PROJECTIVE PSEUDODIFFERENTIAL ANALYSIS AND HARMONIC ANALYSIS

Michael Pevzner (1) , André Unterberger (1)

1 LMR - Laboratoire de Mathématiques de Reims

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 $\pi_{\lambda,epsilon})$ 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 $\pi_{\lambda,epsilon})$.

Mots clés: Pseudodifferential analysis; Covariant Quantization; Para-Hermitian symmetric spaces.; Para-Hermitian symmetric spaces

Domaines: Théorie des représentations [math.RT]

Michael Pevzner, André Unterberger. PROJECTIVE PSEUDODIFFERENTIAL ANALYSIS AND HARMONIC ANALYSIS. Journal of Functional Analysis, 2007, 242, pp.442-485. ⟨10.1016/jfa2006.07.002⟩. ⟨hal-00163344⟩