小特征幂幺元，III

全正性中幂幺元的正则性

Regularity of Unipotent Elements in Total Positivity

本文研究分裂连通约化群全非负幺半群中幂幺元的正则性，证明其在某些 Levi 子群中正则，证实 Lusztig 猜想。

Unipotent elements in small characteristic, IV

We consider the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. We propose a definition of a partition of this variety into smooth locally closed smooth subvarieties indexed by the unipotent classes in the corresponding group over complex numbers. We obtain explicit results in type A,C and D.

$E_8$型和${^2\negmedspace E}_6$型群的幺正特征在幺正元处的值

The values of unipotent characters at unipotent elements for groups of type $E_8$ and ${^2\!E}_6$

本研究通过明确对应单位根，完成了$E_8$型李群对所有素数幂$q$、${^2\negmedspace E}_6$型李群对2的幂次$q$的幺正特征在幺正元处的值计算，了结了例外李型单群该类值的计算工作。

Springer's work on unipotent classes and Weyl group representations

In this paper we discuss some of Springer's work on unipotent elements in a reductive groups and on representations of Weyl groups. Among the topics considered are Springer's bijection from the unipotent variety to the nilpotent variety, Springer's example of nonrationality of certain representations of the Hecke algebras associated to a Weyl group and the Springer representation of the Weyl group associated to a unipotent element.

特征p=2下的椭圆Weyl群元素与幺正等距映射

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。

有限约化群的广义Green函数与幂幺类，IV

Generalized Green functions and unipotent classes for finite reductive groups, IV

本文定义了连通约化群幂幺类的分裂元，将广义Green函数的Lusztig算法计算归结为分裂元存在性，并证明典型群及除E7型一个类外的例外群中分裂元均存在。

幂幺轨道的好位置辫表示与横截面

Good position braid representatives and transversal slices of unipotent orbits

本文利用Weyl群关联辫幺半群中的好元素，为代数闭域上连通约化群的所有幂幺轨道构造横截面，并以此重新表述仿射Springer纤维的维数公式。

On the characters of unipotent representations of a semisimple p-adic group

Let G be a semisimple almost simple algebraic group defined and split over a nonarchimedean local field K and let V be a unipotent representation of G(K) (for example, an Iwahori-spherical representation). We calculate the character of V at compact very regular elements of G(K) at compact very regular elements of G(K).

Unipotent blocks and weighted affine Weyl groups

We show that various properties of unipotent elements in a reductive group over the complex numbers can be recovered purely in terms of the affine Weyl group of the dual group.

幂单表示：将 q 变为 -q，II

Unipotent representations: changing q to -q, II

本文研究有限域上 Chevalley 群中，将 q 变为 -q 对半单元素处幂单表示特征值多项式的影响。

On the Fixed Points of a Regular Unipotent Element

The fixed point variety of a regular unipotent element on a wonderful completion is investigated. For the wonderful completion of the quotient by a symmetric Levi subgroup, it is shown that the fixed point variety is $SL_2$-regular.

Products of Unipotent Elements in Certain Algebras

Let $F$ be a field with at least three elements and $G$ a locally finite group. This paper aims to show that if either $F$ is algebraically closed or the characteristic of $F$ is positive, then an element in the group algebra $FG$ is a product of unipotent elements if, and only if, it? lies in the first derived subgroup of the unit group of $FG$. In addition, it? is a product of at most three unipotent elements. Moreover, we explore some crucial properties satisfied by certain algebras like the connection between unipotent elements of index 2 and commutators as well as we investigate the unipo…

Evaluating Characteristic Functions of Character Sheaves at Unipotent Elements

Assume $\mathbf{G}$ is a connected reductive algebraic group defined over an algebraic closure $\mathbb{K} = \overline{\mathbb{F}}_p$ of the finite field of prime order $p>0$. Furthermore, assume that $F : \mathbf{G} \to \mathbf{G}$ is a Frobenius endomorphism of $\mathbf{G}$. In this article we give a formula for the value of any $F$-stable character sheaf of $\mathbf{G}$ at a unipotent element. This formula is expressed in terms of class functions of $\mathbf{G}^F$ which are supported on a single unipotent class of $\mathbf{G}$. In general these functions are not determined, however we give …

Overgroups of regular unipotent elements in simple algebraic groups

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.

