On one class of unipotent subgroups of semisimple algebraic groups

Picard数为一的光滑射影horospherical簇上的单幂群等变紧化

Equivariant compactifications of a unipotent group by a smooth projective horospherical variety of Picard number one

本文将Cheong关于齐性空间紧化唯一性定理推广至Picard数为一的光滑射影horospherical簇，构造其单幂子群并证明等变紧化的唯一性。

Quotients by Reductive Group, Borel Subgroup, Unipotent Group and Maximal Torus

Consider an algebraic action of a connected complex reductive algebraic group on a complex polarized projective variety. In this paper, we first introduce the nilpotent quotient, the quotient of the polarized projective variety by a maximal unipotent subgroup. Then, we introduce and investigate three induced actions: one by the reductive group, one by a Borel subgroup, and one by a maximal torus, respectively. Our main result is that there are natural correspondences among quotients of these three actions. In the end, we mention a possible application to the moduli spaces of parabolic bundles …

连通嵌套自同构群的结构

Structure of connected nested automorphism groups

本文研究了Affine代数簇X的自同构群Aut(X)中最大单位元子群的结构，证明连通嵌套子群闭合，并扩展了Regeta和van Santen对最大可换单位元子群的描述。

Actions of the derived group of a maximal unipotent subgroup on $G$-varieties

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. We study actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We prove that $k[G/U]^{U'}$ is a polynomial algebra of Krull dimension $2r$, where $r=rk(G)$. A related result is that, for any simple finite-dimensional $G$-module $V$, $V^{U'}$ is a cyclic $U/U'$-module. Second, we study "symmetries" of Poincare series for $U'$-invariants on affine conical $G$-varieties. Third, we obtain a classification of simple $G$-modules $V$ with polyno…

On the maximal solvable subgroups of semisimple algebraic groups

Let $G$ be a semisimple affine algebraic group defined over a field $k$ of characteristic zero. We describe all the maximal connected solvable subgroups of $G$, defined over $k$, up to conjugation by rational points of $G$.

Infinite transitivity, finite generation, and Demazure roots

An affine algebraic variety X of dimension at least 2 is called flexible if the subgroup SAut(X) in Aut(X) generated by the one-parameter unipotent subgroups acts m-transitively on reg(X) for any m $\ge$ 1. In the previous paper we proved that any nondegenerate toric affine variety X is flexible. In the present paper we show that one can find a subgroup of SAut(X) generated by a finite number of one-parameter unipotent subgroups which has the same transitivity property, provided the toric variety X is smooth in codimension 2. For X=$\mathbb{A}^n$ with n$\ge$2, three such subgroups suffice.

Affinity criterion for the quotient of an algebraic group by a one-dimensional subgroup

In this work we show that the homogeneous space of an affine algebraic group $G$ by a one-dimensional unipotent subgroup $H$ is affine if and only if the subgroup is not contained in any reductive subgroup of $G$.

On solvable spherical subgroups of semisimple algebraic groups (report version)

A new approach to classification of solvable spherical subgroups of semisimple algebraic groups is considered. This approach is completely different from the known approach by D. Luna and provides an explicit classification.

微小可交换单位群方案与一维李代数

Infinitesimal commutative unipotent group schemes with one-dimensional Lie algebra

本文研究了特征为p>0的代数闭域上，阶为p^n的微小可交换单位k-群方案，其一维李代数的分类与描述，并应用于超奇异椭圆曲线的子群方案分析及Brion关于单位群方案在曲线上的有理作用问题。

On solvable spherical subgroups of semisimple algebraic groups

We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugation.

Conjugation of semisimple subgroups over real number fields of bounded degree

Let $G$ be a linear algebraic group over a field $k$ of characteristic 0. We show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over an algebraic closure of $k$ are actually conjugate over a finite field extension of $k$ of degree bounded independently of the subgroups. Moreover, if $k$ is a real number field, we show that any two connected semisimple $k$-subgroups of $G$ that are conjugate over the field of real numbers $\mathbb{R}$ are actually conjugate over a finite real extension of $k$ of degree bounded independently of the subgroups.

