New geometric examples of Anosov actions

We study a new class of Anosov actions (in the sense of Hirsch, Pugh and Shub) of reductive Lie groups, which are related to Riemannian symmetric spaces of non-compact type. The orbits of these actions can be identified with unions of parallel geodesics and the resulting orbit spaces are symplectic manifolds. For symmetric spaces of rank 1 all actions coincide with the geodesic flow.     

Simplicity and superrigidity of twin building lattices

Kac–Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type (i.e. they are not Bruhat–Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac–Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac–Moody group is always proper. We also show that Kac–Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices. Dedicated to Jacques Tits with our admiration     

b-Functions of Prehomogeneous Vector Spaces of Dynkin–Kostant Type for Exceptional Groups

The purpose of this paper is to determine the b-functions of all the prehomogeneous vector spaces associated to nilpotent orbits in the Dynkin–Kostant theory for all the complex simple algebraic groups of exceptional type. Our method to calculate b-functions is based on the structure theorem for b-functions of several variables and the functional equations for them.     

Λ-buildings and base change functors

We prove an analog of the base change functor of Λ-trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine buildings is closed under taking ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings.     

Compact Hankel Forms on Planar Domains

A Hankel form on a Hilbert function space is a bounded, symmetric, bilinear form [., .] satisfying [fx, y] = [x, fy] for a class of multipliers f. We prove analogs of Weyl–Horn and Ky Fan inequalities for compact Hankel forms, and apply them to estimate the related eigenvalues, both for Hardy–Smirnov and Bergman spaces norms associated to multiply connected planar domains. In the case of the unit disk, we investigate the asymptotic of some measures constructed by eigenfunctions of Hankel operators with certain Markov functions as symbols. Submitted: May 2, 2008. Accepted: June 28, 2008.     

Semisimple Symplectic Symmetric Spaces

A symplectic symmetric space is a connected affine symmetric manifold M endowed with a symplectic structure which is invariant under the geodesic symmetries. When the transvection group G⁰ of such a symmetric space M is semisimple, its action on (M,) is strongly Hamiltonian; a classical theorem due to Kostant implies that the moment map associated to this action realises a G⁰-equivariant symplectic covering of a coadjoint orbit O in the dual of the Lie algebra of G⁰. We show that this orbit itself admits a structure of symplectic symmetric space whose transvection algebra is . The main result of this paper is the classification of symmetric orbits for any semisimple Lie group. The classification is given in terms of root systems of transvection algebras and therefore provides, in a symplectic framework, a theorem analogous to the Borel–de Siebenthal theorem for Riemannian symmetric spaces. When its dimension is greater than 2, such a symmetric orbit is not regular and, in general, neither Hermitian nor pseudo-Hermitian.     

Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.     

Locally minimal topological groups 1

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G,τ) is called locally minimal if there exists a neighborhood U of 0 in τ such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than τ. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. Unlike the minimal topologies, the locally minimal ones are always available on “large” groups. To support this line, we prove that a bounded abelian group G admits a non-discrete locally minimal and locally GTG group topology iff |G|?c.     

Kostant homology formulas for oscillator modules of Lie superalgebras

We provide a systematic approach to obtain formulas for characters and Kostant u-homology groups of the oscillator modules of the finite-dimensional general linear and ortho-symplectic superalgebras, via Howe dualities for infinite-dimensional Lie algebras. Specializing these Lie superalgebras to Lie algebras, we recover, in a new way, formulas for Kostant homology groups of unitarizable highest weight representations of Hermitian symmetric pairs. In addition, two new reductive dual pairs related to the above-mentioned u-homology computation are worked out.     

Symmetric orbits of orthogonal Plücker actions and triviality of their second order envelopes

The orbits of Lie groups acting in Euclidean spaces by isometries are extrinsically symmetric iff they are parallel, i.e. satisfy % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0. Submanifolds characterized by the integrability condition \-R · h = 0 of this system % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0 are called semi-parallel; they are the second order envelopes of the symmetric orbits. Let the orbit set of an action of SO(n, R) in E ^{% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGOmaaaaaaa!3773!\[\frac{1}{2}\]n(n–1)} contain a Plücker submanifold. It is proved that 1) the only symmetric orbits are Plücker orbits and for n = 2 > 4 the unitary orbits, 2) each of their second order envelopes is trivial, i.e. is a single orbit or its open part.Partially supported by ESF Grant 139/305     

Double exponential map on symmetric spaces

We establish an asymptotic formula for the double exponential map operator on affine symmetric spaces. This operator plays an important role in the geometric calculus of symbols of (pseudo)differential operators on manifolds with connection, whose foundations were laid by Sharafutdinov. To obtain this result, we essentially use the structural theory of symmetric spaces and techniques of the Lie group theory. One of the key moments is an application of the Campbell-Hausdorff series in Dynkin form.     

