Computing in unipotent and reductive algebraic groups

On the coadjoint orbits of maximal unipotent subgroups of reductive groups

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which h…

Covering numbers of unipotent conjugacy classes in simple algebraic groups

For simple algebraic groups defined over algebraically closed fields of good characteristic, we give upper bounds on the covering numbers of unipotent conjugacy classes in terms of their (co)ranks and in terms of their dimensions.

Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with single non-trivial Jordan block

In this paper we prove the following result. Let $G$ be a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed field $F$ of characteristic $p\geq 0$, and let $u\in G$ be a nonidentity unipotent element. Let $ϕ$ be a non-trivial irreducible representation of $G$. Then the Jordan normal form of $ϕ(u)$ contains at most one non-trivial block if and only if $G$ is of type $G_2$, $u$ is a regular unipotent element and $\dim ϕ\leq 7$. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent eleme…

代数群的单势正则子群

Unipotent normal subgroups of algebraic groups

本文研究了代数群的单势正则子群的性质，并探讨了其在光滑连通仿射群中的类比关系。

On the conjugacy of maximal unipotent subgroups of real semisimple Lie groups

The existence of closed orbits of real algebraic groups on real algebraic varieties is established. As an application, it is shown that if G is a real reductive linear group with Iwasawa decomposition G= KAN, then every unipotent subgroup of G is conjugate to a subgroup of N.

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。

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 2

Let k be a separably closed field. Let G be a reductive algebraic k-group. In this paper, we study Serre's notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We also show that a regular reductive k-subgroup of G is G-completely reducible over k. Various open problems concerning complete reducibility are discussed. We present examples where the number of overgroups of irreducible…

单连通代数群中与幂幺元关联的 A1-子群的唯一性

Unicity of A1-subgroups associated to unipotent elements in simple algebraic groups

本文研究了正特征单连通代数群中幂幺元关联的 A1-子群的共轭唯一性问题，确定了满足共轭性的极大 G-稳定子簇，从而明确了 Jacobson-Morozov 定理在正特征代数群中的适用程度。

Topological representation zeta functions of unipotent groups

Inspired by work surrounding Igusa's local zeta function, we introduce topological representation zeta functions of unipotent algebraic groups over number fields. These group-theoretic invariants capture common features of established $p$-adic representation zeta functions associated with pro-$p$ groups derived from unipotent groups. We investigate fundamental properties of the topological zeta functions considered here. We also develop a method for computing them under non-degeneracy assumptions. As an application, among other things, we obtain a complete classification of topological represe…

Towards a Jordan decomposition of blocks of finite reductive groups

\input amssym.def \input amssym.tex Let $G$ be a connected algebraic reductive group over an algebraic closure of a prime field ${\Bbb F}_p$, defined over ${\Bbb F}_q$ thanks to a Frobenius $F$. Let $\ell$ be a prime different from $p$. Let $B$ be an $\ell$-block of the subgroup of rational points $G^F$. Under mild restrictions on $\ell$, we show the existence of an algebraic reductive group $H$ defined over ${\Bbb F}_q$ {\it via} a Frobenius $F$, and of a unipotent $\ell$-block $b$ of $H^F$ such that : the respective defect groups of $b$ and $B$ are isomorphic, the associated Brauer categorie…

永久缠绕幂幺群

Permawound Unipotent Groups

该研究引入永久缠绕幂幺群，利用其“普遍存在性”和“刚性”性质，证明有限生成域上非分裂光滑幂幺群一阶上同调无限，还表明特定域上交换p - 挠缠绕幂幺群是交换伪约化群的最大幂幺商，部分解答了托塔罗的问题。

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.

刻画显著幂幺轨道的不等式

Inequalities characterizing distinguished unipotent orbits

针对特征0代数闭域上的连通约化群，本文结合经典群的组合计算与例外群的计算机验证，提出并证明了显著幂幺轨道的新不等式刻画，可服务于L-参数栈上尖点层的后续研究。

On a question of Külshammer for representations of finite groups in reductive groups

Let $G$ be a simple algebraic group of type $G_2$ over an algebraically closed field of characteristic $2$. We give an example of a finite group $Γ$ with Sylow $2$-subgroup $Γ_2$ and an infinite family of pairwise non-conjugate homomorphisms $ρ\colon Γ\rightarrow G$ whose restrictions to $Γ_2$ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of $Γ$ on a connected unipotent group $V$ such that the map of 1-cohomologies ${\rm H}^1(Γ,V)\rightarrow {\rm H}^1(Γ_p,V)$ induced by restriction of 1-cocycles has an infinite fibre.

