Related papers: Homogeneous spherical data of orbits in spherical …

It known from the work of Feigin-Tsygan, Weibel and Keller that the cohomology groups of a smooth complex variety X can be recovered from (roughly speaking) its derived category of coherent sheaves. In this paper we show that for a finite…

Let G be a semi-simple Lie group and Q a parabilic subgroup of its complexification G^\mathbb C, then Z:=G^\mathbb C/Q is a compact complex homogeneous manifold. Moreover, G as well as K^\mathbb C, the complexification of the maximal…

This article explores the relationship between Schubert varieties and equivariant embeddings, using the framework of homogeneous fiber bundles over flag varieties. We show that the homogenous fiber bundles obtained from…

We study locally conformally homogeneous Lorentzian manifolds of dimension at least $3$, admitting an essential pseudo-group of local conformal transformations. Generalizing a recent result of Alekseevsky and Galaev, we show that any such…

The orbit polytope for a finite group G acting linearly and freely on a sphere S is used to construct a cellularized fundamental domain for the action. A resolution of the integers over G results from the associated G-equivariant…

We detect Hilbert manifolds among isometrically homogeneous metric spaces and apply the obtained results to recognizing Hilbert manifolds among homogeneous spaces of the form G/H where G is a metrizable topological group and H is a closed…

We consider the spherical DG category $Sph_G$ attached to an affine algebraic group $G$. By definition, $Sph_G := IndCoh(LS_G(S^2))$ consists of ind-coherent sheaves of the stack of $G$-local systems on the $2$-sphere $S^2$. The…

A reductive homogeneous space $G/H$ is always diffeomorphic to the normal bundle of an orbit of a maximal compact subgroup of $G$. We prove that if $G/H$ admits compact quotients, then the sphere bundle associated to this normal bundle is…

Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring…

Given a complete nonsingular algebraic variety $X$ and a divisor $D$ with normal crossings, we say that $X$ is log homogeneous with boundary $D$ if the logarithmic tangent bundle $T_X(- \log D)$ is generated by its global sections. We then…

Variables adapted to the quantum dynamics of spherically symmetric models are introduced, which further simplify the spherically symmetric volume operator and allow an explicit computation of all matrix elements of the Euclidean and…

Let H be a spherical subgroup of minimal rank of the semisimple simply connected complex algebraic group G. We define some functions on the homogeneous space G/H that we call generalised spherical minors. When G = H x H, we recover…

In this paper we study the problem of explicitly describing the space of invariant linear forms on induced distinguished representations in terms of invariant linear forms on the inducing representation. More precisely, for certain tempered…

Let $k_0$ be a field of characteristic $0$ with algebraic closure $k$. Let $G$ be a connected reductive $k$-group, and let $Y$ be a spherical variety over $k$ (a spherical homogeneous space or a spherical embedding). Let $G_0$ be a…

For a geometrically finite group Gamma of G=SO(n,1), we survey recent developments on counting and equidistribution problems for orbits of Gamma in a homogeneous space H\G where H is trivial, symmetric or horospherical. Main applications…

An algebraic zip datum is a tuple $\CZ := (G,P,Q,\phi)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\phi\colon P/R_uP\to Q/R_uQ$. We study the action of the group $E := \{(p,q)\in…

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is called spherical provided it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Kr\"amer in 1979. In…

Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The…

Let $T$ be a torus of dimension $n>1$ and $M$ a compact $T-$manifold. $M$ is a GKM manifold if the set of zero dimensional orbits in the orbit space $M/T$ is zero dimensional and the set of one dimensional orbits in $M/T$ is one…

This is a draft of a monograph to appear in the Springer series "Encyclopaedia of Mathematical Sciences", subseries "Invariant Theory and Algebraic Transformation Groups". The subject is homogeneous spaces of algebraic groups and their…