La série discrète de \(SL(2,{\mathbb{R}})\) et les opérateurs pseudo- différentiels sur une demi-droite. (French) Zbl 0549.35119

Unterberger, André; Unterberger, Julianne

Ann. Sci. Éc. Norm. Supér. (4) 17, 83-116 (1984).

The paper describes a symbolic calculus for operators acting on functions defined on a half-lime. The calculus is covariant under a certain representation of \(SL(2,{\mathbb{R}})\) taken from the (projective) discrete series of this group, in exactly the same way that the standard Weyl calculus of pseudodifferential operators is covariant under the metaplectic representations: this requires that the Poincaré half-plane be used as a phase space (the space where symbols live). A few formulas, some possibly new, on the Laplace-Beltrami operator of the half-plane, arise in a natural way in this context.

MSC:

35S05	Pseudodifferential operators as generalizations of partial differential operators
43A85	Harmonic analysis on homogeneous spaces
58J40	Pseudodifferential and Fourier integral operators on manifolds
53C55	Global differential geometry of Hermitian and Kählerian manifolds
47Gxx	Integral, integro-differential, and pseudodifferential operators

Keywords:

quantization; symbolic calculus; Weyl calculus; Poincaré half-plane; Laplace-Beltrami operator

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