### Abstract

If K is a connected subgroup of a nilpotent Lie group G, the irreducible decompositionof the action on L^{2}(KG) has either pure infinite or boundedly finite multiplicities. In the finite case the authors recently proved that the algebra D(KG) of G-invariant differential operators on KG is commutative, even if the action is not multiplicity free, and produced evidence for the conjecture that D(KG) is isomorphic to the algebra of all Ad^{*}(K)-invariant polynomials on the annihilator {A figure is presented}, where {A figure is presented} is the Lie algebra of K. Here the conjecture is proved for a large class of data (K, G). For such pairs an explicit construction of the isomorphism can be found; it is a type of Fourier transform with some unusual nonlinear aspects. Furthermore the operators in D(KG) have tempered fundamental solutions.

Original language | English (US) |
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Pages (from-to) | 374-426 |

Number of pages | 53 |

Journal | Journal of Functional Analysis |

Volume | 108 |

Issue number | 2 |

DOIs | |

State | Published - Sep 1992 |

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### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*108*(2), 374-426. https://doi.org/10.1016/0022-1236(92)90030-M