The Fourier transform on 2-step Lie groups

Abstract

In this paper, we study the Fourier transform on finite dimensional $2$-step Lie groups in terms of its canonical bilinear form (CBF) and its matrix coefficients. The parameter space of these matrix coefficients $\tilde{g}$, endowed with a distance $\rho_E$ which exchanges the regularity of a function with the decay of its Fourier matrix coefficients (cf. the Riemann-Lebesgue lemma for the classical Fourier transform) is however not complete. We compute explicitly its completion $\hat{g}$; the lack of completeness appears exactly when the CBF has nonmaximal rank. We provide an example for which partial degeneracy (partial rank loss) of the canonical form occurs, as opposed to the full degeneracy at the origin. We also compute the kernel $(w,\hat{w}) \mapsto \Theta(w,\hat{w})$ of the matrix-coefficients Fourier transform, the analogue for the $2$-step groups of the classical Fourier kernel $(x,\xi) \mapsto e^{i \langle \xi, x\rangle}$.