## On the Corwin-Greenleaf conjecture. (Sur la conjecture de Corwin-Greenleaf.)(French)Zbl 0878.22009

Let $$G=\exp ({\mathfrak g})$$ be a nilpotent Lie group, $$H=\exp ({\mathfrak h})$$ be a closed connected subgroup of $$G$$. Let $$\chi_{f|{\mathfrak h}} =\chi$$ $$(f\in {\mathfrak g}^*)$$ be a unitary character of $$H$$ and let $$\tau= \text{ind}^G_H\chi$$ be the corresponding induced representation. Suppose that the multiplicities of the irreducible representations occurring in the disintegration of $$\tau$$ are finite. The conjecture of Corwin-Greenleaf says that the algebra $$D_\tau (G/H)$$ of the differential operators which commute with $$\tau$$ is isomorphic to the algebra $$\mathbb{C} [\Gamma_\tau]^H$$ of the $$H$$-invariant polynomials on the space $$\Gamma_f =f+ {\mathfrak h}^\perp$$. This conjecture has been proved by L. Corwin and F. P. Greenleaf in [J. Funct. Anal. 108, 374-426 (1992; Zbl 0784.22003)] in the case where there exists a subalgebra $${\mathfrak b}$$ of $${\mathfrak g}$$, which is a polarization at every $$\varphi$$ in an open Zariski dense subset of $$\Gamma_\tau$$ and which is normalized by $${\mathfrak h}$$. The author shows that the Corwin-Greenleaf conjecture is true if $$\dim H\cdot \varphi$$ is of dimension $$\leq 1$$ for generic $$\varphi$$ in $$\Gamma_\tau$$. This is done by an explicit analysis of the different possible positions of $${\mathfrak h}$$ with respect to the center of $${\mathfrak h}$$. As a corollary it is shown that the conjecture is true if there exists a common polarization $${\mathfrak b}$$ for the generic elements in $$\Gamma_f$$ or if $$\dim ({\mathfrak h}) ={1\over 2} (\dim({\mathfrak g}) +\dim ({\mathfrak g} (\varphi)))-1$$ for generic $$\varphi$$.
Reviewer: J.Ludwig (Metz)

### MSC:

 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 43A85 Harmonic analysis on homogeneous spaces

Zbl 0784.22003
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