Fourier transforms of smooth functions on certain nilpotent Lie groups

AbstractThe authors consider irreducible representations π ϵ N̂ of a nilpotent Lie group and define a Fourier transform for Schwartz class (and other) functions φ on N by forming the kernels Kφ(x, y) of the trace class operations πφ = ∝N φ(n)πn dn, regarding the π as modeled in L2(Rk) 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 Kφ(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 = Kφ(x, y, λ) all factor through A to give “rationalized” Fourier transforms Fφ(u) such that Fφ ∘ A = F1φ. On the rationalized parameter space a function f(u) is of the form Fφ = f ⇔ f is Schwartz class on Rn. If polynomial operators T ϵ P(N) are transferred to operators T̃ on Rn such that F(Tφ) = T̃(Fφ), P(N) is transformed isomorphically to P(Rn)