Special unipotent groups are split

We show that over any field $k$, a smooth unipotent algebraic $k$-group is special if and only if it is $k$-split.

Weakly Closed Unipotent Subgroups in Chevalley Groups

The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group fails to be a so called 2F-module.

On exponentiation and infinitesimal one-parameter subgroups of reductive groups

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and assume $p$ is good for $G$. Let $P$ be a parabolic subgroup with unipotent radical $U$. For $r \ge 1$, denote by $\mathbb{G}_{a(r)}$ the $r$-th Frobenius kernel of $\mathbb{G}_a$. We prove that if the nilpotence class of $U$ is less than $p$, then any embedding of $\mathbb{G}_{a(r)}$ in $U$ lies inside a one-parameter subgroup of $U$, and there is a canonical way in which to choose such a subgroup. Applying this result, we prove that if $p$ is at least as big as the Coxeter number of $G$,…

Unipotent classes in the classical groups parameterized by subgroups

This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.

On conjugacy of unipotent elements in finite groups of Lie type

Let $\bfG$ be a connected reductive algebraic group defined over $\F_q$, where $q$ is a power of a prime $p$ that is good for $\bfG$. Let $F$ be the Frobenius morphism associated with the $\FF_q$-structure on $\bfG$ and set $G = \bfG^F$, the fixed point subgroup of $F$. Let $\bfP$ be an $F$-stable parabolic subgroup of $\bfG$ and let $\bfU$ be the unipotent radical of $\bfP$; set $P = \bfP^F$ and $U = \bfU^F$. Let $G_\uni$ be the set of unipotent elements in $G$. In this note we show that the number of conjugacy classes of $U$ in $G_\uni$ is given by a polynomial in $q$ with integer coefficien…

On the cohomology of integral $p$-adic unipotent radicals

Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.

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…

Cohomology for infinitesimal unipotent algebraic and quantum groups

In this paper we study the structure of cohomology spaces for the Frobenius kernels of unipotent and parabolic algebraic group schemes and of their quantum analogs. Given a simple algebraic group $G$, a parabolic subgroup $P_J$, and its unipotent radical $U_J$, we determine the ring structure of the cohomology ring $H^\bullet((U_J)_1,k)$. We also obtain new results on computing $H^\bullet((P_J)_1,L(λ))$ as an $L_J$-module where $L(λ)$ is a simple $G$-module with high weight $λ$ in the closure of the bottom $p$-alcove. Finally, we provide generalizations of all our results to the quantum situat…

单酉特征的一个局部-全局原理

A local-global principle for unipotent characters

本文针对类型A、B、C的单连通有限约化群，将Dade猜想和Späth特征三元组猜想适配到单酉特征，给出计数公式，并构造与特征三元组同构相容的参数化，提供局部-全局原理的几何版本。

伽罗华自同构与块覆盖单幂块

Galois automorphisms and blocks covering unipotent blocks

本文证明了有限简单群中关于特征值延伸的条件成立，从而为有限群具有正常ℓ-补提供了新特征化，并研究了单幂块的特征分布及块级Galois-McKay猜想。

A classification of unipotent spherical conjugacy classes in bad characteristic

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.

Arithmetic lattices in unipotent algebraic groups

Fixing an arithmetic lattice $Γ$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ with $[Γ: Γ\cap Δ] [Δ: Γ\cap Δ] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith's "Subgroups of finite index in nilpotent groups" apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $\mathbb{Z}$-group with arithmetic lattice $Γ$, the Dirichlet function associated to the commensurability growth function satisfies an Eule…

On unipotent algebraic G-groups and 1-cohomology

In this paper we consider non-abelian 1-cohomology for groups with coefficients in other groups. We prove versions of the `five lemma' arising from this situation. We go on to show that a connected unipotent algebraic group Q acted on morphically by a connected algebraic group G admits a filtration with successive quotients having the structure of G-modules. From these results we deduce extensions to results due to Cline, Parshall, Scott and van der Kallen. Firstly, if G is a connected, reductive algebraic group with Borel subgroup B and Q a unipotent algebraic G-group, we show the restriction…

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 $\gl^*$.

