Related papers: Spherical gradient manifolds

Let $G$ be a complex semisimple algebraic group and $X$ be a complex symmetric homogeneous $G$-variety. Assume that both $G$, $X$ as well as the $G$-action on $X$ are defined over real numbers. Then $G(\mathbb{R})$ acts on $X(\mathbb{R})$…

Given an action of a complex reductive Lie group G on a normal variety X, we show that every analytically Zariski-open subset of X admitting an analytic Hilbert quotient with projective quotient space is given as the set of semistable…

Let $k$ be a field. We investigate the relationship between subgroups of a pseudo-reductive $k$-group $G$ and its maximal reductive quotient $G'$, with applications to the subgroup structure of $G$. Let $k'/k$ be the minimal field of…

The description of irreducible representations of a group G can be seen as a question in harmonic analysis; namely, decomposing a suitable space of functions on G into irreducibles for the action of G x G by left and right multiplication.…

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna's slice theorem as well as a version of the Hilbert-Mumford criterion. A…

A section of a Riemannian $G$-manifold $M$ is a closed submanifold $\Sigma$ which meets each orbit orthogonally. It is shown that the algebra of $G$-invariant differential forms on $M$ which are horizontal in the sense that they kill every…

We construct an analogue of Whittaker reduction for Poisson actions of a semisimple complex Poisson-Lie group G. The reduction takes place along a class of transversal slices to unipotent orbits in G, which are generalizations of the…

We study a class of group actions on hyperk\"ahler manifolds which we call actions of linear type. If $M$ is a hyperk\"ahler manifold possessing such a $G$-action, the hyperk\"ahler Kirwan map is surjective if and only if the natural…

Using the analytic assembly map that appears in the Baum-Connes conjecture in noncommutative geometry, we generalise the $\Spin^c$-version of the Guillemin-Sternberg conjecture that `quantisation commutes with reduction' to (discrete series…

Let $K$ be a compact connected Lie group acting unitarily on a finite-dimensional complex vector space $V$. One calls this a {\em multiplicity-free} action whenever the $K$-isotypic components of $\C[V]$ are $K$-irreducible. We have shown…

We determine the possible finite groups $G$ of symplectic automorphisms of hyperk\"ahler manifolds which are deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that $G$ has such an action if, and only if, it is…

We consider compact symplectic manifolds acted on effectively by a compact connected Lie group $K$ in a Hamiltonian fashion. We prove that the squared moment map $||\mu||^2$ is constant if and only if $K$ is semisimple and the manifold is…

A real form $G_0$ of a complex semisimple Lie group $G$ has only finitely many orbits in any given compact $G$-homogeneous projective algebraic manifold $Z=G/Q$. A maximal compact subgroup $K_0$ of $G_0$ has special orbits $C$ which are…

Let G be an n-dimensional torus and $\tau$ a Hamiltonian action of G on a compact symplectic manifold, M. If M is pre-quantizable one can associate with $\tau$ a representation of G on a virtual vector space, Q(M), by…

In this paper we deal with a Hamiltonian action of a reductive algebraic group $G$ on an irreducible normal affine Poisson variety $X$. We study the invariant moment map $\psi_{G,X}:X\to \g$, that is, the composition of the moment map…

Let a reductive group $G$ act on a smooth affine complex algebraic variety $X.$ Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mu:T^*(X)\to \mathfrak{g}$ be the moment map. If the moment map is flat, and for a generic character…

Let $X=G/H$ be a spherical homogeneous variety for a complex reductive algebraic group $G$. We prove that the orbit space of $X$ under the action of a maximal compact subgroup $K\subset G$ is homeomorphic to the valuation cone of $X$. We…

Let k be a local field and G the set of k-points of a connected semisimple algebraic k-group of rank one. We describe all torsion-free discrete subgroups of G\times G acting properly discontinuously on G by left and right multiplication. To…

Consider a holomorphic torus action on a possibly non-compact K\"ahler manifold. We show that the higher cohomology groups appearing in the geometric quantization of the symplectic quotient are isomorphic to the invariant parts of the…

Let $G$ be a compact Lie group. We study a class of Hamiltonian $(G \times S^{1})$-manifolds decorated with a function $s$ with certain equivariance properties, under conditions on the $G$-action which we call of (semi-)linear type. In this…