of admissible representations is an analogue of the category of finite dimensional representations of a semisimple Lie algebra. The monoid associated to this category and the category of restricted duals by a generalized Tannaka-Krein reconstruction contains the Kac-Moody group as open dense unit group and has similar properties as a reductive algebraic monoid. In particular there are Bruhat and Birkhoff decompositions, the Weyl group replaced by the Weyl monoid, [M 1]. We determine the closure relations of the Bruhat and Birkhoff cells, which…
The coinvariant algebra of a Weyl group plays a fundamental role in several areas of mathematics. The fake degrees are the graded multiplicities of the irreducible modules of a Weyl group in its coinvariant algebra, and they were computed by Steinberg, Lusztig and Beynon-Lusztig. In this paper we formulate a notion of spin coinvariant algebra for every Weyl group. Then we compute all the spin fake degrees for each classical Weyl group, which are by definition the graded multiplicities of the simple modules of a spin Weyl group in the spin coinvariant algebra. The spin fake degrees for the exce…
Casselman--Shahidi conjecture on the singularity of intertwining operators: groups of exceptional type
本文修改基于 Heiermann--Opdam 引理的方法，统一证明了例外型群中 Casselman--Shahidi 全纯性猜想的某些情形，核心贡献是提出一种生成余秩一相对最长 Weyl 元既约分解的算法。
We study the exceptional theta correspondence for real groups obtained by restricting the minimal representation of the split exceptional group of the type E_n, to a split dual pair where one member is the exceptional group of the type G_2. We prove that the correspondence gives a bijection between spherical representations if n=6,7, and a slightly weaker statement if n=8.
Given a simple Lie algebra $\mathfrak{g}$, Kostant's weight $q$-multiplicity formula is an alternating sum over the Weyl group whose terms involve the $q$-analog of Kostant's partition function. For $ξ$ (a weight of $\mathfrak{g}$), the $q$-analog of Kostant's partition function is a polynomial-valued function defined by $\wp_q(ξ)=\sum c_i q^i$ where $c_i$ is the number of ways $ξ$ can be written as a sum of $i$ positive roots of $\mathfrak{g}$. In this way, the evaluation of Kostant's weight $q$-multiplicity formula at $q = 1$ recovers the multiplicity of a weight in a highest weight represen…
Highest weight category structures on $rep(B)$ and full exceptional collections on generalized flag varieties over $\mathbb Z$
本文通过构造 $\mathbb P$ 的诺特表示的有界导出范畴的半正交分解，得到了 $\mathbb Z$ 上广义旗簇的全异常集合，该分解依赖于细化 Bruhat 序的 Weyl 群全序。
Polynomial invariants are fundamental objects in analysis on Lie groups and symmetric spaces. Invariant differential operators on symmetric spaces are described by Weyl group invariant polynomial. In this article we give a simple criterion that ensure that the restriction of invariant polynomials to subspaces is surjective. We apply our criterion to problems in Fourier analysis on projective/injective limits, specifically to theorems of Paley--Wiener type.
We construct a full strongly exceptional collection in the triangulated category of graded matrix factorizations of a polynomial associated to a non-degenerate regular system of weights whose smallest exponents are equal to -1. In the associated Grothendieck group, the strongly exceptional collection defines a root basis of a generalized root system of sign (l,0,2) and a Coxeter element of finite order, whose primitive eigenvector is a regular element in the expanded symmetric domain of type IV with respect to the Weyl group.
The hyperbolic cover of an elliptic Weyl group
本文研究了椭圆韦伊系统双曲覆盖的结构及其与扩展柯西特系统的同构性，并建立了加权射影线的厚子范畴与偏序集之间的双射关系。
We construct a new isomorphism between the endomorphism algebra of an induced cuspidal character sheaf and the group algebra of the relative Weyl group involved. We show it differs from Lusztig one by a linear character, and we relate this linear character to some stabilizers. Some consequences on characteristic functions of character sheaves are obtained. In the forthcoming part II, we will apply these methods to obtain a precision on Digne, Lehrer and Michel's theorem on Lusztig restriction of Gel'fand-Graev characters.
Action of $w_0$ on $V^L$ for orthogonal and exceptional groups
本文针对实单李群中限制Weyl群最长元$w_0$在极大分裂环面中心化子不变子空间$V^L$上非平凡作用的表示集合问题，给出正交群与例外群情形下的猜想及支撑实验结果，部分解答了该问题。
The Saito determinant for extended affine Weyl discriminant strata
该研究计算扩展仿射Weyl群判别层的斋藤行列式，将Weyl分母公式因式分解推广到更高余维数层，得到Antoniou-Feigin-Strachan结果的q-模拟细化。
Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case
本文研究扩张 Weyl 群与加权射影线上厚子范畴的组合对应，证明管状类型下 Hurwitz 作用在 Coxeter 元约化反射分解上的传递性，从而建立保序双射。
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