# Quantization and Coorbit Spaces for Nilpotent Groups

## TL;DR

This paper explores the quantization of symbols on nilpotent Lie groups using harmonic analysis tools, introducing coorbit spaces and phase-space shifts to advance understanding of non-commutative harmonic analysis.

## Contribution

It introduces a new class of coorbit spaces for symbols on nilpotent groups, linking them with coorbit spaces of vectors through Fourier-Wigner transforms.

## Key findings

- Development of phase-space shifts for nilpotent groups
- Introduction of coorbit spaces of symbols
- Analysis of the relationship between symbol and vector coorbit spaces

## Abstract

We reconsider the quantization of symbols defined on the product between a nilpotent Lie algebra and its dual. To keep track of the non-commutative group background, the Lie algebra is endowed with the Baker-Campbell-Hausdorff product, making it via the exponential diffeomorphism a copy of its unique connected simply connected nilpotent Lie group. Using harmonic analysis tools, we emphasize the role of a Weyl system, of the associated Fourier-Wigner transformation and, at the level of symbols, of an important family of exponential functions. Such notions also serve to introduce a family of phase-space shifts. These are used to define and briefly study a new class of coorbit spaces of symbols and its relationship with coorbit spaces of vectors, defined via the Fourier-Wigner transform.