## Abstract

The authors consider irreducible representations π ε{lunate} N ̂ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels K_{φ}(x, y) of the trace class operations π_{φ} = ∝_{N} φ(n)π_{n} dn, regarding the π as modeled in L^{2}(R^{k}) for all π in general position. For a special class of groups they show that the models, and parameters λ labeling the representations in general position, can be chosen so the joint behavior of the kernels K_{φ}(x, y, λ) can be interpreted in a useful way. The variables (x, y, λ) run through a Zariski open set in R^{n}, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: R^{n} → R^{n} with the following properties. The Fourier transforms F_{1}φ = K_{φ}(x, y, λ) all factor through A to give "rationalized" Fourier transforms Fφ(u) such that Fφ {ring operator} A = F_{1}φ. On the rationalized parameter space a function f(u) is of the form F_{φ} = f ⇔ f is Schwartz class on R^{n}. If polynomial operators T ε{lunate} P(N) are transferred to operators T ̃ on R^{n} such that F(Tφ) = T ̃(Fφ), P(N) is transformed isomorphically to P(R^{n}).

Original language | English (US) |
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Pages (from-to) | 203-217 |

Number of pages | 15 |

Journal | Journal of Functional Analysis |

Volume | 37 |

Issue number | 2 |

DOIs | |

State | Published - Jun 15 1980 |

## ASJC Scopus subject areas

- Analysis