Link to this article: http://journals.cambridge.org/abstract_S0143385701001511How to cite this article: TATYANA FOTH and SVETLANA KATOK (2001). Spanning sets for automorphic forms and dynamics of the frame ow on complex hyperbolic spaces.Abstract. Let G be a semisimple Lie group with no compact factors, K a maximal compact subgroup of G, and a lattice in G. We study automorphic forms for if G is of real rank one with some additional assumptions, using a dynamical approach based on properties of the homogeneous flow on \G and a Livshitz type theorem we prove for such a flow. In the Hermitian case G = SU (n, 1) we construct relative Poincaré series associated to closed geodesics on \G/K for one-dimensional representations of K, and prove that they span the corresponding spaces of holomorphic cusp forms.

Abstract. Hamiltonian quantization of an integral compact symplectic manifold M depends on a choice of compatible almost-complex structure, J. For open sets U in the set of compatible almost complex structures and small enough values of Planck's constant, the Hilbert spaces, HJ , of the quantization form a bundle over U with a natural (L 2 ) connection. In this paper we examine the dependence of the Hilbert spaces on the choice of J, by computing the semi-classical limit of the curvature of this connection. We also show that parallel transport provides a link between the action of the group Symp(M ) of symplectic transformations of M and the Schrödinger equation.

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Abstract. We show that any holomorphic automorphic form of sufficiently large weight on an irreducible bounded symmetric domain in C n , n > 1, is the Poincaré series of a polynomial in z 1 ,. . . ,z n and give an upper bound for the degree of this polynomial. We also give an explicit construction of a basis in the space of holomorphic automorphic forms.

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