The Cox ring of a spherical embedding

Given a spherical homogeneous space G/H of minimal rank, we provide a simple procedure to describe its embeddings as varieties with torus action in terms of divisorial fans.The torus in question is obtained as the identity component of the quotient group N/H , where N is the normalizer of H in G.The resulting Chow quotient is equal to (a blowup of) the simple toroidal compactification of G/(H N • ).In the horospherical case, for example, it is equal to a flag variety, and the slices (coefficients) of the divisorial fan are merely shifts of the colored fan along the colors.

According to Sakellaridis, many zeta integrals in the theory of automorphic forms can be produced or explained by appropriate choices of a Schwartz space of test functions on a spherical homogeneous space, which are in turn dictated by the geometry of affine spherical embeddings. We pursue this perspective by developing a local counterpart and try to explicate the functional equations. These constructions are also related to the L 2 -spectral decomposition of spherical homogeneous spaces in view of the Gelfand-Kostyuchenko method. To justify this viewpoint, we prove the convergence of p-adic local zeta integrals under certain premises, work out the case of prehomogeneous vector spaces and re-derive a large portion of Godement-Jacquet theory. Furthermore, we explain the doubling method and show that it fits into the paradigm of L-monoids developed by L. Lafforgue, B. C. Ngo et al., by reviewing the constructions of Braverman and Kazhdan (2002). In the global setting, we give certain speculations about global zeta integrals, Poisson formulas and their relation to period integrals.

First we give a formula of spherical functions on certain spherical homogeneous spaces. Then, applying it, we complete the theory of the spherical functions on the space X of nondegenerate unramified hermitian forms on a p-adic number field. More precisely, we give an explicit expression for the spherical functions, prove theorems on the spherical Fourier transforms on the space of Schwartz-Bruhat functions on X, and parametrize of all spherical functions on X. Finally, as an application, we give explicit expressions of local densities of representations of hermitian forms.

Reductive homogeneous Lorentzian manifolds

A spherical homogeneous spaceG/H of a connected semisimple algebraic group G is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group G. All the excellent affine spherical homogeneous spaces are classified up to isomorphism.

Under study are equivariant projective compactifications of reductive groups that can be obtained as the closure of the image of the group in the space of projective linear operators of a representation. The structure and the mutual position of the orbits of the action of the direct square of the group acting by left/right multiplication and the local structure of the compactification in the neighbourhood of a closed orbit are described. Several conditions for the normality and smoothness of a compactification are obtained. The methods used are based on the theory of equivariant embeddings of spherical homogeneous spaces and reductive algebraic semigroups.

Let G be a connected, complex reductive group and B, a Borel subgroup. The homogeneous space G/B is called the (full) flag variety of G; for GL n this is the ordinary variety of flags of subspaces in ℂ n . The properties of this variety have many implications in the study of reductive groups and their representations. For example, all the irreducible finite dimensional G-modules can be obtained as global sections of line bundles on the flag variety. More generally, any of its (G-linearized) line bundles has at most one non-vanishing sheaf-cohomology space, having a natural structure of a simple G-module if G is simply connected. This is part of the content of the Borel-Weil-Bott theorem and can be seen as a geometric interpretation of E. Cartan’s theory of highest weights.

In this paper we develop fundamental tools and methods to study meromorphic functions in an equivariant setup. As our main result we construct quotients of Rosenlicht-type for Stein spaces acted upon holomorphically by complex-reductive Lie groups and their algebraic subgroups. In particular, we show that in this setup invariant meromorphic functions separate orbits in general position. Applications to almost homogeneous spaces and principal orbit types are given. Furthermore, we use the main result to investigate the relation between holomorphic and meromorphic invariants for reductive group actions. As one important step in our proof we obtain a weak equivariant analogue of Narasimhan’s embedding theorem for Stein spaces.

We study an algebraic varieties with the action of a reductive group G. The relation is elucidated between the notions of complexity and rank of an arbitrary G-variety and the structure of stabilizers of general position of some actions of G itself and its Borel subgroup. The application of this theory to homogeneous spaces provides the explicit formulas for the rank and the complexity of quasiaffine G/H in terms of co-isotropy representation of H. The existence of Cartan subspace (and hence the freeness of algebra of invariants) for co-isotropy representation of a connected observable spherical subgroup H is proved.

We present a systematic approach to studying the geometric aspects of Vinberg theta-representations. The main idea is to use the Borel-Weil construction for representations of reductive groups as sections of homogeneous bundles on homogeneous spaces, and then to study degeneracy loci of these vector bundles. Our main technical tool is to use free resolutions as an "enhanced" version of degeneracy loci formulas. We illustrate our approach on several examples and show how they are connected to moduli spaces of Abelian varieties. To make the article accessible to both algebraists and geometers, we also include background material on free resolutions and representation theory.

Smooth embeddings of SL(2) and PGL(2)

Let $X=G/\Gamma $ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X . We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H -orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, $\Gamma $ is a non-cocompact lattice of G , and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H -orbit to a single compact set in X . This complements our results in the case of an arithmetic lattice.

On inhomogeneous spherical space forms

We investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

We prove the LeBrun–Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold $X$ of dimension $2n+1$ that has reductive automorphism group of rank at least $n-2$ is necessarily homogeneous. This implies that any positive quaternion-Kähler manifold of real dimension at most $16$ is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most $9$ with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Białynicki-Birula decomposition and equivariant Riemann–Roch theorems. From the point of view of $T$ varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most $\frac{1}{2} (\operatorname{dim}X+5)$ with $\operatorname{dim}$ arbitrarily high, but we require this special (contact) structure of $X$. Previous methods for studying $T$-varieties in general usually only apply for complexity at most $2$ or $3$.