Spherical Subgroups Research Articles

Chains in weak order posets associated to involutions

The W-set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W-sets of three different weak order posets: the set of all involutions in the symmetric group, the set of fixed point free involutions of the symmetric group, and the set of charged involutions. These distinguished sets of involutions parametrize Borel orbits in the classical symmetric spaces associated to the general linear group. In particular, we characterize the maximal chains of an arbitrary lower order ideal in any of these three posets.

Formula omitted]-Finite distributions on p-adic groups

For a real reductive group G, the center z(U(g)) of the universal enveloping algebra of the Lie algebra g of G acts on the space of distributions on G. This action proved to be very useful (see e.g. [17,18,32,8]).Over non-Archimedean local fields, one can replace the action of z(U(g)) by the action of the Bernstein center z of G, i.e. the center of the category of smooth representations. However, this action is not well studied. In this paper we provide some tools to work with this action and prove the following results.•The wavefront set of any z-finite distribution ξ on G over any point g∈G lies inside the nilpotent cone of Tg⁎G≅g.•Let H1,H2⊂G be symmetric subgroups. Consider the space J of H1×H2-invariant distributions on G. We prove that the z-finite distributions in J form a dense subspace. In fact we prove this result in wider generality, where the groups Hi are spherical subgroups of certain type and the invariance condition is replaced by equivariance.Further we apply those results to density and regularity of relative characters.The first result can be viewed as a version of Howe's expansion of characters.The second result can be viewed as a spherical space analog of a classical theorem on density of characters of finite length representations. It can also be viewed as a spectral version of Bernstein's localization principle.In the Archimedean case, the first result is well-known and the second remains open.

Strongly solvable spherical subgroups and their combinatorial invariants

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna's general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna's 1993 classification of strongly solvable wonderful subgroups, and the author's 2011 classification of strongly solvable spherical subgroups. We give a detailed presentation of all the three classifications and exhibit interrelations between the corresponding combinatorial invariants, which enables one to pass from one of these classifications to any other.

Spherical subgroups in simple algebraic groups

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

Vanishing of certain equivariant distributions on spherical spaces

We prove vanishing of distributions on a split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup, and eigen with respect to the center of the universal enveloping algebra of the Lie algebra of G. This is a generalization of a result by Shalika, that concerned the group case. Shalika's result was crucial in the proof of his multiplicity one theorem. We view our result as a step in the study of multiplicities of quasi-regular representations on spherical varieties. As an application we prove non-vanishing of spherical Bessel functions.

Normalizers of solvable spherical subgroups

For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G, we compute the group N G (H), the normalizer of H in G. Therebywe complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups.

On Frobenius splitting of orbit closures of spherical subgroups in flag varieties

Let H denote a spherical subgroup within a semisimple algebraic group G. In this paper we study the closures of the finitely many H-orbits in the flag variety of G. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.

Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group

This paper represents the nal step in solving the problem, posed by Siegel in 1945, of determining the minimal co-volume lattices of hyperbolic 3-space H (also Problem 3.60 (F) in the Kirby problem list from 1993). Here we identify the two smallest co-volume lattices. Both these groups are two-generator arithmetic lattices, generated by two elements of nite orders 2 and 3. Their co-volumes are 0:0390::: and 0:0408::: ; the precise values are given in terms of the Dedekind zeta function of a number eld via a formula of Borel. Our earlier work dealt with the cases when there is a nite spherical subgroup or high order torsion in the lattice. Thus, here we are concerned with the study of simple torsion of low order and the geometric structure of Klein 4-subgroups of a Kleinian group. We also identify certain universal geometric constraints imposed by discreteness on Kleinian groups which are of independent interest. To obtain these results we use a range of geometric and arithmetic criteria to obtain information on the structure of the singular set of the associated orbifold and then co-volume bounds by studying equivariant neighbourhoods of xed point sets, together with a rigorous computer search of certain parameter spaces for two-generator Kleinian groups.

A classification of unipotent spherical conjugacy classes in bad characteristic

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of every involutorial automorphism (involution) of G is a spherical subgroup of G.

Sphericity and multiplication of double cosets for infinite-dimensional classical groups

We construct spherical subgroups in infinite-dimensional classical groups $G$ (usually they are not symmetric and their finite-dimensional analogs are not spherical). We present a structure of a semigroup on double cosets $L\setminus G/L$ for various subgroups $L$ in $G$, moreover these semigroups act in spaces of $L$-fixed vectors in unitary representations of $G$. We also obtain semigroup envelops of groups $G$ generalizing constructions of operator colligations.

Parabolically connected subgroups

All reductive spherical subgroups of the group SL(n) are found for which the intersections with every parabolic subgroup of SL(n) are connected. This condition guarantees that open equivariant embeddings of the corresponding homogeneous spaces into Moishezon spaces are algebraic. Bibliography: 6 titles.

