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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

Citations:

Zbl 0784.22003
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