分支通过正则单位元的唯一性

Uniqueness of Branching through regular unipotent elements

本文研究了在特定子群下有限维不可约表示的唯一性问题，通过Dynkin分类证明了表示在子群上的限制在保子群的外自同构下唯一。

有限一般线性群和酉群的实值特征标在正则幂幺元上的推广

Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

当n是奇数时，考虑秩为n的有限一般线性酉群，由逆转置自同构扩张。扩张群中存在平方为正则幂幺元的元素，并给出了扩张群在这些元素上的不可约特征标的值。在此过程中，得到了关于这些群的实共轭类和实值特征标的若干中间结果。

The Based Rings of Two-sided cells in an Affine Weyl group of type $\tilde B_3$, III

We compute the based rings of two-sided cells corresponding to the unipotent class in $Sp_6(\mathbb C)$ with Jordan blocks (2211). The results also verify Lusztig's conjecture on the structure of the based rings of the two-sided cells of an affine Weyl group.

Quadratic unipotent blocks in general linear, unitary and symplectic groups

An irreducible ordinary character of a finite reductive group is called quadratic unipotent if it corresponds under Jordan decomposition to a semisimple element $s$ in a dual group such that $s^2=1$. We prove that there is a bijection between, on the one hand the set of quadratic unipotent characters of $GL(n,q)$ or $U(n,q)$ for all $n \geq 0$ and on the other hand, the set of quadratic unipotent characters of $Sp(2n,q)$ for all $n \geq 0$. We then extend this correspondence to $\ell$-blocks for certain $\ell$ not dividing $q$.

A Murnaghan--Nakayama rule for values of unipotent characters in classical groups

We derive a Murnaghan--Nakayama type formula for the values of unipotent characters of finite classical groups on regular semisimple elements. This relies on Asai's explicit decomposition of Lusztig restriction. We use our formula to show that most complex irreducible characters vanish on some $\ell$-singular element for certain primes $\ell$. As an application we classify the simple endotrivial modules of the finite quasi-simple classical groups. As a further application we show that for finite simple classical groups and primes $\ell\ge3$ the first Cartan invariant in the principal $\ell$-bl…

关于李型有限群中幂幺元的共轭性

On conjugacy of unipotent elements in finite groups of Lie type

设$\bfG$是定义在$\F_q$上的连通约化代数群，其中$q$是适合于$\bfG$的素数$p$的幂。令$F$是与$\bfG$上的$\FF_q$ -结构相关联的Frobenius态射，并集合$G = \bfG^F$，$F$的不动点子群。设$\bfP$是$\bfG$的$F$ -稳定抛物子群，$\bfU$是$\bfP$的幂幺根，集合$P = \bfP^F$和$U = \bfU^F$ .令$G_\uni$是$G$中的幂幺元的集合。本文证明了$G_\uni$中$U$的共轭类数由$q$中的一个整数系数多项式给出。

Unipotent elements forcing irreducibility in linear algebraic groups

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p > 0$. We consider connected reductive subgroups $X$ of $G$ that contain a given distinguished unipotent element $u$ of $G$. A result of Testerman and Zalesski (Proc. Amer. Math. Soc., 2013) shows that if $u$ is a regular unipotent element, then $X$ cannot be contained in a proper parabolic subgroup of $G$. We generalize their result and show that if $u$ has order $p$, then except for two known examples which occur in the case $(G, p) = (C_2, 2)$, the subgroup $X$ cannot be contained in a proper para…

Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

Let $G$ be a classical group with natural module $V$ over an algebraically closed field of good characteristic. For every unipotent element $u$ of $G$, we describe the Jordan block sizes of $u$ on the irreducible $G$-modules which occur as composition factors of $V \otimes V^*$, $\wedge^2(V)$, and $S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $u$, for which recursive formulae are known.

单幂表示：将 q 变为 -q

Unipotent representations: changing q to -q

本文针对有限域上 Chevalley 群（Weyl 群最长元素中心）构造单幂表示的对合，使次数多项式在符号意义下通过 q 变 -q 相互关联。

具有交换幂幺根的抛物子代数的对称对的交换簇的不可约性

On the irreducibility of the commuting variety of a symmetric pair associated to a parabolic subalgebra with abelian unipotent radical

本文研究了单复李代数中具有交换幂幺根的抛物子代数的对称对的交换簇。通过使用“级联”的建设Kostant，我们构造了一个Cartan子空间，这反过来又提供了，在某些情况下，有用的信息，非$\mathfrak{p}$ -正则半单元的中心化子。在秩2对称对$(\mathrm{so}_{p+2},\mathrm{so}_{p}\times \mathrm{so}_{2})$、$p\geq 2$的情况下，鉴于作者之前的结果，这允许我们应用归纳法，并将可交换簇的不可约性问题简化为对$\mathfrak{p}$的均匀性的考虑-杰出元素。最后，通过Kostant-Sekiguchi的对应，我们检查，在这种情况下，$\mathfrak{p}$ -区别元素确实是甚至，因此，交换品种是不可约的。

Finding Characters Satisfying a Maximal Condition for Their Unipotent Support

In this article we extend independent results of Lusztig and Hézard concerning the existence of irreducible characters of finite reductive groups, (defined in good characteristic and arising from simple algebraic groups), satisfying a strong numerical relationship with their unipotent support. Along the way we obtain some results concerning quasi-isolated semisimple elements.

