Related papers: Real group orbits on flag ind-varieties of $\mathr…

Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$, and $G$ is not metacyclic. Then $G$ always has a regular orbit on $V$ except for a few "small" cases. We completely…

We give examples of groups G such that G^00 is different from G^000. We also prove that for groups G definable in an o-minimal structure, G has a "bounded orbit" iff G is definably amenable. These results answer questions of Gismatullin,…

Let $G$ be a permutation group acting on $[n]=\{1, ..., n\}$ and $\mathcal{V}=\{V_{i}: i=1, ..., n\}$ be a system of $n$ subsets of $[n]$. When is there an element $g \in G$ so that $g(i) \in V_{i}$ for each $i \in [n]$? If such $g$ exists,…

We study representations of the classical infinite dimensional real simple Lie groups $G$ induced from factor representations of minimal parabolic subgroups $P$. This makes strong use of the recently developed structure theory for those…

Suppose that $G$ is a finite solvable group and let $H$ be a Hall $\pi$-subgroup, let $b(H)$ be the largest character degree of $H$, we show that $|G:O_{\pi' \pi}(G)|_{\pi} \leq b(H)^2$.

Let $G$ be a connected semi-simple algebraic group of adjoint type over an algebraically closed field, and let $\overline{G}$ be the wonderful compactification of $G$. For a fixed pair $(B, B^-)$ of opposite Borel subgroups of $G$, we look…

We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$, where $G = GL(n,\C)$, and where $K$ is one of…

Let A be a finitely generated associative algebra over an algebraically closed field. We characterize the finite dimensional modules over A whose orbit closures are regular varieties.

Let $G$ be any group. The quotient group $T(G)$ of the multiple holomorph by the holomorph of $G$ has been investigated for various families of groups $G$. In this paper, we shall take $G$ to be a finite $p$-group of class two for any odd…

In this article, we prove that every finite abelian group $G$ of odd order occurs as a subgroup of the class group of infinitely many real cyclotomic fields.

This article gives a description of the diagonal $GL_3$-orbits on the quadruple projective variety $(\mathbb P^2)^4$. We give explicit representatives of orbits, and describe the closure relations of orbits. A distinguished feature of our…

Let $P$ be a parabolic subgroup in $SL_n(\mathbb C)$. We show that there is a $SL_n(\mathbb C)$-stable closed subvariety of an affine Schubert variety in an infinite dimensional partial Flag variety (associated to the Kac-Moody group…

Let $G$ be a permutation group of degree $n$ and let $s(G)$ denote the number of set-orbits of $G$. We determine $\inf(\frac {\log_2 s(G)} n)$ over all groups $G$ that satisfy certain restrictions on composition factors (i.e. $Alt(k), k >…

We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic,…

We show that any soluble group $G$ of type Bredon-$\FP_{\infty}$ with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type $\FP_{\infty}$ must be virtually cyclic. To prove…

We continue our study of the inclusion posets of diagonal $SL(n)$-orbit closures in a product of two partial flag varieties. We prove that, if the diagonal action is of complexity one, then the poset is isomorphic to one of the 28 posets…

The purpose of the present paper is twofold: to introduce the notion of a generalized flag in an infinite dimensional vector space $V$ (extending the notion of a flag of subspaces in a vector space), and to give a geometric realization of…

We prove a generalization of a theorem of Borel-Harish-Chandra on closed orbits of linear actions of reductive groups. Consider a real reductive algebraic group $G$ acting linearly and rationally on a real vector space $V$. $G$ can be…

Let $M$ be a finite volume analytic pseudo-Riemannian manifold that admits an isometric $G$-action with a dense orbit, where $G$ is a connected non-compact simple Lie group. For low-dimensional $M$, i.e. $\dim(M) < 2\dim(G)$, when the…

Let $ G $ be a Lie group with Lie algebra $ \mathfrak{g} $. An element $ X \in \mathfrak{g} $ is called $\mathrm{Ad}_G$-real if $ -X=gXg^{-1} $ for some $ g \in G $. Moreover, if $ -X=gXg^{-1} $ holds for some involution $ g\in G $, then $…