Riemannian Symmetric Spaces Research Articles

Bourgain–Brezis inequalities on symmetric spaces of non-compact type

Let M be a global Riemannian symmetric space of non-compact type. We prove a duality estimate, for pairings of divergence-free L1 vector fields, with vector fields in a critical Sobolev space on M:|∫M〈f,ϕ〉dV|≤C‖f‖L1(dV)‖∇ϕ‖Lm(dV). This estimate provides a remedy for the failure of a critical Sobolev embedding on such symmetric spaces.

Quantization of compact Riemannian symmetric spaces

The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of Kähler polarizations parametrized by the upper half plane S . Using this family, geometric quantization, including the half-form correction, produces the field H c o r r → S of quantum Hilbert spaces. We show that projective flatness of H c o r r implies, that the symmetric space must be isometric to a compact Lie group equipped with a biinvariant metric. In the latter case the flatness of H c o r r was previously established.

On the chaotic behavior of the Dunkl heat semigroup on weighted L p spaces

In this paper we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian on weighted L-p spaces. In the case of the heat semigroup associated to the standard Laplacian we obtain a complete picture on the spaces L-p(R-n, (phi(i rho)(x))(2)dx) where phi(i rho) is the Euclidean spherical function. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar.

The absolute continuity of convolutions of orbital measures in symmetric spaces

We characterize the absolute continuity of convolution products of orbital measures on the classical, irreducible Riemannian symmetric spaces G/K of Cartan type III, where G is a non-compact, connected Lie group and K is a compact, connected subgroup. By the orbital measures, we mean the uniform measures supported on the double cosets, KzK, in G. The characterization can be expressed in terms of dimensions of eigenspaces or combinatorial properties of the annihilating roots of the elements z. A consequence of our work is to show that the convolution product of any rank G/K, continuous, K-bi-invariant measures is absolutely continuous in any of these symmetric spaces, other than those whose restricted root system is type An or D3, when rank G/K +1 is needed.

Regression models on Riemannian symmetric spaces

Regression models on Riemannian symmetric spaces

The normal holonomy of $CR$-submanifolds

We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a $CR$-submanifold of a complex space form. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation of a Riemannian symmetric space. In case of a totally real submanifold we give two results about reduction of codimension. We describe explicitly the action of the normal holonomy in the case in which the totally real submanifold is contained in a totally real totally geodesic submanifold. In such a case we prove the compactness of the normal holonomy group.

Resonances for the Laplacian on products of two rank one Riemannian symmetric spaces

Let X=X1×X2 be a direct product of two rank-one Riemannian symmetric spaces of the noncompact type. We show that when at least one of the two spaces is isomorphic to a real hyperbolic space of odd dimension, the resolvent of the Laplacian of X can be lifted to a holomorphic function on a Riemann surface which is a branched covering of C. In all other cases, the resolvent of the Laplacian of X admits a singular meromorphic lift. The poles of this function are called the resonances of the Laplacian. We determine all resonances and show that the corresponding residue operators are given by convolution with spherical functions parameterized by the resonances. The ranges of these operators are finite dimensional and explicitly realized as direct sums of finite-dimensional irreducible spherical representations of the group of the isometries of X.

Characterization of Dini Lipschitz Functions in Terms of Their Helgason Transform

Abstract In this paper, using a generalized translation operator, we obtain an analog of Younis Theorem 5.2 in [6] for the Helgason Fourier transform of a set of functions satisfying the Dini Lipschitz condition in the space L2 for functions on noncompact rank one Riemannian symmetric spaces.

Conical representations for direct limits of symmetric spaces

We extend the definition of conical representations for Riemannian symmetric spaces to a certain class of infinite-dimensional Riemannian symmetric spaces. Using a certain infinite-dimensional version of Weyl’s Unitary Trick, we show that there is a correspondence between smooth representations of infinite-dimensional noncompact-type Riemannian symmetric spaces and smooth representations of infinite-dimensional compact-type symmetric spaces. We classify all smooth conical representations which are unitary on the compact-type side. Finally, a new class of non-smooth unitary conical representations appears on the compact-type side which has no analogue in the finite-dimensional case. We classify these representations and show how to decompose them into direct integrals of irreducible conical representations.

Classification of flat connected quandles

Let [Formula: see text] be an additive abelian group. Then the binary operation [Formula: see text] gives a quandle structure on [Formula: see text], denoted by [Formula: see text], and called the Takasaki quandle of [Formula: see text]. Viewing quandles as generalization of Riemannian symmetric spaces, Ishihara and Tamaru [Flat connected finite quandles, to appear in Proc. Amer. Math. Soc. (2016)] introduced flat quandles, and classified quandles which are finite, flat and connected. In this note, we classify all flat connected quandles. More precisely, we prove that a quandle [Formula: see text] is flat and connected if and only if [Formula: see text], where [Formula: see text] is a 2-divisible group.

Maximal totally geodesic submanifolds and index of symmetric spaces

Let $M$ be an irreducible Riemannian symmetric space. The index $i(M)$ of $M$ is the minimal codimension of a totally geodesic submanifold of $M$. In [1] we proved that $i(M)$ is bounded from below by the rank $\mathrm{rk}(M)$ of $M$, that is, $\mathrm{rk}(M) \leq i(M)$. In this paper we classify all irreducible Riemannian symmetric spaces $M$ for which the equality holds, that is, $\mathrm{rk}(M) = i(M)$. In this context we also obtain an explicit classification of all nonsemisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric $\mathrm{R}$-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with $i(M) \in \{4, 5, 6 \}$.