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…

次正规化子猜想与幺模特征

The subnormaliser conjecture and unipotent characters

本文通过推导李型近单群的幺模特征的泛型双射，证明了有限群特征双射的次正规化子猜想的若干实例。

Component groups of unipotent centralizers in good characteristic

Let G be a connected, reductive group over an algebraically closed field of good characteristic. For u in G unipotent, we describe the conjugacy classes in the component group A(u) of the centralizer of u. Our results extend work of the second author done for simple, adjoint G over the complex numbers. When G is simple and adjoint, the previous work of the second author makes our description combinatorial and explicit; moreover, it turns out that knowledge of the conjugacy classes suffices to determine the group structure of A(u). Thus we obtain the result, previously known through case-checki…

Waring's problem for unipotent algebraic groups

In this paper, we formulate an analogue of Waring's problem for an algebraic group $G$. At the field level we consider a morphism of varieties $f\colon \mathbb{A}^1\to G$ and ask whether every element of $G(K)$ is the product of a bounded number of elements $f(\mathbb{A}^1(K)) = f(K)$. We give an affirmative answer when $G$ is unipotent and $K$ is a characteristic zero field which is not formally real. The idea is the same at the integral level, except one must work with schemes, and the question is whether every element in a finite index subgroup of $G(\mathcal{O})$ can be written as a produc…

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible $k$-subgroups of $G$ which are $G$-completely reducible over $k$. Our construction is based on that of subgroups of $G$ acting non-separably on the unipotent radical of a proper parabolic subgroup of $G$ in our previous work. We also present examples with the same property for a non-connected reductive group $G$. Along the way, several general results concerning compl…

Conjugacy of Levi subgroups of reductive groups and a generalization to linear algebraic groups

We investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate. These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.

Bounding Harish-Chandra series

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees.

不可约幂零数值幺半群

Irreducible Unipotent Numerical Monoids

本文延续此前对幂零线性代数群整点群的补有限子幺半群的研究，开发相关分析工具技术，提出幂零数值幺半群的理想理论。

The set of fixed points of a unipotent group

Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected unipotent group, which acts effectively on an affine variety $X.$ Then every non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled variety, i.e, there exists an affine cylinder $W\times K$ and a dominant, generically-finite polynomial mapping $ϕ:W\times K\rightarrow R.$ We show also that if an arbitrary infinite algebraic group $G$ acts effectively on $K^n$ and the set of fixed points contains a hypersurface $H$, then this hypersurface is $K$-uniruled.

Unipotent differential algebraic groups as parameterized differential Galois groups

We deal with aspects of the direct and inverse problems in parameterized Picard-Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) G is a PPV Galois group over these fields if and only if G contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs G, including unipotent groups, G is such a group if and only if it has differential type 0. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a par…

Asymptotics of commuting probabilities in reductive algebraic groups

Let $G$ be an algebraic group. For $d\geq 1$, we define the commuting probabilities $cp_d(G) = \frac{dim(\mathfrak C_d(G))}{dim(G^d)}$, where $\mathfrak C_d(G)$ is the variety of commuting $d$-tuples in $G$. We prove that for a reductive group $G$ when $d$ is large, $cp_d(G)\sim \fracα{n}$ where $n=\dim(G)$, and $α$ is the maximal dimension of an Abelian subgroup of $G$. For a finite reductive group $G$ defined over the field $\mathbb F_q$, we show that $cp_{d+1}(G(\mathbb F_q))\sim q^{(α-n)d}$, and give several examples.

几乎可约与几乎代数的 Leibniz 代数

Almost-reductive and almost-algebraic Leibniz algebras

本文探讨将 Lie 代数的几乎代数概念引入 Leibniz 代数，提出几乎可约与几乎代数两类代数，证明后者严格包含前者，并研究其性质及与其他代数类的关系。

矩阵元素、单流形和汇合的线性性

Matrix entries, unipotents, and linearity of amalgams

研究代数群子群汇合的线性性，发现某些线性群的'双'具有线性性，同时获得新的有限生成残余有限群非线性例子。

A^1-connectedness in reductive algebraic groups

Using sheaves of A^1-connected components, we prove that the Morel-Voevodsky singular construction on a reductive algebraic group fails to be A^1-local if the group does not satisfy suitable isotropy hypotheses. As a consequence, we show the failure of A^1-invariance of torsors for such groups on smooth affine schemes over infinite perfect fields. We also characterize A^1-connected reductive algebraic groups over a field of characteristic 0.