Fourier transforms of smooth functions on certain nilpotent Lie groups

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 L²(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 Rⁿ, n = dim N. The authors show there is a polynomial map u = A(x, y, λ) that is a birational isomorphism A: Rⁿ → Rⁿ with the following properties. The Fourier transforms F₁φ = K_φ(x, y, λ) all factor through A to give "rationalized" Fourier transforms Fφ(u) such that Fφ {ring operator} A = F₁φ. On the rationalized parameter space a function f(u) is of the form F_φ = f ⇔ f is Schwartz class on Rⁿ. If polynomial operators T ε{lunate} P(N) are transferred to operators T ̃ on Rⁿ such that F(Tφ) = T ̃(Fφ), P(N) is transformed isomorphically to P(Rⁿ).