Let $G$ be a connected semisimple algebraic group over an algebraically closed field $k$. In 1965 Steinberg proved that if $G$ is simply connected, then in $G$ there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary $G$ such a cross-section exists if and only if the universal covering isogeny $τ\colon \tG\to G$ is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for ${\rm char}\,k=0$, the converse to Steinberg's theorem holds. The existence of a cross-section in $G$ implies, at least for ${\rm char}\,k=0$, that the algebra $k[G]^G$ of class functions on $G$ is generated by ${\rm rk}\,G$ elements. We describe, for arbitrary $G$, a minimal generating set of $k[G]^G$ and that of the representation ring of $G$ and answer two Grothendieck's questions on constructing generating sets of $k[G]^G$. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary $G$ and the existence of a rational cross-section in $G$ (for ${\rm char}\,k=0$, this has been proved earlier); this answers the other Grothendieck's question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational $W$-equivariant map $T\dashrightarrow G/T$ where $T$ is a maximal torus of $G$ and $W$ the Weyl group.
Let $G= GL_n(\mathbb{C})$, the general linear group over the complex numbers, and let $B$ be the set of invertible upper triangular matrices in $G$. Let $\mathfrak{b}=\text{Lie}(B)$. For $μ:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*$, where $\mathfrak{b}^*\cong \mathfrak{g}/\mathfrak{u}$ and $\mathfrak{u}$ being strictly upper triangular matrices in $\mathfrak{g}=\text{Lie}(G)$, we prove that the Hamiltonian reduction $μ^{-1}(0)^{rss}/\!\!/B$ of the extended regular semisimple locus $\mathfrak{b}^{rss}$ of the Borel subalgebra is smooth, affine, reduced, and scheme-theoreti…
We calculate explicitly an adelic quotient group for an excellent Noetherian normal integral two-dimensional separated scheme. An application to an irreducible normal projective algebraic surface over a field is given.
We produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko and Devyatov as well as we produce new bounds (e.g. for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).
Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open set of stable points, through which every invariant morphism factors uniquely.
In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumford's geometric invariant theory (GIT). The article surveys some recent work on geometric invariant theory and quotients of varieties by linear algebraic group actions, as well as background material on linear algebraic groups, Mumford's GIT and some of the challenges that the non-reductive setting presents. The earlier work of two of the authors in the setting of unipotent group a…
We discuss the basic properties of various versions of two variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the elliptic genera of Witten's phases on $N=2$ theories.
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.
In this paper we consider the Brauer groups of algebraic stacks and GIT quotients: the only algebraic stacks we consider in this paper are quotient stacks [X/G], where X is a smooth scheme of finite type over a field k, and G is a linear algebraic group over k and acting on X, as well as various moduli stacks of principal G-bundles on a smooth projective curve X, associated to a reductive group G. We also consider the Brauer groups of the corresponding coarse moduli spaces, which most often identify with the corresponding GIT-quotients. One conclusion that we seem to draw then is that the Brau…
Based on the work of Conrad-Gabber-Prasad, the paper deals with the geometry of particular pseudo-semisimple groups, namely those which can be written as quotient of Weil restriction of semisimple groups. We establish that these groups are retract rational when their are split, and give results on their Picard groups.
The Decomposition Theorem and the Intersection Cohomology of Quotients in Algebraic Geometry
本文将交上同调的分解定理应用于几何不变论商，将商的交上同调与作用量的半稳定点的交上同调联系起来。设一个连通的约化复代数群$G$线性作用在复射影簇$X$上.证明了：若$1 \to N \to G \to H \to 1$是连通约化群的短正合序列，$X^{ss}$是$N$作用于$X$的半稳定点集，则几何不变理论商$X^{ss}//N$的$H$ -等变交上同调是$X^{ss}$的$G$ -等变交上同调的直和项.
In this paper we prove that the etale sheafification of the functor arising from the quotient of an algebraic supergroup by a closed subsupergroup is representable by a smooth superscheme.
In this paper we consider the orbifold curve, which is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold curve via the product of the Gromov-Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Givental's action giving the CY/LG correspondence between the Gromov-Witten theory of the orbifold curve $\mathcal{E} / \mathbb{Z}_4$ and FJRW theory of the pair defined by the …
Holomorphic functions on an algebraic group invariant under a Zariski-dense subgroup
设G是复线性代数群，H是子群，H在G中的Zerkiki-拓扑是稠密的.设G/[G，G]是约化的，且（1）G是可解的，或（2）G '=[G，G]中的半单元素是稠密的.则G上的每个H-不变全纯函数都是常数。若G= G '，则H-不变亚纯函数或多重次调和函数都是常数.最后用Margulis的一个例子证明了存在一个代数群G，使得G=G'，使得存在一个不含任何半单元的Zagliki稠密离散子群。
We prove some fundamental results like localization, excision, Nisnevich descent and the Mayer-Vietoris property for equivariant regular blow-up for the equivariant K-theory of schemes with an affine group scheme action. We also show that the equivariant K-theory with finite coefficients is invariant under equivariant vector bundle morphisms. We show that the equivariant homotopy K-theory is invariant under equivariant vector bundle morphisms and satisfies all the above properties, including nil-invariance.
