Trigonometric polynomials over homogeneous spaces of compact groups

This paper presents a systematic theoretical study for the abstract notion of operator-valued Fourier transforms over homogeneous spaces of compact groups. Let G be a compact group, H be a closed subgroup of G , and \mu be the normalized G -invariant measure over the left coset space G/H associated to the Weil's formula. We introduce the generalized notions of abstract dual homogeneous space \widehat{G/H} for the compact homogeneous space G/H and also the operator-valued Fourier transform over the Banach function space L^1(G/H,\mu) . We prove that the abstract Fourier transform over G/H satisfies the Plancherel formula and the Poisson summation formula.

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

We consider invariant Riemannian metrics on compact homogeneous spaces $G/H$ where an intermediate subgroup $K$ between $G$ and $H$ exists. In this case, the homogeneous space $G/H$ is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups $(H,K,G)$ whether nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachh\"ofer and K. Tapp \cite{ST}, we examine all $G$-invariant fibration metrics on $G/H$ for $G$ a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of nonnegative sectional curvature.

The first part of this paper describes the construction of pseudo-Riemannian homogeneous spaces with special curvature properties such as Einstein spaces, using corresponding known compact Riemannian ones. This construction is based on the notion of a certain duality between compact and non-compact homogeneous spaces. In the second part we apply this method to obtain pseudo-Riemannian homogeneous manifolds with real Killing spinors. We will prove that under a certain additional condition a dual pseudo-Riemannian space (G′/H′, g′) of a compact Riemannian homogeneous space (G/H, g) with homogeneousSpin-structure admits a homogeneousSpin+-structure and theG_invariant Killing spinors on (G/H, g) correspond toG′-invariant Killing spinors on (G′/H′, g′). We can ensure that in most cases the hypothesis onG-invariance is satisfied.

Le K be a compact connected Lie group, L be a connected closed subgroup of K. It is well known that L is a subgroup of maximal rank of K if and only if the Euler characteristic of the manifold M = K/L is positive. Such homogeneous spaces M have been classified in [7, 10]. However, their topological classification was unknown. This classification is obtained in the present article. We show tha two compact homogeneous spaces M = K/L and M′ = K′/L′ of positive Euler characteristic are diffeomorphic if and only if the graded rings H *(M,Z) and H *(M′,Z) are isomorphic. We also obtain the rational homotopy classification of such homogeneous spaces which is not equivalent to the differential one. These results were announced in [15].

In this work, we introduce and solve a new functional equation on compact homogeneous spaces based on the d’Alembert’s functional equation. Our work extends the result of D.Yang, in which the d’Alembert functional equation on compact groups was solved with tools from harmonic analysis.

Compact Lie groups and homogeneous spaces

A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c ⋅ g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.

Preface.- Programme of the Workshop on Boundaries.- Programme of the Alp-Workshop 2009.- Publications of D.I. Cartwright.- Publications of M.A. Picardello.- Publications of V.A. Kaimonvich.- M.J. Dunwoody, An Inaccessible Graph.- J. Parkinson and B. Schapira, A Local Limit Theorem for Random Walks on the Chambers of A2 Buildings.- A. Erschler, On Continuity of Range, Entropy and Drift for Random Walks on Groups.- Y. Guivarc'h and C.R.E. Raja, Polynomial Growth, Recurrence and Ergodicity for Random Walks on Locally Compact Groups and Homogeneous Spaces.- M. Bjoerklund, Ergodic Theorems for Homogeneous Dilations.- A. Gnedin, Boundaries from Inhomogeneous Bernoulli Trials.- P.E.T. Jorgensen and E.P.J. Pearse, Resistance Boundaries of Infinite Networks.- M. Arnaudon and A. Thalmaier, Brownian Motion and Negative Curvature.- R.K. Wojciechowski, Stochastically Incomplete Manifolds and Graphs.- S. Haeseler and M. Keller, Generalized Solutions and Spectrum for Dirichlet Forms on Graphs.- R. Froese, D. Hasler and W. Spitzer, A Geometric Approach to Absolutely Continuous Spectrum for Discrete Schroedinger Operators.- A. Bendikov, B. Bobikau and C. Pittet, Some Spectral and Geometric Aspects of Countable Groups.- P. Muller and P. Stollmann, Percolation Hamiltonians.- T.S. Turova, Survey of Scalings for the Largest Connected Component in Inhomogeneous Random Graphs.- D. D'Angeli, A. Donno and T. Nagnibeda, Partition Functions of the Ising Model on Some Self-similar Schreier Graphs.- I. Krasovsky, Aspects of Toeplitz Determinants.

