Isometric actions of simple Lie groups on pseudoRiemannian manifolds

Let M be a connected compact pseudoRiemannian manifold acted upon topologically transitively and isometrically by a connected noncompact simple Lie group G. If m_0, n_0 are the dimensions of the maximal lightlike subspaces tangent to M and G, respectively, where G carries any bi-invariant metric, then we have n_0 \leq m_0. We study G-actions that satisfy the condition n_0 = m_0. With no rank restrictions on G, we prove that M has a finite covering \hat{M} to which the G-action lifts so that \hat{M} is G-equivariantly diffeomorphic to an action on a double coset K\backslash L/\Gamma, as considered in Zimmer's program, with G normal in L (Theorem A). If G has finite center and \rank_\R(G)\geq 2, then we prove that we can choose \hat{M} for which L is semisimple and \Gamma is an irreducible lattice (Theorem B). We also prove that our condition n_0 = m_0 completely characterizes, up to a finite covering, such double coset G-actions (Theorem C). This describes a large family of double coset G-actions and provides a partial positive answer to the conjecture proposed in Zimmer's program.Comment: 29 pages, published versio

Radial Toeplitz operators on the weighted Bergman spaces of Cartan domains

Let $D$ be an irreducible bounded symmetric domain with biholomorphism group $G$ with maximal compact subgroup $K$. For the Toeplitz operators with $K$-invariant symbols we provide explicit simultaneous diagonalization formulas on every weighted Bergman space. The expressions are given in the general case, but are also worked out explicitly for every irreducible bounded symmetric domain including the exceptional ones

Commuting Toeplitz operators and representation theory

An important pair of objects to study are given by the Bergman spaces on bounded symmetric domains and the Toeplitz operators on these spaces. It is well known that the bounded symmetric domains are closely related to the semisimple Lie groups of Hermitian type, the latter providing the biholomophisms of the former. Furthermore, this relationship goes as far as representation theory since the simple Lie groups of Hermitian type admit representations on the Bergman spaces that yield their holomorphic discrete series. We will explain how this interplay allows to obtain and understand large and nontrivial commutative C*-algebras generated by Toeplitz operators

On low-dimensional manifolds with isometric Ũ (p, q) - Actions

Denote by Ũ(p, q) the universal covering group of Ũ(p, q), the linear group of isometries of the pseudo-Hermitian space Cp,q of signature p, q. Let M be a connected analytic complete pseudo-Riemannian manifold that admits an isometric Ũ(p, q)-action and that satisfies dim M ≤ n(n + 2) where n = p + q. We prove that if the action of SU(p, q) (the connected derived group of eU(p, q)) has a dense orbit and the center of eU(p, q) acts non-trivially, then M is an isometric quotient of manifolds involving simple Lie groups with bi-invariant metrics. Furthermore, the Ũ(p, q)-action is lifted to M∼to natural actions on the groups involved. As a particular case, we prove that when M∼is not a pseudo-Riemannian product, then its geometry and Ũ(p, q)-action are obtained from one of the symmetric pairs (su(p, q + 1), u(p, q)) or (su(p + 1, q), u(p, q))