Infinite-dimensional Symmetric Space Research Articles

The Novikov conjecture, the group of volume preserving diffeomorphisms and Hilbert-Hadamard spaces

We prove that the Novikov conjecture holds for any discrete group admitting an isometric and metrically proper action on an admissible Hilbert-Hadamard space. Admissible Hilbert-Hadamard spaces are a class of (possibly infinite-dimensional) non-positively curved metric spaces that contain dense sequences of closed convex subsets isometric to Riemannian manifolds. Examples of admissible Hilbert-Hadamard spaces include Hilbert spaces, certain simply connected and non-positively curved Riemannian-Hilbertian manifolds and infinite-dimensional symmetric spaces. Thus our main theorem can be considered as an infinite-dimensional analogue of Kasparov’s theorem on the Novikov conjecture for groups acting properly and isometrically on complete, simply connected and non-positively curved manifolds. As a consequence, we show that the Novikov conjecture holds for geometrically discrete subgroups of the group of volume preserving diffeomorphisms of a closed smooth manifold. This result is inspired by Connes’ theorem that the Novikov conjecture holds for higher signatures associated to the Gelfand-Fuchs classes of groups of diffeormorphisms.

BC type z-measures and determinantal point processes

The (BC type) z-measures are a family of four parameter z,z′,a,b probability measures on the path space of the nonnegative Gelfand–Tsetlin graph with Jacobi-edge multiplicities. We can interpret the z-measures as random point processes Pz,z′,a,b on the punctured positive real line X=R>0∖{1}. Our main result is that these random processes are determinantal and moreover we compute their correlation kernels explicitly in terms of hypergeometric functions.For very special values of the parameters z,z′, the processes Pz,z′,a,b on X are essentially scaling limits of Racah orthogonal polynomial ensembles and their correlation kernels can be computed simply from some limits of the Racah polynomials. Thus, in the language of random matrices, we study certain analytic continuations of processes that are limits of Racah ensembles, and such that they retain the determinantal structure. Another interpretation of our results, and the main motivation of this paper, is the representation theory of big groups. In representation-theoretic terms, this paper solves a natural problem of harmonic analysis for several infinite-dimensional symmetric spaces.

Topology of exceptional orbit hypersurfaces of prehomogeneous spaces

We consider the topology for a class of hypersurfaces with highly nonisolated singularities which arise as ‘exceptional orbit varieties’ of a special class of prehomogeneous vector spaces, which are representations of linear algebraic groups with open orbits. These hypersurface singularities include both determinantal hypersurfaces and linear free (and free*) divisors. Although these hypersurfaces have highly nonisolated singularities, we determine the topology of their Milnor fibers, complements, and links. We do so by using the action of linear algebraic groups beginning with the complement, instead of using Morse-type arguments on the Milnor fibers. This includes replacing the local Milnor fiber by a global Milnor fiber which has a ‘complex geometry’ resulting from a transitive action of an appropriate algebraic group, yielding a compact ‘model submanifold’ for the homotopy type of the Milnor fiber. The topology includes the (co)homology (in characteristic 0, and 2-torsion in one family) and homotopy groups, and we deduce the triviality of the monodromy transformations on rational (or complex) cohomology. Unlike isolated singularities, the cohomology of the Milnor fibers and complements are isomorphic as algebras to exterior algebras or for one family, modules over exterior algebras; and cohomology of the link is, as a vector space, a truncated and shifted exterior algebra, for which the cohomology product structure is essentially trivial. We also deduce from Bott's periodicity theorem, the homotopy groups of the Milnor fibers for determinantal hypersurfaces in the ‘stable range’ as the stable homotopy groups of the associated infinite-dimensional symmetric spaces. Lastly, we combine the preceding with a Theorem of Oka to obtain a class of ‘formal linear combinations’ of exceptional orbit hypersurfaces which have Milnor fibers that are homotopy equivalent to joins of the compact model submanifolds. It follows that Milnor fibers for all of these hypersurfaces are essentially never homotopy equivalent to bouquets of spheres (even allowing differing dimensions).

Infinite dimensional symmetric spaces and Lax equations compatible with the infinite Toda chain

In this paper we present a natural embedding of the infinite Toda chain in a set of Lax equations in the algebra L T consisting of Z × Z -matrices that possess only a finite number of nonzero diagonals above the main central diagonal. This hierarchy of Lax equations describes the evolution of deformations of a set of commuting anti-symmetric matrices and corresponds to splitting this algebra into its anti-symmetric part and the subalgebra of matrices in L T that have no component above the main diagonal. We show that the projections of these deformations satisfy a set of zero curvature relations, which demonstrates the compatibility of the system. Further we introduce a suitable L T -module in which we can distinguish elements, the so-called wave matrices, that will lead you to solutions of the hierarchy. We conclude by showing how wave matrices of the infinite Toda chain hierarchy can be constructed starting from an infinite dimensional symmetric space.

