In this paper, we use the theory of deformation quantization to understand Connes' and Moscovici's results [5]. We use Fedosov's method of deformation quantization of symplectic manifolds to reconstruct Zagier's deformation [11] of modular forms, and relate this deformation to the Weyl-Moyal product. We also show that the projective structure introduced by Connes and Moscovici is equivalent to the existence of certain geometric data in the case of foliation groupoids. Using the methods developed by the second author [9], we reconstruct a universal deformation formula of the Hopf algebra H1 associated to codimension one foliations. In the end, we prove that the first Rankin-Cohen bracket RC1 defines a noncommutative Poisson structure for an arbitrary H1 action. *

We introduce a Hopf algebroid associated to a proper Lie group action on a smooth manifold. We prove that the cyclic cohomology of this Hopf algebroid is equal to the de Rham cohomology of invariant differential forms. When the action is cocompact, we develop a generalized Hodge theory for the de Rham cohomology of invariant differential forms. We prove that every cyclic cohomology class of the Hopf algebroid is represented by a generalized harmonic form. This implies that the space of cyclic cohomology of the Hopf algebroid is finite dimensional. As an application of the techniques developed in this paper, we discuss properties of the Euler characteristic for a proper cocompact action.Theorem 1.1. Let G be a Lie group acting properly and cocompactly on a smooth manifold M . The cyclic cohomology groups of H(G, M ) are of finite dimension.The above result allows us to introduce the Euler characteristic for a proper cocompact action of a Lie group G as the alternating sum of the dimensions of the de Rham cohomology groups of G-invariant differential forms. We are able to generalize the following two classical results about Euler characteristic to the case of a proper cocompact G-action.(1) The Poincaré duality theorem holds for twisted de Rham cohomology groups of Ginvariant differential forms. In particular, when the dimension of M is odd, the Euler characteristic of a proper cocompact G-action on M is 0; (2) When there is a nowhere vanishing G-invariant vector field on M , the Euler characteristic of a proper cocompact G-action is also 0.

A conjecture by Mackey and Higson claims that there is close relationship between irreducible representations of a real reductive group and those of its Cartan motion group. The case of irreducible tempered unitary representations has been verified recently by Afgoustidis. We study the admissible representations of SL(2, R) by considering families of D-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makcey-Higson bijection in the general case.

Abstract. This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms.

We find a universal deformation formula for Connes-Moscovici's Hopf algebra H 1 without any projectivity assumption using Fedosov's quantization of symplectic diffeomorphisms.