Representations of pseudo-orthogonal groups associated with a~cone

V. F. Molchanov

Sbornik. Mathematics, Tome 10 (1970) no. 3, pp. 333-347

Résumé

We study representations of the group $SO_0(p,q)$, $p>1$, $q>1$, in the spaces $D_\chi$, $\chi=(\sigma,\varepsilon)$ ($\sigma$ is a complex number; $\varepsilon=0$ or 1), of $C^\infty$-functions $\varphi(x)$ on the cone $-x_1^2-\dots-x_p^2+x_{p+1}^2+\dots+x_{p+q}^2=0$, $x\ne0$, of homogeneous degree $\sigma$ and parity $\varepsilon$: $\varphi(tx)=|t|^\sigma{\operatorname{sign}}^\varepsilon t\cdot\varphi(x)$. We consider the structure of the invariant subspaces, irreducibility, the operators which commute with the group (the intertwining operators), invariant Hermitian forms, and unitarity. Figures: 1. Bibliography: 12 titles.