Journal of Lie Theory Volume 7 (1997) 121–146 \begin{matrix}c \\ ◯ \end{matrix}1997 Heldermann Verlag

Sur la conjecture de Corwin-Greenleaf

A la mémoire du Professeur Lawrence J. Corwin

Fujiwara Hidenori

Communicated by J. Ludwig

Abstract. Let G=exp((𝔤) be a nilpotent Lie group, H=exp((𝔥) a closed subgroup of G. Let χ=χ_{f} be a unitary character of H and τ= ind_{H}^{G}χ the induced representation of G. Let us suppose that the multiplicities of the irreducible representations occuring in the disintegration of τ are finite. We prove in this note the conjecture of Corwin-Greenleaf, which says that the algebra D_{τ} (G/H) of the differential operators which commute with τ is isomorhic to the algebra ℂ[Γ_{τ}]^{H} under the condition that the H -orbits in Γ_{τ} are of dimension ≤1.

Faculté de Technologie à Kyusyu Université de Kinki Iizuka 820 Japan fujiwara@fuk.kindai.ac.jp

Received July 1, 1996 and in final form October 30, 1996

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag