Projective Pseudodifferential Analysis and Harmonic Analysis

Abstract

We consider pseudodifferential operators on functions on $\R^{n+1}$ which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. Their symbols can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space $G_n/H_n=SL(n+1,\R)/GL(n,\R)$, and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space $P_n(\R)$ : these spaces are the representation spaces of the maximal degenerate series $(\pi_{i\lambda,\epsilon})$ of $G_n$ . This new approach to the quantization of $G_n/H_n$, already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of $L^2(G_n/H_n)$ under the quasiregular action of $G_n$ . We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation $\pi_{i\lambda,\epsilon}$ .