The Selberg–Weil–Kobayashi rigidity theorem: The rank one solvable case

A local rigidity theorem was proved by Selberg and Weil for Riemannian symmetric spaces and generalized by Kobayashi for a non-Riemannian homogeneous space [Formula: see text], determining explicitly which homogeneous spaces [Formula: see text] allow nontrivial continuous deformations of co-compact discontinuous groups. When [Formula: see text] is assumed to be exponential solvable and [Formula: see text] is a maximal subgroup, an analog of such a theorem states that the local rigidity holds if and only if [Formula: see text] is isomorphic to the group Aff([Formula: see text]) of affine transformations of the real line (cf. [L. Abdelmoula, A. Baklouti and I. Kédim, The Selberg–Weil–Kobayashi rigidity theorem for exponential Lie groups, Int. Math. Res. Not. 17 (2012) 4062–4084.]). The present paper deals with the more general context, when [Formula: see text] is a connected solvable Lie group and [Formula: see text] a maximal nonnormal subgroup of [Formula: see text]. We prove that any discontinuous group [Formula: see text] for a homogeneous space [Formula: see text] is abelian and at most of rank 2. Then we discuss an analog of the Selberg–Weil–Kobayashi local rigidity theorem in this solvable setting. In contrast to the semi-simple setting, the [Formula: see text]-action on [Formula: see text] is not always effective, and thus the space of group theoretic deformations (formal deformations) [Formula: see text] could be larger than geometric deformation spaces. We determine [Formula: see text] and also its quotient modulo uneffective parts when the rank [Formula: see text]. Unlike the context of exponential solvable case, we prove the existence of formal colored discontinuous groups. That is, the parameter space admits a mixture of locally rigid and formally nonrigid deformations.

Fourier Transforms of Dini–Lipschitz Functions on Rank 1 Symmetric Spaces

In this paper, we prove an analog of Younis’s result [Int J Math Math Sci 9(2): 301–312 1986, Theorem 5.2] on the image under the Fourier–Helgason transform of a set of functions satisfying the Dini–Lipschitz functions in \({L^{p} (1 < p \leq 2)}\) for functions on noncompact rank 1 Riemannian symmetric spaces.

Flat connected finite quandles

Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of Riemannian symmetric spaces, and classify flat connected finite quandles.

Special Lagrangian submanifolds invariant under the isotropy action of symmetric spaces of rank two

We study special Lagrangian submanifolds of the cotangent bundle $T^*S^n$ of the sphere in the tangent space of Riemannian symmetric space of rank two. We show that the special Lagrangian submanifolds correspond to the solution of a differential equation on $\mathbb{R}^2$ under the assumption that the submanifold is of cohomogeneity one. Our result is the generalization of the former work of Sakai and the first author [5]. We study the qualitative properties of the solution for the special Lagrangian submanifolds and give some examples.

Regression Models on Riemannian Symmetric Spaces.

The aim of this paper is to develop a general regression framework for the analysis of manifold-valued response in a Riemannian symmetric space (RSS) and its association with multiple covariates of interest, such as age or gender, in Euclidean space. Such RSS-valued data arises frequently in medical imaging, surface modeling, and computer vision, among many others. We develop an intrinsic regression model solely based on an intrinsic conditional moment assumption, avoiding specifying any parametric distribution in RSS. We propose various link functions to map from the Euclidean space of multiple covariates to the RSS of responses. We develop a two-stage procedure to calculate the parameter estimates and determine their asymptotic distributions. We construct the Wald and geodesic test statistics to test hypotheses of unknown parameters. We systematically investigate the geometric invariant property of these estimates and test statistics. Simulation studies and a real data analysis are used to evaluate the finite sample properties of our methods.

Biharmonic hypersurfaces in Riemannian symmetric spaces I

We classify biharmonic geodesic spheres in the Cayley projective plane. Our results completes the classification of all biharmonic homogeneous hypersurfaces in simply connected compact Riemannian symmetric spaces of rank 1. In addition we show that complex Grassmannian manifolds, and exceptional Lie groups $F_4$ and $G_2$ admit proper biharmonic real hypersurfaces.

Poisson transforms for differential forms

We give a construction of a Poisson transform mapping density valued differential forms on generalized flag manifolds to differential forms on the corresponding Riemannian symmetric spaces, which can be described entirely in terms of finite dimensional representations of reductive Lie groups. Moreover, we will explicitly generate a family of degree-preserving Poisson transforms whose restriction to real valued differential forms has coclosed images. In addition, as a transform on sections of density bundles it can be related to the classical Poisson transform, proving that we produced a natural generalization of the classical theory.

Mini-Workshop: Coideal Subalgebras of Quantum Groups

Coideal subalgebras of quantized enveloping algebras appear naturally if one considers quantum group analogs of Lie subalgebras. Examples appear in the theory of quantum integrable systems with boundary and in harmonic analysis on quantum group analogs of Riemannian symmetric spaces. Recently, much progress has been made to develop a deeper representation theoretic understanding of these examples. On the other hand, coideal subalgebras play a fundamental role in the theory of Nichols algebras. The workshop aimed to discuss these theories in view of the recent developments.

On the index of symmetric spaces

Abstract Let M be an irreducible Riemannian symmetric space. The index of M is the minimal codimension of a (nontrivial) totally geodesic submanifold of M. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less than or equal to 3.