## Abstract

We show that generalised time-frequency shifts on the Heisenberg group H_{n} – R^{2n}`1 give rise to a novel type of function spaces on R^{2n}`^{1}. Similarly to classical modulation spaces and Besov spaces on R^{2n}`^{1}, these spaces can be characterised in terms of specific frequency partitions of the Fourier domain R^{p2n}`^{1} as well as decay of the matrix coefficients of specific Lie group representations. The representations in question are the generic unitary irreducible representations of the 3-step nilpotent Dynin-Folland group, also known as the Heisenberg group of the Heisenberg group or the meta-Heisenberg group. By realising these representations as nonstandard time-frequency shifts on the phase space R^{4n}`^{2} – H_{n} x R^{2n}`^{1}, we obtain a Fourier analytic characterisation, which from a geometric point of view locates the spaces somewhere between modulation spaces and Besov spaces. A conclusive comparison with the latter and some embeddings are given by using novel methods from decomposition space theory.

Original language | English |
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Pages (from-to) | 51-92 |

Number of pages | 42 |

Journal | Journal of Lie Theory |

Volume | 34 |

Issue number | 1 |

Publication status | Published - 1 Apr 2024 |

## Keywords

- Besov space
- coorbit theory
- decomposition space
- Dynin-Folland group
- flat orbit condition
- Heisenberg group
- Kirillov theory
- meta-Heisenberg group
- modulation space
- Nilpotent Lie group
- square-integrable representation

## ASJC Scopus subject areas

- Algebra and Number Theory