##
**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.

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.

### MSC:

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 |