Propriétés de l'intégrale de Cauchy Harish-Chandra pour certaines paires duales d'algèbres de Lie

Abstract

We consider a symplectic group Sp and an reductive and irreductible dual pair ( G, G′) in Sp in the sense of R. Howe. Let g (resp. g′ ) be the Lie algebra of G (resp. G′). T. Przebinda has defined a map Chc , called the Cauchy Harish-Chandra integral from the space of smooth compactly supported functions of g to the space of functions defined on the open set g ′ reg of semisimple regular elements of g′ . We prove that these functions are invariant integrals if G and G′ are linear groups and they behave locally like invariant integrals if G and G′ are unitary groups of same rank. In this last case, we obtain the jump relations up to a multiplicative constant which only depends on the dual pair. To cite this article: F. Bernon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 945–948.