Fourier Inversion on the Group of Signatures

Abstract

The main objective of this work is to develop a framework for Fourier analysis on the group of signatures, $G_N(\mathbb{R}^d)$. Employing Kirillov's orbit method, we define the Fourier transform on this group via irreducible unitary representations. Our main contribution is the derivation of necessary and sufficient conditions for identifying coadjoint orbits in general position of $G_N(\mathbb{R}^d)$. This enables the computation of the set of \textit{jump-indices} of generic orbits, crucial for the Fourier inversion theorem. We also obtain an explicit construction of a polarization for any linear functional in general position, which allows a concrete description of the Fourier transform of functions on $G_N(\mathbb{R}^d)$. This framework yields an explicit Fourier inversion formula for the group of signatures in arbitrary dimension. Furthermore, we show that the theoretical framework developed here extends naturally to a broader class of graded Lie groups, including the group generated by the truncated tensor algebra $T_0^N(\mathbb{R}^d)$.