正交群与辛群中指数为$2$的幂幺元的乘积

Products of unipotent elements of index $2$ in orthogonal and symplectic groups

该研究刻画了正交群与辛群中可表为两个指数2的幂幺等距元乘积的元素，给出辛群情形的充要条件，证明辛群元素均可表为三个指数2幂幺元的乘积，并指出正交群相关分类与平方零斜自伴算子和的分类高度吻合。

Representatives for unipotent classes and nilpotent orbits

Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…

Hesselink normal forms of unipotent elements in some representations of classical groups in characteristic two

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic two. Any non-trivial self-dual irreducible $K[G]$-module $W$ admits a non-degenerate $G$-invariant alternating bilinear form, thus giving a representation $f: G \rightarrow \operatorname{Sp}(W)$. In the case where $G = \operatorname{SL}_n(K)$ and $W$ has highest weight $\varpi_1 + \varpi_{n-1}$, and in the case where $G = \operatorname{Sp}_{2n}(K)$ and $W$ has highest weight $\varpi_2$, we determine for every unipotent element $u \in G$ the conjugacy class of $f(u)$ in $\operatorname{Sp}(W)$. As…

幂幺元生成子群作用在齐性空间上的不变测度与轨道闭包

Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements

M. Ratner描述了李群齐性空间上幂幺流的有限遍历不变测度和轨道闭包，推广到由幂幺元生成的子群的作用.更精确地说：设G是一个李群（不一定是连通的），Gamma是G的一个闭子群。设W是G的一个子群，使得Ad（W）包含在Ad（W）的幂幺元生成的子群的Zapriki闭包中（在G的李代数的自同构群中）.则W对G/Gamma的作用的任何有限遍历不变测度是齐次测度（即，它被支持在保持该测度的子群的闭轨道上）。此外，如果G/Gamma具有有限体积（即，具有有限的G-不变测度），则W在G/Gamma上的任何轨道的闭包是齐次集（即，包含W的子群的有限体积闭轨道）。如果W被W的任意子群λ替换，使得W/λ具有有限体积，则上述两个结果都成立。

Remarks on quantum unipotent subgroup and dual canonical basis

We prove the tensor product decomposition of the half of quantized universal enveloping algebra associated with a Weyl group element which was conjectured by Berenstein and Greenstein.

Unipotent Elements and Twisting in Link Homology

Let $\mathcal{U}$ be the unipotent variety of a complex reductive group $G$. Fix opposed Borel subgroups $B_\pm \subseteq G$ with unipotent radicals $U_\pm$. The map that sends $x_+x_- \mapsto x_+x_-x_+^{-1}$ for all $x_\pm \in U_\pm$ restricts to a map from $U_+U_- \cap gB_+$ into $\mathcal{U} \cap gB_+$, for any $g$. We conjecture that the restricted map forms half of a homotopy equivalence between these varieties, and thus, induces a weight-preserving isomorphism between their compactly-supported cohomologies. Noting that the map is equivariant with respect to certain actions of $B_+ \cap g…

模仿射 Hecke 范畴与正则幂零中心化子

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。

On the values of unipotent characters of finite Chevalley groups of type $E_6$ in characteristic 3

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power o…

双向量的幂幺元的反向分解

The reverse decomposition of unipotents for bivectors

本文针对交换环上一般线性群的第二基本表示，将初等生成元构造为任意矩阵及其逆的初等共轭乘积的多项式表达，并得到了相关矩阵任意列的稳定化定理。

The character table of the finite Chevalley group $F_4(q)$ for $q$ a power of~$2$

Let $q$ be a prime power and $F_4(q)$ be the Chevalley group of type $F_4$ over a finite field with $q$ elements. Marcelo--Shinoda (1995) determined the values of the unipotent characters of $F_4(q)$ on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that $q$ is a power of~$2$, we explain how to construct the complete character table of~$F_4(q)$.

