Related papers: Spherical orbit closures in simple projective spac…

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of…

Let $G$ be a connected linear algebraic group, let $V$ be a finite dimensional algebraic $G$-module, and let $\mathcal O_1$, $\mathcal O_2$ be two $G$-orbits in $V$. We describe a constructive way to find out whether $\mathcal O_1$ lies in…

We prove that a finite coprime linear group G in characteristic p>=(|G|-1)/2 has a regular orbit. This bound on p is best possible. We also give an application to blocks with abelian defect groups.

Orbits of coadjoint representations of classical compact Lie groups have a lot of applications. They appear in representation theory, geometrical quantization, theory of magnetism, quantum optics etc. As geometric objects the orbits were…

A morphism of linear algebraic groups $\phi:K\rightarrow G$ is called an epimorphism if it admits right cancellation. A subgroup $H\leq G$ is epimorphic if the inclusion map is an epimorphism. For $G$ a simple algebraic group over an…

Let G be a connected, reductive algebraic group over an algebraically closed field of characteristic zero or good and odd. We characterize the spherical conjugacy classes of G as those intersecting only Bruhat cells corresponding to…

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is…

In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type $E_6$, $F_4$ and $G_2$. The methods for classifying the orbits for…

Given a complex simply connected simple algebraic group $G$ of exceptional type and a maximal parabolic subgroup $P \subset G$, we classify all triples $(G,P,H)$ such that $H \subset G$ is a maximal reductive subgroup acting spherically on…

We prove that the orbit closure of the determinant is not normal. A similar result is obtained for the orbit closure of the permanent multiplied by a power of a linear form.

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $\sigma$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy…

Given two elements of a vector space acted on by a reductive group, we ask whether they lie in the same orbit, and if not, whether one lies in the orbit closure of the other. We develop techniques to optimize the orbit and orbit closure…

Let M_d(k) denote the space of dxd-matrices with coefficients in an algebraically closed field k. Let X be an orbit closure in the product [M_d(k)]^t equipped with the action of the general linear group GL_d(k) by simultaneous conjugation.…

We study the regular function ring $R(\mathcal{O})$ for all symplectic nilpotent orbits $\mathcal{O}$ with even column sizes. We begin by recalling the quantization model for all such orbits by Barbasch using unipotent representations. With…

Let $G$ be the group $GL_r(C) \times (C^\times)^n$. We conjecture that the finely-graded Hilbert series of a $G$ orbit closure in the space of $r$-by-$n$ matrices is wholly determined by the associated matroid. In support of this, we prove…

Let $S$ be a finitely generated abelian semigroup of invertible linear operators on a finite dimensional real or complex vector space $V$. We show that every coarsely dense orbit of $S$ is actually dense in $V$. More generally, if the orbit…

We discuss a particular class of rational Gorenstein singularities, which we call symplectic. A normal variety V has symplectic singularities if its smooth part carries a closed symplectic 2-form whose pull-back in any resolution X --> V…

We will investigate the norm closure of the unitary and similarity orbits of normal operators in unital, simple, purely infinite C*-algebras. An operator theoretic proof will be given to the classification of when two normal operators are…

The aim of this note is to present some new explicit examples of $O(d,d)$-generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits $\mathcal{O}$ of some semi-simple Lie group $G$. Using this construction, an…

Let G be a semisimple complex algebraic group, and H a wonderful subgroup of G. We prove several results relating the subgroup H to the properties of a combinatorial invariant S of G/H, called its spherical system. It is also possible to…