Quasicrystallographic groups on Minkowski spaces

We generalize the quasicrystallographic groups in the sense of Novikov and Veselov from Euclidean spaces to pseudo-Euclidean and affine spaces. We prove that the quasicrystallographic groups on Minkowski spaces whose rotation groups satisfy an additional assumption are projections of crystallographic groups on pseudo-Euclidean spaces. An example shows that the assumption cannot be dropped. We prove that each quasicrystallographic group is a projection of a crystallographic group on an affine space.     

Iterations by odd functions with two extrema

Let I = [?1, 1] and fI → I be continuous, piecewise monotone and odd with two extrema. A periodic orbit is called symmetric if ?x is in the orbit when x is in the orbit. A periodic orbit which is not symmetric is called asymmetric. The first result of this paper proves an ordering of the periods for the symmetric orbits. There are two possibilities depending on how f behaves in a neighbourhood of 0. The second result of this paper proves that for a one-parameter family of odd functions with negative Schwarzian derivative there are three different types of nondegenerate bifurcations: saddle node, period-doubling pitchfork and period-preserving pitchfork. The last type of bifurcation occurs exactly when a symmetric orbit bifurcates to two asymmetric orbits.     

On the exponentiality of affine symmetric spaces

An affine symmetric space G/H is said to be exponential if every two points of this space can be joined by a geodesic and weakly exponential if the union of all geodesics issuing from one point is everywhere dense in G/H. For the group space (G × G)/G _diag of a Lie group G, these properties are equivalent to the exponentiality and weak exponentiality of G, respectively. We generalize known theorems on the image of the exponential mapping in Lie groups to the case of affine symmetric spaces. We prove the weak exponentiality of the symmetric spaces of solvable Lie groups, and in the semisimple case we obtain criteria for exponentiality and weak exponentiality.     

Champs affines

The purpose of this work is to introduce a notion of affine stacks, which is a homotopy version of the notion of affine schemes, and to give several applications in the context of algebraic topology and algebraic geometry. As a first application we show how affine stacks can be used in order to give a new point of view (and new proofs) on rational and p-adic homotopy theory. This gives a first solution to A. Grothendieck’s schematization problem described in [18]. We also use affine stacks in order to introduce a notion of schematic homotopy types. We show that schematic homotopy types give a second solution to the schematization problem, which also allows us to go beyond rational and p-adic homotopy theory for spaces with arbitrary fundamental groups. The notion of schematic homotopy types is also used in order to construct various homotopy types of algebraic varieties corresponding to various co-homology theories (Betti, de Rham, l-adic, ...), extending the well known constructions of the various fundamental groups. Finally, just as algebraic stacks are obtained by gluing affine schemes we define $$ \infty $$-geometric stacks as a certain gluing of affine stacks. Examples of $$ \infty $$-geometric stacks in the context of algebraic topology (moduli spaces of dga structures up to quasi-isomorphisms) and Hodge theory (non-abelian periods) are given.     

Minimum L1-Distance Projection onto the Boundary of a Convex Set: Simple Characterization

We show that the minimum distance projection in the L ₁-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec (Ref. 1).     

On the Mean Exit Time for Compact Symmetric Spaces

Given a compact symmetric space, M, we obtain the mean exit time function from a principal orbit, for a Brownian particle starting and moving in a generalized ball whose boundary is the principal orbit. We also obtain the mean exit time flmction of a tube of radius r around special totally geodesic submanifolds P of M. Finally we give a comparison result for the mean exit time function of tubes around submanifolds in Riemannian manifolds, using these totally geodesic submanifolds in compact symmetric spaces as a model.     

Hilbert-generated spaces

We classify several classes of the subspaces of Banach spaces X for which there is a bounded linear operator from a Hilbert space onto a dense subset in X. Dually, we provide optimal affine homeomorphisms from weak star dual unit balls onto weakly compact sets in Hilbert spaces or in c₀(Γ) spaces in their weak topology. The existence of such embeddings is characterized by the existence of certain uniformly Gâteaux smooth norms.     

Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t₁, ..., t_n) of WDVV equations of associativity polynomial in t₁, ..., t_n-1, exp t_n.     

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\G/L for various subgroups L in G; 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.