On character sheaves and characters of reductive groups at unipotent classes

With a view to determining character values of finite reductive groups at unipotent elements, we prove a number of results concerning inner products of generalised Gelfand-Graev characters with characteristic functions of character sheaves, here called Lusztig functions. These are used to determine projections of generalised Gelfand-Graev characters to the space of unipo- tent characters, and to the space of characters with a given wave front set. Such projections are expressed largely in terms of Weyl group data. We show how the values of characters at their unipotent support or wave front se…

Characters of unipotent groups over finite fields

Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of L-packets in terms of the so-called "admissible pairs" for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of G(F_q) is a power of q, confirming a conjecture of V. Drinfeld. This paper is the first in a series of three papers exploring the relatio…

Unipotent elements and generalized exponential maps

Let $G$ be a simple and simply connected algebraic group over an algebraically closed field $\Bbbk$ of characteristic $p>0$. Assume that $p$ is good for the root system of $G$ and that the covering map $G_{sc} \rightarrow G$ is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for $G$ that behaved like the exponential map on the resticted nullcone of $G$. In the present paper we give a formal definition of these maps, which we call `generalized exponential maps.' We provide an explicit and uniform construction of such maps for all root syste…

2-Calabi-Yau范畴和幂幺群的簇结构

Cluster structures for 2-Calabi-Yau categories and unipotent groups

我们调查集群倾斜对象（和子类别）在三角形2-卡-丘范畴和相关的类别。特别是，我们构建了一类新的此类范畴相关的预投射代数的非Dynkin箭图与元素的Coxeter群。这类2-Calabi-Yau范畴包含Dynkin图的预投射代数的簇范畴和稳定范畴作为特例。对于这些2-Calabi-Yau类别，我们构建与每个简化表达式相关的集群倾斜对象。相关联的双曲正切描述的减少的表达。受簇代数理论的启发，我们提出了（弱）簇结构和子结构的概念，并给出了几个例子。我们给出了与幂幺群相关的簇代数和子簇代数的应用，包括Dynkin和非Dynkin情形。

Dual canonical bases for unipotent groups and base affine spaces

Denote by $N \subset SL_k$ the subgroup of unipotent upper triangular matrices. In this paper, we show that the dual canonical basis of $\mathbb{C}[N]$ (and base affine spaces) can be parameterized by semi-standard Young tableaux. Moreover, we give an explicit formula for every element in the the dual canonical basis using the data of the corresponding semistandard Young tableau. We apply our results to study cluster variables in $\mathbb{C}[N]$.

On Lusztig's map for spherical unipotent conjugacy classes

We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Psi from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.

约化群的Abel幂幺子群

Abelian Unipotent Subgroups of Reductive Groups

设G是定义在特征p > 0的代数闭域k上的连通约化群.本文件有两个目的。首先，当p是好素数时，我们给出了D的“序公式”的一个新的证明。Testerman关于G中幂幺元的p-幂零度的公式，并证明了同样的公式决定了G的李代数中相应的幂零元的p-幂零度.其次，如果G是半单的且p足够大，我们证明了G总是有一个忠实表示（r，V），并且对Lie（G）中的每个p-幂零X，dr（X）的指数都在r（G）中.这个性质简化了Suslin，Friedlander和Bendel对Frobenius核G_d，d > 1的（偶）上同调环的描述.先前的作者已经观察到典型群的自然表示具有上述性质（对p没有限制）。我们的方法适用于任何Chevalley群，因此也给出了具有“例外型”根系的拟单群的结果。这些方法给出了关于p的显式充分条件，对于Coxeter数为h的伴随半单G，条件p > 2 h-2总是足够好的.

挑剔元素、次正规化子与特征标对应

Picky elements, subnormalisers, and character correspondences

本文通过研究有限李型群及约化线性代数群中幂幺、半单元素的次正规化子与挑剔元素，为 Moretó–Rizo 的不可约特征标局部—整体猜想提供证据。

Representations of unipotent groups over local fields and Gutkin's conjecture

Let F be a finite field or a local field of any characteristic. If A is a finite dimensional associative nilpotent algebra over F, the set 1+A of all formal expressions of the form 1+x, where x ranges over the elements of A, is a locally compact group with the topology induced by the standard one on F and the multiplication given by (1+x)(1+y)=1+(x+y+xy). We prove a result conjectured by Eugene Gutkin in 1973: every unitary irreducible representation of 1+A can be obtained by unitary induction from a 1-dimensional unitary character of a subgroup of the form 1+B, where B is an F-subalgebra of A…

The Hesselink stratification of nullcones and base change

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p \ge 0$. We give a case-free proof of Lusztig's conjectures [Unipotent elements in small characteristic, {\em Transform. Groups} 10 (2005), 449--487] on so-called unipotent pieces. This presents a uniform picture of the unipotent elements of $G$ which can be viewed as an extension of the Dynkin--Kostant theory, but is valid without restriction on $p$. We also obtain analogous results for the adjoint action of $G$ on its Lie algebra $\gl$ and the coadjoint action of $G$ on $\gl^*$.

