Volume 2, Issue 3-4

Let G be a connected, simply connected nilpotent Lie group and Γ a lattice subgroup of G . The quasi regular representation decomposes as a direct sum of unitary irreducible representations of G . The nilmanifold G /Γ is called nilmanifold with flat orbits if the coadjoint orbit corresponding to any element of the spectrum of ℛ Γ is flat. In this note, we give a new multiplicity formula related to the decomposition into irreducibles of ℛ Γ in the case when G /Γ is a nilmanifold with flat orbits. As an application, we first prove that every nilmanifold with flat orbits satisfies the Moore formula. We also give partial answers to some related questions proposed by Brezin and by Corwin and Greenleaf.

We define conjugate function operators (called also “Hilbert transform”), Poisson and conjugate Poisson integrals associated with the Jacobi–Dunkl Laplacians on and on ℝ. We prove that these operators satisfy L p -boundedness, 1 < p < +∞, in appropriate Lebesgue weighted spaces.

Let (ℝ + , Δ( x ) dx ) be a Jacobi hypergroup with weight function Δ( x ) = c (sinh x ) 2 α +1 (cosh x ) 2 β +1 . As in the Euclidean case, the real Hardy space H 1 (Δ) for (ℝ + , Δ( x ) dx ) is defined as the set of all locally integrable functions on ℝ + whose radial maximal functions belong to L 1 (Δ). In this paper we give a characterization of H 1 (Δ) in terms of weighted Triebel–Lizorkin spaces on ℝ via the Abel transform. As an application, we introduce three types of atoms for (ℝ + , Δ), one of them is smooth, and give an atomic decomposition of H 1 (Δ).

We consider the unitarizability of multiplier representations of transformation groups defined on Hilbert spaces of holomorphic functions on a homogeneous bounded domain. In particular, for the Iwasawa subgroup of the holomorphic automorphism group the classification of the unitary multiplier representations is accomplished by making use of results in [Ishi, J. Funct. Anal. 167: 425–462, 1999]. As an application, the Wallach set of the homogeneous bounded domain is described.

We outline an approach of the Kirillov map for exponential groups based on deformations, and suggest a possible way towards an alternative proof of the Leptin–Ludwig bicontinuity theorem [Unitary representation theory of exponential Lie groups, W.De Gruyter, 1994.] along these lines.

A holomorphic action of a Lie group G on a connected complex manifold D is called strongly visible with slice S if G · S is open in D and if there exists an antiholomorphic diffeomorphism σ of D preserving each G -orbit in D such that σ | s = id s . This paper deals with the non-symmetric spherical variety SO(8, ℂ)/ G 2 (ℂ). We prove that a maximal compact subgroup of SO(8, ℂ) acts on SO(8, ℂ)/ G 2 (ℂ) in a strongly visible fashion. Moreover, we can take a slice to be of dimension three which coincides with the rank of this spherical variety.

We prove a general version of Paley–Wiener theorem characterising functions with finite Fourier expansions. The result covers Fourier series on compact symmetric spaces, Hermite and special Hermite expansions as particular cases. Gutzmer's formula plays an important role in the formulation and proof of the theorem.

Let G be a connected and simply connected nilpotent Lie group, and m the dimension of the generic coadjoint orbits of G . Then it is proved in [Baklouti, Smaoui, Ludwig, J. Funct. Anal. 199: 508–520, 2003] that the operator value Fourier transform norm satisfies , where and q being the conjugate of p . When the assumption on G to be simply connected is removed, we prove an analogue estimate of type , where H designates the maximal compact central subgroup of G and m the dimension of generic coadjoint orbits of the universal covering of G . The case of non-simply connected Heisenberg groups is treated as an example.