H-type Groups Research Articles

Spherical functions on harmonic extensions ofH-type groups

We consider the harmonic extension AN of an H-type group N with Lie algebra n = v + z, and [v, v] = z. We characterize the positive definite spherical functions on AN.

Modified H-type groups and symmetric-like Riemannian spaces

We give a complete classification of the modified H-type groups which belong to each one of the following classes of symmetric-like Riemannian spaces: naturally reductive spaces. Riemannian g.o. spaces, weakly symmetric spaces, commutative spaces and D'Atri spaces.

A Paley-Wiener theorem on NA harmonic spaces

Let N be an H-type group and consider its one-dimensional solvable extension NA, equipped with a suitable left-invariant Riemannian metric. We prove a Paley-Wiener theorem for nonradial functions on NA supported in a set whose boundary is a horocycle of t

Weakly symmetric groups of Heisenberg type

We prove that an H-type group is weakly symmetric if and only if it is commutative. As a consequence we get the classification of all weakly symmetric H-type groups.

The Genus-zero problem for n|h-type groups

The Genus-zero problem for n|h-type groups

Harmonic analysis on solvable extensions of H-type groups

To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space. This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric. We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra. This makesS an example of a harmonic manifold, not necessarily symmetric. In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context. We finally determine the spherical functions for the groupsS and their Martin boundary.

A class of nonsymmetric harmonic Riemannian spaces

Certain solvable extensions of H-type groups provide noncompact counterexamples to a conjecture of Lichnerowicz, which asserted that "harmonic" Riemannian spaces must be rank 1 symmetric spaces.

H-type groups and Iwasawa decompositions

Since their introduction by A. Kaplan [Kpl] some ten years ago, generalised Heisenberg groups, also known as groups of Heisenberg type or H-type groups, have provided a framework in which to construct interesting examples in geometry and analysis (see, for instance, [C2], [Kp2], [Kp3], [KpR], [K2], [Rl], [R2], [TL], [TV]). The Iwasawa N-groups associated to all the real rank one simple Lie groups are H-type, so one has a convenient vehicle for studying these in a unified way: many problems on these simple Lie groups can be reduced to a problem on H-type groups, via the so-called noncompact picture, and often problems on H-type groups can be solved on all the groups of the family in one fell swoop (as in, for example, [CH], [CK], [DR]). Out of this approach to studying simple Lie groups several problems arise, such as why only some H-type groups correspond to simple Lie groups of real rank one. In this paper, we discuss various features of Iwasawa N-groups which distinguish them in the class of all H-type groups. We shall show that all H-type groups which possess certain geometric properties, clearly possessed by Iwasawa N-groups, satisfy a Lie-algebraic condition (implicit in the work of B. Kostant [Kt2]) that we shall call the J’-condition. We shall also use elementary Clifford algebra to classify the

A multiplier theorem for H-type groups

We prove an $L^p$-boundedness result for a convolution operator with rough kernel supported on a hyperplane of a group of Heisenberg type.