Quelques propriétés des représentations sphériques pour les espaces symétriques réductifs. (Some properties of spherical representations of reductive symmetric spaces)
The authors study the reductive real Lie group G and the symmetric space G/H. Suppose V is an admissible (\({\mathfrak g},K)\)-module of finite length with a linear functional T which is fixed by \({\mathfrak h}\) and \(H\cap K\). They show that V can be mapped equivariantly into \(C^{\infty}(G/H)\) so that T becomes the pull-back of the Dirac measure at the origin. The proof makes use of the fact that formal power series for matrix coefficients of V satisfy a system of differential equations with regular singularities. The methods begin with work of Casselman and make use of a result of Korevaar and Wiegerinck as well as earlier work appearing in a paper of \textit{P. Delorme} [in Lect. Notes Math. 1243, 108-143 (1987; Zbl 0658.22003)].
scientific article; zbMATH DE number 4081858
scientific article