Isometry theorem for the Segal-Bargmann transform on a noncompact symmetric space of the complex type

We consider the Segal-Bargmann transform on a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the dual compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.     

Geometric flow and rigidity on symmetric spaces of noncompact type

In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of , where is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit.

     

Perelomov problem and inversion of the Segal-Bargmann transform

We reconstruct a function from the values of its Segal-Bargmann transform at lattice points.     

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.     

Uncertainty Principles for the Segal-Bargmann Transform

We recall some properties of the Segal-Bargmann transform; and we establish for this transform qualitative uncertainty principles: local uncertainty principle, Heisenberg uncertainty principle, Donoho-Stark''s uncertainty principle and Matolcsi-Sz\"ucs uncertainty principle.     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

Horospherical transform on Riemannian symmetric manifolds of noncompact type

We discuss I. M. Gelfand’s project of rebuilding the representation theory of semisimple Lie groups on the basis of integral geometry. The basic examples are related to harmonic analysis and the horospherical transform on symmetric manifolds. Specifically, we consider the inversion of this transform on Riemannian symmetric manifolds of noncompact type. In the known explicit inversion formulas, the nonlocal part essentially depends on the type of the root system. We suggest a universal modification of this operator.     

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let (M,g) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We introduce a new method to analyze Δ and the resolvent (Δ-σ)^-1; this has origins in quantum N-body scattering, but is independent of the ‘classical’ theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.     

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-K-invariant $L^1$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-K-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-K-invariance condition and strong $L^{\infty }$ convergence.     

An inversion formula for the Segal–Bargmann transform on a symmetric space of non-compact type

We prove analogs of the heat kernel transform inversion formulae of Golse, Leichtnam and the author [E. Leichtnam, F. Golse, M. Stenzel, Intrinsic microlocal analysis and inversion formulae for the heat equation on compact real-analytic Riemannian manifolds, Ann. Sci. École Norm. Sup. (4) 29 (6) (1996) 669–736. MR1422988 (97h:58153), Theorems 0.3, 0.4] in the setting of a Riemannian symmetric space of Helgason's non-compact type.     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

Scattering theory for the Laplacian on symmetric spaces of noncompact type and its application to a conjecture of Strichartz

In this paper we develop the scattering theory for the Laplacian on symmetric spaces of noncompact type. We study the asymptotic properties of the resolvent in the framework of the Agmon–Hörmander space. Our approach is based on a detailed analysis of the Helgason Fourier transform and generalized spherical functions on symmetric spaces of noncompact type. As an application of our scattering theory, we prove a conjecture by Strichartz concerning a characterization of a family of generalized eigenfunctions of the Laplacian.     

Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

It is established that if and are connected locally symmetric spaces of noncompact type where has finite volume, and is a totally geodesic immersion, then the closure of in is an immersed ``algebraic' submanifold. It is also shown that if in addition, the real ranks of and are equal, then the the closure of in is a totally geodesic submanifold of The proof is a straightforward application of Ratner's Theorem combined with the structure theory of symmetric spaces.

     

Self-adjointness of Toeplitz operators on the Segal-Bargmann space

We prove a new criterion that guarantees self-adjointness of Toeplitz operators with unbounded operator-valued symbols. Our criterion applies, in particular, to symbols with Lipschitz continuous derivatives, which is the natural class of Hamiltonian functions for classical mechanics. For this we extend the Berger-Coburn estimate to the case of vector-valued Segal-Bargmann spaces. Finally, we apply our result to prove self-adjointness for a class of (operator-valued) quadratic forms on the space of Schwartz functions in the Schrödinger representation.     

Cohomogeneity one actions on noncompact symmetric spaces of rank one

We classify, up to orbit equivalence, all cohomogeneity one actions on the hyperbolic planes over the complex, quaternionic and Cayley numbers, and on the complex hyperbolic spaces , . For the quaternionic hyperbolic spaces , , we reduce the classification problem to a problem in quaternionic linear algebra and obtain partial results. For real hyperbolic spaces, this classification problem was essentially solved by Élie Cartan.

     

Uniqueness of symmetric basis in quasi-Banach spaces

We show that if X is a nonlocally convex natural quasi-Banach space with symmetric basis whose Banach envelope is isomorphic to ?₁, then all symmetric bases of X are equivalent. The scope of this result is quite ample since the Banach envelopes of natural quasi-Banach spaces with basis always exhibit an ?₁-like behavior, in the sense that they contain copies of 's uniformly complemented.     

The uncertainty principle on Riemannian symmetric spaces of the noncompact type

The uncertainty principle in says that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. A quantitative assertion of this principle is Hardy's theorem. In this article we prove various generalisations of Hardy's theorem for Riemannian symmetric spaces of the noncompact type. In the case of the real line these results were obtained by Morgan and Cowling-Price.

     

Fundamental groups of compact manifolds and symmetric geometry of noncompact type

We introduce the notion of geometrical engagement for actions of semisimple Lie groups and their lattices as a concept closely related to Zimmer's topological engagement condition. Our notion is a geometrical criterion in the sense that it makes use of Riemannian distances. However, it can be used together with the foliated harmonic map techniques introduced in [8] to establish foliated geometric superrigidity results for both actions and geometric objects. In particular, we improve the applications of the main theorem in [9] to consider nonpositively curved compact manifolds (not necessarily with strictly negative curvature). We also establish topological restrictions for Riemannian manifolds whose universal cover have a suitable symmetric de Rham factor (Theorem B), as well as geometric obstructions for nonpositively curved compact manifolds to have fundamental groups isomorphic to certain groups build out of cocompact lattices in higher rank simple Lie groups (Corollary 4.5). Received: October 22, 1997