Parabolically connected subgroups

All reductive spherical subgroups of the group are found for which the intersections with every parabolic subgroup of are connected. This condition guarantees that open equivariant embeddings of the corresponding homogeneous spaces into Moishezon spaces are algebraic.Bibliography: 6 titles.

On solvable spherical subgroups of semisimple algebraic groups

We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugation.

Minimal co-volume hyperbolic lattices, I: The spherical points of a Kleinian group

We identify the two minimal co- volume lattices of the isometry group of hyperbolic 3-space that contain a finite spherical triangle group. These two groups are arithmetic and are in fact the two minimal co- volume lattices. Our results here represent the key step in establishing this fact, thereby solving a problem posed by Siegel in 1945. As a consequence we obtain sharp bounds on the order of the symmetry group of a hyperbolic 3-manifold in terms of its volume, analogous to the Hurwitz 84g - 84 theorem of 1892. The finite spherical subgroups of a Kleinian group give rise to the vertices of the singular graph in the quotient orbifold. We identify the small values of the discrete spectrum of hyperbolic distances between these vertices and show these small values give rise to arithmetic lattices. Once vertices are sufficiently separated, one obtains volume bounds by studying equivariant sets.

Conjugacy of 2–spherical subgroups of Coxeter groups and parallel walls

Let (W,S) be a Coxeter system of finite rank (ie |S| is finite) and let A be the associated Coxeter (or Davis) complex. We study chains of pairwise parallel walls in A using Tits' bilinear form associated to the standard root system of (W,S). As an application, we prove the strong parallel wall conjecture of G Niblo and L Reeves [J Group Theory 6 (2003) 399--413]. This allows to prove finiteness of the number of conjugacy classes of certain one-ended subgroups of W, which yields in turn the determination of all co-Hopfian Coxeter groups of 2--spherical type.

About Knop's Action of the Weyl Group on the Set of Orbits of a Spherical Subgroup in the Flag Manifold

Let G be a connected reductive algebraic group defined on an algebraically closed field k of characteristic different from 2. Let B denote the flag variety of G. Let H be a spherical subgroup of G. F. Knop defined an action of the Weyl group W of G on the finite set of the H-orbits in B. Here, we define an invariant, namely the type, separating the orbits of W.

On a class of double cosets in reductive algebraic groups

We study a class of double coset spaces RA \ G1 × G2 / RC, where G1 and G2 are connected reductive algebraic groups, and RA and RC are certain spherical subgroups of G1 × G2 obtained by “identifying” Levi factors of parabolic subgroups in G1 and G2. Such double cosets naturally appear in the symplectic leaf decompositions of Poisson homogeneous spaces of complex reductive groups with the Belavin-Drinfel'd Poisson structures. They also appear in orbit decompositions of the De Concini-Procesi compactifications of semisimple groups of adjoint type. We find explicit parametrizations of the double coset spaces and describe the double cosets as homogeneous spaces of RA × RC. We further show that all such double cosets give rise to set-theoretical solutions to the quantum Yang-Baxter equation on unipotent algebraic groups.

Infiniteness of double coset collections in algebraic groups

Let G be a linear algebraic group defined over an algebraically closed field. The double coset question addressed in this paper is the following: Given closed subgroups X and P, is the double coset collection X⧹ G/ P finite or infinite? We limit ourselves to the case where X is maximal rank and reductive and P parabolic. This paper presents a criterion for infiniteness which involves only dimensions of centralizers of semisimple elements. This result is then applied to finish the classification of those X which are spherical subgroups. Finally, excluding a case in F 4, we show that if X⧹ G/ P is finite then X is spherical or the Levi factor of P is spherical. This places great restrictions on X and P for X⧹ G/ P to be finite. The primary method is to descend to calculations at the finite group level and then to use elementary character theory.

Recent Results in the Geometry of Kleinian Groups

A spherical point of a Kleinian group Γ is a point of ℍ3 that is stabilized by a spherical triangle subgroup of Γ. Such points appear as vertices in the singular graph of the quotient hyperbolic 3-orbifold. We announce here sharp lower bounds for the hyperbolic distances between such points in H3. These bound from below the edge lengths of the singular graph. An elliptic element of a Kleinian group is simple if the translates of its axis under the group Γ form a disjoint collection of hyperbolic lines. Here we announce that the minimal covolume Kleinian group contains no simple elliptics of order p ≥ 3.

Quotients of group completions by spherical subgroups

Let G be a semi-simple algebraic group and let H be a spherical subgroup. The ground field k is algebraically closed and of characteristic zero. This article is concerned with projective embeddings Y of spherical homogeneous spaces G/ H. Our approach in the study of such a variety Y is to realize them as quotients under the action of H of projective embeddings of G. First, we give a more precise sense to this project by defining the quotient of a G-variety by a spherical subgroup H. Then, we give a condition, in terms of G-invariant valuations, under which Y can be obtained by quotient of an embedding of G. Finally, if the index of H in its normalizer is finite, we show that an important class of embeddings of G/ H (toroidal and liftable) geometric quotients of embeddings of G.