On the transverse Scalar Curvature of a Compact Sasaki Manifold

Abstract We show that the standard picture regarding the notion of stability of constant scalar curvature metrics in Kähler geometry described by S.K. Donaldson [10, 11], which involves the geometry of infinitedimensional groups and spaces, can be applied to the constant scalar curvature metrics in Sasaki geometry with only few modification. We prove that the space of Sasaki metrics is an infinite dimensional symmetric space and that the transverse scalar curvature of a Sasaki metric is a moment map of the strict contactomophism group

The Paley–Wiener Theorem and Limits of Symmetric Spaces

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.

Bergman metrics and geodesics in the space of Kähler metrics on principally polarized abelian varieties

AbstractIt is well known in Kähler geometry that the infinite-dimensional symmetric space $\mathcal{H}$ of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds $\mathcal{B}_k\subset\mathcal{H}$ of Bergman metrics of height k. Then it is natural to ask whether geodesics in $\mathcal{H}$ can be approximated by Bergman geodesics in $\mathcal{B}_k$. For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.

Bergman metrics and geodesics in the space of Kähler metrics on toric varieties

Geodesics on the infinite dimensional symmetric space $\hcal$ of K\"ahler metrics in a fixed K\"ahler class on a projective K\"ahler manifold X are solutions of a homogeneous complex Monge-Amp\`ere equation in $X \times A$, where $A \subset \C$ is an annulus. They are analogues of 1PS (one-parameter subgroups) on symmetric spaces $G_{\C}/G$. Donaldson, Arezzo-Tian and Phong-Sturm raised the question whether Monge-Amp\`ere geodesics can be approximated by 1PS geodesics in the symmetric spaces of Bergman metrics. Phong-Sturm proved weak C^0 convergence of Bergman to Monge-Amp\`ere geodesics on a general \kahler manifold. In this article we prove convergence in $C^2(A \times X)$ in the case of toric K\"ahler metrics, extending our earlier result on $\CP^1$.

MODULI OF QUANTA

The classical phase of the matrix model of 11-dimensional M-theory is complex, infinite-dimensional Hilbert space. As a complex manifold, the latter admits a continuum of nonequivalent, complex-differentiable structures that can be placed in 1-to-1 correspondence with families of coherent states in the Hilbert space of quantum states. The moduli space of nonbiholomorphic complex structures on classical phase space turns out to be an infinite-dimensional symmetric space. We argue that each choice of a complex differentiable structure gives rise to a physically different notion of an elementary quantum.

Schwarz reflection geometry I: Continuous iteration of reflection

Differential equations are derived for a continous limit of iterated Schwarzian reflection of analytic curves, and solutions are interpreted as geodesics in an infinite-dimensional symmetric space geometry.

Asymptotics of Jack polynomials as the number of variables goes to infinity

In this paper we study the asymptotic behavior of the Jack rational functions as the number of variables grows to infinity. Our results generalize the results of A. Vershik and S. Kerov obtained in the Schur function case (theta=1). For theta=1/2,2 our results describe approximation of the spherical functions of the infinite-dimensional symmetric spaces $U(\infty)/O(\infty)$ and $U(2\infty)/Sp(\infty)$ by the spherical functions of the corresponding finite-dimensional symmetric spaces.

Hidden-symmetry algebra for the self-dual Yang-Mills equation

We find that the infinite-parameter hidden-symmetry algebra for the SL(N, C ) self-dual Yang-Mills field is the Kac-Moody algebra sl(N, C ) ⊗ C(λ,λ −1) . There is no center term for the algebra. The corresponding hidden-symmetry algebra for the real self-dual SU( N) Yang-Mills field is also derived. It is related to an infinite-dimensional symmetric space over a subalgebra su( N) ⊗ R ( λ), or can be viewed as a real form su( N) ⊗ C R( λ, λ −1) of the complex algebra sl(N, C ) ⊗ C(λ, λ −1) . A precise definition is given in the text.

More about hidden-symmetry algebra for the self-dual Yang-Mills system

The infinite-parameter "hidden symmetry" algebra for the complex self-dual $\mathrm{SU}(N)$ Yang-Mills fields is extended from $\mathrm{SL}(N, C)\ensuremath{\bigotimes}C[\ensuremath{\lambda}]$ to $\mathrm{SL}(N, C)\ensuremath{\bigotimes}C[\ensuremath{\lambda}, {\ensuremath{\lambda}}^{\ensuremath{-}1}]\ensuremath{\bigotimes}\mathrm{SL}(N, C)$. Furthermore, we also find an infinite set of infinitesimal "hidden symmetry" transformations, indexed by all integers, which can maintain the reality of the self-dual Yang-Mills potentials. Their infinite-parameter Lie algebra is shown to be related to an infinitedimensional symmetric space with isotropy group $\mathrm{SU}(N)\ensuremath{\bigotimes}R[\ensuremath{\lambda}]$.