Formule de Gutzmer pour la complexification d'une espace Riemannien symétrique

Jacques, Faraut

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 13 (2002) no. 3-4, pp. 233-241.

Résumé

A Gutzmer formula for the complexification of a Riemann symmetric space. We consider a complex manifold $\Omega$ and a real Lie group $G$ of holomorphic automorphisms of $\Omega$. The question we study is, for a holomorphic function $f$ on $\Omega$, to evaluate the integral of $|f|^{2}$ over a $G$-orbit by using the harmonic analysis of $G$. When $\Omega$ is an annulus in the complex plane and $G$ the rotation group, it is solved by a classical formula which is sometimes called Gutzmer’s formula. We establish a generalization of it when $\Omega$ is a $G$-invariant domain in the complexification of a Riemannian symmetric space $G/K$.