Distributions coniques associées au groupe orthogonal O(p,q) (Q760630)

Let \(G=O(p,q)\) and denote by \(\Xi\) the isotropic cone in \({\mathbb{R}}^{p+q}\) of the quadratic form with signature (p,q). Then \(\Xi =G/MN\) with \(M\cong {\mathbb{R}}^{p+q-2}\) and conical distributions are defined as MN-invariant homogeneous distributions on \(\Xi\). Homogeneity is meant with respect to the action of \(A={\mathbb{R}}^*\) on \(\Xi\) by dilations. The authors construct for every character of A two independent conical distributions by regularization of certain integrals over orbits of MN in \(\Xi\) and show in Théorème IV.1 that for most characters these span the space of conical distributions with homogeneity determined by that character. In the remaining cases however, the dimension is 3 and an another basis element is constructed. To prove this use is made of the mean value map, introduced by \textit{A. Tengstrand} [Math. Scand. 8, 201-218 (1960; Zbl 0104.334)]. An application is given to the construction of operators intertwining representations of G in spaces of homogeneous functions on the cone thus reobtaining certain results of \textit{V. F. Molchanov} and [Mat. Sb., Nov. Ser. 81(123), 358-375 81970; Zbl 0219.22015] and \textit{R. S. Strichartz} [J. Funct. Anal. 12, 341-383 (1973; Zbl 0253.43013)].