Gaussian processes on compact symmetric spaces

Paul Lévy studied Gaussian processes (a) with the parameter a running over Euclidean d-space R ^d and he also studied the case when a runs over the d-sphere S ^d . His results were extended by Gangolli in a number of directions, one being the extension to the cases where the parameter a lies in the other two-point homogeneous Riemannian manifolds. In the compact cases Gangolli showed there was a distinction between spheres and projective spaces, in that the process discovered by Lévy which he called Brownian motion parametrized by spheres does not exist for projective spaces. However many interesting Gaussian process exist with parameters running through projective spaces as we show.Sponsored in part by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation Grant MPS75-06687AO2     

Eigenvalue pinching theorems on compact symmetric spaces

We prove two first eigenvalue pinching theorems for Riemannian symmetric spaces (Theorems 1 and 2). As their application, we answer negatively a question raised by Elworthy and Rosenberg, who proposed to show that for every compact simple Lie group with a bi-invariant Riemannian metric on with respect to , being the Killing form of the Lie algebra , the first eigenvalue would satisfy

for all orthonormal bases of tangent spaces of (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that for a Lie group .

     

The Lichnerowicz Laplacian on compact symmetric spaces

We show that, on a compact symmetric space, the Lichnerowicz Laplacian acting on the space of covariant tensor fields coincides with the Casimir operator and we deduce that, on a compact semisimple Lie group, the Lichnerowicz Laplacian is the mean of the left invariant Casimir operator and the right invariant Casimir operator. To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Holomorphic Sobolev spaces associated to compact symmetric spaces

Using Gutzmer's formula, due to Lassalle, we characterise the images of Sobolev spaces under the Segal-Bargmann transform on compact Riemannian symmetric spaces. We also obtain necessary and sufficient conditions on a holomorphic function to be in the image of smooth functions and distributions under the Segal-Bargmann transform.     

Berezin transform on¶compact Hermitian symmetric spaces

Let X=G ^* be a compact Hermitian symmetric space. We study the Berezin transform on L ²(X) and calculate its spectrum under the decomposition of L ²(X) into the irreducible representations of G ^*. As applications we find the expansion of powers of the canonical polynomial (Bergman reproducing kernel for the canonical line bundle) in terms of the spherical polynomials on X, and we find the irreducible decomposition of tensor products of Bergman spaces on X. Received: 10 September 1996 / Revised version: 10 September 1997     

Completely positive compact operators on non-commutative symmetric spaces

Under natural conditions, it is shown that a completely positive operator between two non-commutative symmetric spaces of τ-measurable operators which is dominated in the sense of complete positivity by a completely positive compact operator is itself compact.     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.     

Biharmonic homogeneous hypersurfaces in compact symmetric spaces

In this paper, we study biharmonic hypersurfaces in Einstein manifolds. Then, we determine all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one.     

Qualitative behavior of special functions on compact symmetric spaces

For a symmetric space $\text{G}\text{K}$ of compact type, the highest-weight vectors for representations of G occurring in $\text{L}{}^2\text{(G}\text{K)}$ become heavily concentrated near certain submanifolds of $\text{G}\text{K}$ as the highest weight goes to infinity. This fact is applied to obtain estimates for the spectral measures of the operators qλ = P_λqP_λ, where $\text{P}{}_{\mathrm{\lambda }}\text{: L}{}^2\text{(}\text{G}\text{K}\text{) → V}{}_{\mathrm{\lambda }}$ is an orthogonal projection onto a G-irreducible summand, and q: G/K → $\text{R}$ is a continuous function acting on $\text{L}{}^2\text{(G}\text{K)}$ by multiplication.     

Minimal hyperspheres in rank two compact symmetric spaces

We describe a method to construct embedded, minimal hyperspheres in rank two compact symmetric spaces which are equivariant under the isotropy action of the symmetric space, and we supply the details of the construction for the exceptional Lie groupG ₂.Partially supported by CNPq (brazil)     

A local Paley-Wiener theorem for compact symmetric spaces

The Fourier coefficients of a smooth K-invariant function on a compact symmetric space M=U/K are given by integration of the function against the spherical functions. For functions with support in a neighborhood of the origin, we describe the size of the support by means of the exponential type of a holomorphic extension of the Fourier coefficients.     

Virtual Betti numbers of compact locally symmetric spaces

We show that the virtual Betti number of a compact locally symmetric space with arithmetic fundamental group is either the Betti number of the compact dual or else is infinite.     

On secant varieties of compact Hermitian symmetric spaces

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three—with one exception, the secant variety of the 21-dimensional spinor variety in P⁶³ where we show that the ideal is generated in degree four. We also discuss the coordinate rings of secant varieties of compact Hermitian symmetric spaces.     

Polar actions on compact rank one symmetric spaces are taut

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z ₂-coefficients.The second author was supported in part by FAPESP and CNPq.