H-type groups and Iwasawa decompositions

Abstract

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