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This book presents a number of important contributions focusing on harmonic analysis and representation theory of Lie groups. All were originally presented at the 5th Tunisian–Japanese conference “Geometric and Harmonic Analysis on Homogeneous Spaces and Applications”, which was held at Mahdia in Tunisia from 17 to 21 December 2017 and was dedicated to the memory of the brilliant Tunisian mathematician Majdi Ben Halima. The peer-reviewed contributions selected for publication have been modified and are, without exception, of a standard equivalent to that in leading mathematical periodicals. Highlighting the close links between group representation theory and harmonic analysis on homogeneous spaces and numerous mathematical areas, such as number theory, algebraic geometry, differential geometry, operator algebra, partial differential equations and mathematical physics, the book is intended for researchers and students working in the area of commutative and non-commutative harmonic analysis as well as group representations.



Monomial Representations of Discrete Type of an Exponential Solvable Lie Group

Let G be an exponential solvable Lie group, H an analytic subgroup of G and \(\chi \) a unitary character of H. We study some problems related to the induced representation \(\tau = \text {ind}_H^G \chi \) of G when \(\tau \) has multiplicities either finite or infinite of discrete type. In particular, we are interested in the Plancherel formula for \(\tau \) and the commutativity problem due to Duflo (Open problems in representation theory of Lie groups, edited by T. Oshima, Katata, Japan 1986, [12]) for the algebra \(D_{\tau }(G/H)\) of G-invariant differential operators on the fiber space associated to the data \((H,\chi )\) over the base space G / H. We give in particular an example where this problem can admit a negative solution in the frame of exponential solvable Lie groups.
Ali Baklouti, Hidenori Fujiwara, Jean Ludwig

Self-Chabauty-isolated Locally Compact Groups

Let G be a locally compact group. We denote by \({\mathcal {SUB}}\left( G\right) \) the space of closed subgroups of G equipped with the Chabauty topology. The group G is called self-Chabauty-isolated if the point G is isolated in \({\mathcal {SUB}}\left( G\right) \). In this paper we are interested in the following question: Give necessary and sufficient conditions for the group G to be a self-Chabauty-isolated.
Hatem Hamrouni, Firas Sadki

Quantization of Color Lie Bialgebras

The main purpose of this paper is to study Quantization of color Lie bialgebras, generalizing to the color case the approach by Etingof–Kazhdan which was considered for superbialgebras by Geer. Moreover we discuss Drinfeld category, Quantization of Triangular color Lie bialgebras and Simple color Lie bialgebras of Cartan type.
Benedikt Hurle, Abdenacer Makhlouf

Harmonic Analysis for 4-Dimensional Real Frobenius Lie Algebras

A real Frobenius Lie algebra is characterized as the Lie algebra of a real Lie group admitting open coadjoint orbits. In this paper, we study irreducible unitary representations corresponding to open coadjoint orbits for each of 4-dimensional Frobenius Lie algebras. We show that such unitary representations are square-integrable, and their Duflo–Moore operators are closely related to the Pfaffian of the Frobenius Lie algebra.
Edi Kurniadi, Hideyuki Ishi

An Example of Holomorphically Induced Representations of Exponential Solvable Lie Groups

We discuss a holomorphically induced representation \(\rho =\rho (f,\mathfrak h)\) of Boidol’s group (split oscillator group) G from a real linear form f of the Lie algebra \(\mathfrak g\) of G and a one-dimensional complex subalgebra \(\mathfrak h\) of \(\mathfrak g_\mathbb C\) given by (2.2) in Sect. 2.\(\rho \) is a subrepresentation of the regular representation of G with the Plancherel measure \(\nu \). For \(\nu \)-almost all irreducible representations \(\pi \) of G, the spaces of generalized vectors satisfying the semi-invariance associated with f and \(\mathfrak h\) are one-dimensional subspaces. On the other hand, according to the choice of f, there are two cases that (1) \(\rho \) vanishes, and (2) \(\rho \) is non-zero.
Junko Inoue

Spherical Functions for Small K-Types

For a connected semisimple real Lie group G of non-compact type, Wallach introduced a class of K-types called small. We classify all small K-types for all simple Lie groups and prove except just one case that each elementary spherical function for each small K-type \((\pi ,V)\) can be expressed as a product of hyperbolic cosines and a Heckman–Opdam hypergeometric function. As an application, the inversion formula for the spherical transform on \(G\times _K V\) is obtained from Opdam’s theory on hypergeometric Fourier transforms.
Hiroshi Oda, Nobukazu Shimeno

A Cartan Decomposition for Non-symmetric Reductive Spherical Pairs of Rank-One Type and Its Application to Visible Actions

A Cartan decomposition for symmetric pairs plays an important role to study not only orbit geometry of the symmetric spaces but also harmonic analysis on them. For non-symmetric reductive pairs, there are examples of generalizations of Cartan decompositions for some spherical complex homogeneous spaces such as complex line bundles over the complexified Hermitian symmetric spaces and triple spaces. This paper provides new examples of a Cartan decomposition for non-symmetric reductive pairs, namely, reductive non-symmetric spherical pairs of rank-one type. We also show that the action of some compact group on a non-symmetric reductive spherical homogeneous space of rank-one type is strongly visible.
Atsumu Sasaki

Lagrangian Submanifolds of Standard Multisymplectic Manifolds

We give a detailed, self-contained proof of Geoffrey Martin’s normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein’s famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential topology, i.e., a foliated tubular neighborhood theorem and a foliated relative Poincaré lemma.
Gabriel Sevestre, Tilmann Wurzbacher

The Poisson Characteristic Variety of Unitary Irreducible Representations of Exponential Lie Groups

We recall the notion of Poisson characteristic variety of a unitary irreducible representation of an exponential solvable Lie group, and conjecture that it coincides with the Zariski closure of the associated coadjoint orbit. We prove this conjecture in some particular situations, including the nilpotent case.
Ali Baklouti, Sami Dhieb, Dominique Manchon
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