紧化Coxeter簇中的幂么商

Unipotent quotients in compactified Coxeter varieties

本文推广了Lusztig关于Coxeter簇幂么商的描述，证明有限域上连通约化群的相关结果，并应用于证明Wang的上同调不变量猜想及导出Alvis-Curtis对偶的性质。

Double cosets $NgN$ of normalizers of maximal tori of simple algebraic groups and orbits of partial actions of Cremona subgroups

Let $G$ be a simple algebraic group over an algebraically closed field $K$ and let $N = N_G(T)$ be the normalizer of a fixed maximal torus $T\leq G$. Further, let $U$ be the unipotent radical of a fixed Borel subgroup $B$ that contains $T$ and let $U^-$ be the unipotent radical of the opposite Borel subgroup $B^-$. The Bruhat decomposition implies the decomposition $G = NU^-UN$. The Zariski closed subset $U^-U\subset G$ is isomorphic to the affine space $A_K^m$ where $m = \dim G -\dim T$ is the number of roots in the corresponding root system. Here we construct a subgroup $\mathcal{N}\leq \ope…

Flag varieties as equivariant compactifications of G_a^n

Let G be a semisimple affine algebraic group and P a parabolic subgroup of G. We classify all flag varieties G/P which admit an action of the commutative unipotent group G_a^n with an open orbit.

Flexible varieties and automorphism groups

Given an affine algebraic variety X of dimension at least 2, we let SAut (X) denote the special automorphism group of X i.e., the subgroup of the full automorphism group Aut (X) generated by all one-parameter unipotent subgroups. We show that if SAut (X) is transitive on the smooth locus of X then it is infinitely transitive on this locus. In turn, the transitivity is equivalent to the flexibility of X. The latter means that for every smooth point x of X the tangent space at x is spanned by the velocity vectors of one-parameter unipotent subgroups of Aut (X). We provide also different variatio…

Strongly dense free subgroups of semisimple algebraic groups

We show that (with one possible exception) there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski dense. As a consequence, we get new generating results for finite simple groups of Lie type and a strengthening of a theorem of Borel related to the Hausdorff-Banach-Tarski paradox. In a sequel to this paper, we use this result to also establish uniform expansion properties for random Cayley graphs over finite simple groups of Lie type.

Bass' triangulability problem

Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem is the affirmative. To this end we prove a theorem on invariant subfields of $1$-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationall…

Quotient of Deligne-Lusztig varieties

We study the quotient of parabolic Deligne-Lusztig varieties by a finite unipotent group U^F where U is the unipotent radical of a rational parabolic subgroup P = L U. We show that in some particular cases the cohomology of this quotient can be expressed in terms of "smaller" parabolic Deligne-Lusztig varieties associated to the Levi subgroup L.

作用于代数曲面的某些群的 Tits 型二择一定理

Tits-type alternative for certain groups acting on algebraic surfaces

本文将此前环面仿射簇上由幂单子群生成的自同构群的 Tits 型二择一结果，推广到复仿射曲面情形，证明此类群或含非阿贝尔自由子群，或为亚阿贝尔幂单代数群。

Normalizers of solvable spherical subgroups

For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups.

仿射簇自同构群的Borel子群

Borel subgroups of the automorphism group of an affine variety

本文研究仿射簇自同构群的Borel子群，建立其与极大嵌套幂幺子群的双射，并证明低维阶乘簇的李代数具“积分”性，且Nagata自同构含于唯一Borel子群。

有限群的三维上三角表示的不变式环

Rings of invariants of three-dimensional upper-triangular representations of finite groups

本文证明了奇特征域上有限三维上三角群的不变式环为多项式环的充要条件，即极大幂幺子群由伪反射生成或不含平延。

由半单元素生成的代数群

Algebraic groups generated by semisimple elements

本文刻画了特征零代数闭域上连通线性代数群中由所有半单元素生成的子群。

Holomorphic functions on an algebraic group invariant under a Zariski-dense subgroup

Let G be complex linear-algebraic group, H a subgroup, which is dense in G in the Zariski-topology. Assume that G/[G,G] is reductive and furthermore that (1) G is solvable, or (2) the semisimple elements in G'=[G,G] are dense. Then every H-invariant holomorphic function on G is constant. If G=G', furthermore every H-invariant meromorphic or plurisubharmonic function is constant. Finally an example of Margulis is used to show the existence of an algebraic group G with G=G' such that there exists a Zariski-dense discrete subgroup without any semisimple element.

Degenerate $SL_n$: representations and flag varieties

The degenerate Lie group is a semidirect product of the Borel subgroup with the normal abelian unipotent subgroup. We introduce a class of the highest weight representations of the degenerate group of type A, generalizing the PBW-graded representations of the classical group. Following the classical construction of the flag varieties, we consider the closures of the orbits of the abelian unipotent subgroup in the projectivizations of the representations. We show that the degenerate flag varieties $\Fl^a_n$ and their desingularizations $R_n$ can be obtained via this construction. We prove that …

Tits type alternative for groups acting on toric affine varieties

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth.

关于具有极大幂单根的动机的Ext群的注记

A remark on Ext groups for motives with maximal unipotent radicals

本文研究具有极大幂单根的动机的Tannakian子范畴中的Ext^1群，并应用于1-动机和混合Tate动机，推导出Grothendieck周期猜想的若干推论。

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…