L’opérateur de Laplace-Beltrami du demi-plan et les quantifications linéaire et projective de SL(2,\({\mathbb{R}})\). (Laplace-Beltrami operator on the half-plane and linear and projective quantization of SL(2,\({\mathbb{R}})\)). (French) Zbl 0592.35119

Colloq. Honneur L. Schwartz, Éc. Polytech. 1983, Vol. 1, Astérisque 131, 255-275 (1985).
[For the entire collection see Zbl 0566.00010.]
This paper was written as an attempt to explore certain connections between non-commutative harmonic analysis and pseudo-differential operator theory. The metaplectic representation in \(L^ 2({\mathbb{R}})\) of the twofold covering of \(G=SL(2,{\mathbb{R}})\) splits into the even and odd parts of \(L^ 2({\mathbb{R}})\) where, say, the odd part may be identified with the space \(H_{}\) taken from the projective holomorphic discrete series \(H_{\lambda}\) of representations of G. Now the Weyl calculus of pseudo-differential operators on \(L^ 2({\mathbb{R}})\) is well-known to be covariant under the metaplectic representation. In a similar way, using the Poincaré half-plane G/K as a phase space, one can define and study a calculus of pseudo-differential operators on \(H_{\lambda}\) covariant under the representation referred to above: this was done in a joint paper of J. Unterberger with the author. Restricting on one hand the Weyl calculus to \(L^ 2_{odd}({\mathbb{R}})\), and taking \(\lambda =1/2\) on the other hand, one can analyse the operator A that expresses the relationship between the two species of symbols (one living on \({\mathbb{R}}^ 2\), the other on the Poincaré half-plane) which give rise to the ”same” operator.
This is the subject of the present paper: we give the kernel of A as well as an expression of this operator connecting it the Radon transformation from G/K to G/MN (in standard notations). It is the author’s belief that some well-known deep aspects of the Radon transformation (e.g. the symmetry of functions in its range) are best grasped through their connections with matters discussed in this paper; also, that the whole scheme should generalize to various groups and symbolic operator calculi.


35S99 Pseudodifferential operators and other generalizations of partial differential operators
43A85 Harmonic analysis on homogeneous spaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
58J40 Pseudodifferential and Fourier integral operators on manifolds


Zbl 0566.00010