A SURVEY ON RANKIN-COHEN DEFORMATIONS

This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms. The famous Erlanger Programm of Klein says that geometry is about to study the transfor- mation groups of various spaces, or more precisely the properties invariant under the actions of such groups, i.e., the symmetries. Noncommutative geometry(NCG), which originated from Connes' study in operator alge- bras in 1970's, brought the landscape of geometry many new objects and some astonishing phenomena. Back to the early 1990s, Connes and Moscovici pointed out that in noncommutative geome- try(NCG) while noncommutative spaces are represented by the algebras (usually noncommuta- tive C ∗ -algebras) of "continuous functions" over noncommutative spaces, the local symmetries are reflected in some Hopf algebras. One of the first noncommutative spaces studied in NCG is the C ∗ -algebra of a foliated space. In the case of codimension n foliations, Connes and Moscovici discovered a Hopf algebra Hn, which governs the local symmetry of leaf spaces of foliations of codimension n. The Hopf algebra Hn is universal in the sense that it depends only on the codimension of a foliation. This family of Hopf algebras {Hn} is very useful in the study of transverse index theory, and later was found to have connections with various different areas of mathematics, c.f. (10), (13). In this paper, we review the application of the Hopf algebra H1 in Rankin-Cohen deformations, which was initiated by Connes and Moscovici (14). We start by recalling the general setting of transverse geometry. Let M be a smooth manifold and F be a foliation on M of codimension n. Let X be a complete flat transversal of F, and F + X be the oriented frame bundle of X. The holonomy pseudogroup acts on X and therefore F + X by transforming X parallelly along paths in leaves of F. The "transverse geometry" is to study the transversal X along with the action by the holonomy pseudogroup . In what follows we focus on the case when n = 1, and define Connes-Moscovici's Hopf algebra H1. Now the complete transversal X is a flat 1-dim manifold; and the oriented frame bundle F + X is diffeomorphic to X × R + , and is a discrete holonomy pseudogroup acting on X as local diffeomorphisms. We introduce coordinates x on X and y on R + . The lifted action of on F + X is