A generalization of Steinberg's cross-section

Let G be a semisimple group over an algebraically closed field. Steinberg has associated to a Coxeter element w of minimal length r a subvariety V of G isomorphic to an affine space of dimension r which meets the regular unipotent class Y in exactly one point. In this paper this is generalized to the case where w is replaced by any elliptic element in the Weyl group of minimal length d in its conjugacy class, V is replaced by a subvariety V' of G isomorphic to an affine space of dimension d and Y is replaced by a unipotent class Y' of codimension d in such a way that the intersection of V' and…

关于好的 $A_1$ 子群、Springer 映射以及约化群中特殊幂幺元素的上群

On good $A_1$ subgroups, Springer maps, and overgroups of distinguished unipotent elements in reductive groups

本文利用好的 $A_1$ 子群给出 Korhonen 关于含特殊幂幺元素的连通约化子群为 $G$-不可约定理的简短统一证明，并将其推广到 Lie 型有限上群、任意阶特殊幂幺元素及 Lie 代数情形。

Lusztig Induction, Unipotent Supports, and Character Bounds

Recently, a strong exponential character bound has been established in [3] for all elements $g \in \mathbf{G}^F$ of a finite reductive group $\mathbf{G}^F$ which satisfy the condition that the centraliser $C_{\mathbf{G}}(g)$ is contained in a $(\mathbf{G},F)$-split Levi subgroup $\mathbf{M}$ of $\mathbf{G}$ and that $\mathbf{G}$ is defined over a field of good characteristic. In this paper, assuming a weak version of Lusztig's conjecture relating irreducible characters and characteristic functions of character sheaves holds, we considerably generalize this result by removing the condition that…

有限域上Howe对偶III：完整计算与Gurevich-Howe猜想

Howe duality over finite fields III: Full computation and the Gurevich-Howe conjectures

本文研究有限域上Howe对偶的完整计算，探讨对称群和正交群的振荡表示限制，并证明Gurevich-Howe猜想。

正则幂零元在 $\mathrm{GL}_3(\mathbb{C})$ 的一个表示下的像

Image of Regular Unipotent under a Representation of $\mathrm{GL}_3(\mathbb{C})$

研究正则幂零元在 $\mathrm{GL}_3(\mathbb{C})$ 的有限维不可约多项式表示下的像，等价于将不可约表示的特定复合分解为 $\mathrm{SL}_2(\mathbb{C})$ 模，并给出其 Jordan 分解的显式分解。

Elliptic elements in a Weyl group: a homogeneity property

Let G be a reductive group over an algebraically closed field whose characteristic is not a bad prime for G. Let w be an elliptic element of the Weyl group which has minimal length in its conjugacy class. We show that there exists a unique unipotent class X in G such that the following holds: if V is the variety of pairs consisting of an element g in X and a Borel subgroup B such that B,gBg^{-1} are in relative position w, then V is a homogeneous G-space.

On the cohomological spectrum and support varieties for infinitesimal unipotent supergroup schemes

We show that if $G$ is an infinitesimal elementary supergroup scheme of height $\leq r$, then the cohomological spectrum $|G|$ of $G$ is naturally homeomorphic to the variety $\mathcal{N}_r(G)$ of supergroup homomorphisms $ρ: \mathbb{M}_r \rightarrow G$ from a certain (non-algebraic) affine supergroup scheme $\mathbb{M}_r$ into $G$. In the case $r=1$, we further identify the cohomological support variety of a finite-dimensional $G$-supermodule $M$ as a subset of $\mathcal{N}_1(G)$. We then discuss how our methods, when combined with recently-announced results by Benson, Iyengar, Krause, and Pe…

Intersections of Deligne--Lusztig varieties and Springer fibres

In this paper we prove a direct geometric relation between Deligne--Lusztig varieties and Springer fibres in type $\mathsf{A}$: For any rational unipotent element, the Springer fibre cuts out a unique component of a specific Deligne--Lusztig variety; moreover, this component forms an open dense subset of a component of the Springer fibre. This boils down to a map from the unipotent variety to the Weyl group, and combines several constructions with a combinatorial flavour (like Weyr normal forms, Robinson--Schensted correspondence, and Spaltenstein's and Steinberg's labellings); it also provide…

关于整群环中幂零元素的积分分解

On integral decomposition of unipotent elements in integral group rings

本文回答了Jespers和Sun关于有限群整群环性质ND的猜想，证明除一个例外和特殊群系列外成立，并研究了相关的SN性质及单位群截面等对象。