Hermitian nilpotent lie groups: Harmonic analysis as spectral theory of Laplacians

In this research, we present some basic facts about Lie algebra and Lie groups. We shall require only elementary facts about the general definition and knowledge of a few of the more basic groups, such as Euclidean groups. Then we introduce the Heisenberg group which is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis.

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Spectral theory of approximate lattices in nilpotent Lie groups

Mathematische Annalen ◽

10.1007/s00208-021-02329-w ◽

2021 ◽

Author(s):

Michael Björklund ◽

Tobias Hartnick

Keyword(s):

Density Function ◽

Lie Groups ◽

Spectral Theory ◽

Regularity Conditions ◽

Nilpotent Lie Groups ◽

Large Collection ◽

Fiber Density ◽

System A ◽

Central Characters ◽

Parametric Version

AbstractWe consider approximate lattices in nilpotent Lie groups. With every such approximate lattice one can associate a hull dynamical system and, to every invariant measure of this system, a corresponding unitary representation. Our results concern both the spectral theory of the representation and the topological dynamics of the system. On the spectral side we construct explicit eigenfunctions for a large collection of central characters using weighted periodization against a twisted fiber density function. We construct this density function by establishing a parametric version of the Bombieri–Taylor conjecture and apply our results to locate high-intensity Bragg peaks in the central diffraction of an approximate lattice. On the topological side we show that under some mild regularity conditions the hull of an approximate lattice admits a sequence of continuous horizontal factors, where the final horizontal factor is abelian and each intermediate factor corresponds to a central extension. We apply this to extend theorems of Meyer and Dani–Navada concerning number-theoretic properties of Meyer sets to the nilpotent setting.

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Harmonic Analysis and Fundamental Solutions on Nilpotent Lie Groups

Analysis and Partial Differential Equations ◽

10.1201/9781482276923-26 ◽

1989 ◽

pp. 285-312

Keyword(s):

Harmonic Analysis ◽

Lie Groups ◽

Fundamental Solutions ◽

Nilpotent Lie Groups

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Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Entropy ◽

10.3390/e21030250 ◽

2019 ◽

Vol 21 (3) ◽

pp. 250

Author(s):

Frédéric Barbaresco ◽

Jean-Pierre Gazeau

Keyword(s):

Fourier Analysis ◽

Heat Equation ◽

Harmonic Analysis ◽

Lie Groups ◽

Coherent States ◽

Geometric Theory ◽

Plancherel Formula ◽

Group Representations ◽

Modern Development ◽

Xxist Century

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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