Let $\alpha_0$ be an affine action of a discrete group $\Gamma$ on a compact homogeneous space $X$ and $\alpha_1$ a smooth action of $\Gamma$ on $X$ which is $C^1$-close to $\alpha_0$. We show that under some conditions, every topological conjugacy between $\alpha_0$ and $\alpha_1$ is smooth. In particular, our results apply to Zariski-dense subgroups of $SL_d(\mathbb{Z})$ acting on the torus $\mathbb{T}^d$ and Zariski-dense subgroups of a simple noncompact Lie group $G$ acting on a compact homogeneous space $X$ of $G$ with an invariant measure.

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−1,1]), where the weight w(x)=wα,β(x)=(1−x)α(1+x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1,1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.

The aim of this paper is to study further the universal toric genus of compact homogeneous spaces and their homogeneous fibrations. We consider the homogeneous spaces with positive Euler characteristic. It is well known that such spaces carry many stable complex structures equivariant under the canonical action of the maximal torus $T^k$. As the torus action in this case has only isolated fixed points it is possible to effectively apply localization formula for the universal toric genus. Using this we prove that the famous topological results related to rigidity and multiplicativity of a Hirzebruch genus can be obtained on homogeneous spaces just using representation theory. In that context for homogeneous $SU$-spaces we prove the well known result about rigidity of the Krichever genus. We also prove that for a large class of stable complex homogeneous spaces any $T^k$-equivariant Hirzebruch genus given by an odd power series vanishes. Related to the problem of multiplicativity we provide construction of stable complex $T^k$-fibrations for which the universal toric genus is twistedly multiplicative. We prove that it is always twistedly multiplicative for almost complex homogeneous fibrations and describe those fibrations for which it is multiplicative. As a consequence for such fibrations the strong relations between rigidity and multiplicativity for an equivariant Hirzebruch genus is established. The universal toric genus of the fibrations for which the base does not admit any stable complex structure is also considered. The main examples here for which we compute the universal toric genus are the homogeneous fibrations over quaternionic projective spaces.

Reductive homogeneous Lorentzian manifolds

The operators \( \mathfrak{U}_\delta\) which we studied in Chapter 3 have analogues in the compact case. In this chapter we study their properties for compact two-point homogeneous spaces of dimension > 1. These are the Riemannian manifolds M with the property that for any two pairs of points \((p_1,\,p_2)\,{\rm and}\,(q_1,q_2)\,{\rm satisfying}\, d((p_1,\,p_2)\,=\,(q_1,q_2)\,{\rm where}\, d{\rm \,is\, the \,distance \,on}\, M\), there exists an isometry mapping \((p_1\,to\,p_2)\,{\rm and}\,(p_1\,to\,p_2).\) By virtue of Wang’s classification (see Helgason [H5, Chapter 1, § 4]) these are also the compact symmetric spaces of rank one. Unlike the non-compact case, the treatment in this chapter is based on the realizations of the spaces under consideration. Accordingly, the use of Lie theory is minimal.KeywordsSymmetric SpaceHarmonic PolynomialRiemannian Symmetric SpaceComplex QuadraticWiener TheoremThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let G be a compact Lie group and H be closed subgroup of G.In this dissertation, we will associate to the compact homogeneous space G/H a abstract simplicial complex T G/H .It was proved that the non-contractility of T G/H implies that the existence of a G-invariant Einstein metric in G/H.It will be shown a way to compute a non-zero reduced homology class of T G/H for intermediate subgroups H < K < G under some hypothesis, which implies the non-contractility of T G/H .This method will be applied to case where G is classical simple and H has maximal rank.