A. H. Dooley

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

An approach to symmetric spaces of rank one via groups of Heisenberg type

We give an elementary unified approach to rank one symmetric spaces of the noncompact type, including proofs of their basic properties and of their classification, with the development of a formalism to facilitate future computations.Our approach is based on the theory of Lie groups of H-type. An algebraic condition of H-type algebras, called J2,is crucial in the description of the symmetric spaces. The classification of H-type algebras satisfying J2 leads to a very simple description of the rank one symmetric spaces of the noncompact type.We also prove Kostant’s double transitive theorem; we describe explicitly the Riemannian metric of the space and the standard decompositions of its isometry group.Examples of the use of our theory include the description of the Poisson kernel and the admissible domains for convergence of Poisson integrals to the boundary.

Sums of adjoint orbits

We investigate a natural generalization of the problem of the description of the eigenvalues of the sum of two Hermitian matrices both of whose eigenvalues are known. We describe more generally the convolution of the invariant probability measures supported on any two adjoint orbits of a compact Lie group. Our techniques utilize the convexity results of Guillemin and Sternberg and Kirwan on the one hand, and the character formulae of Weyl and Kirillov on the other. Applications to representation theory are discussed.

Norms of Characters and Lacunarity for Compact Lie Groups

Abstract This article gives upper and lower estimates for the p -norms of irreducible characters of compact Lie groups in terms of their dimension. These estimates are applied to give some new results on lacunary sets.

The Rokhlin lemma for homeomorphisms of a Cantor set

For a Cantor set X, let Homeo(X) denote the group of all homeomorphisms of X. The main result of this note is the following theorem. Let T ∈ Homeo(X) be an aperiodic homeomorphism, let μ 1 ,μ 2 ,...,μ k be Borel probability measures on X, and let e > 0 and n > 2. Then there exists a clopen set E C X such that the sets E,TE....,T n-1 E are disjoint and μ i (E ∪ TE ∪... ∪ T n-1 E) > 1-e, i = 1,..., k. Several corollaries of this result are given. In particular, it is proved that for any aperiodic T ∈ Homeo(X) the set of all homeomorphisms conjugate to T is dense in the set of aperiodic homeomorphisins.

Contractions of rotation groups and their representations

It is a classical result in the theory of special functions that Bessel functions are limits in an appropriate sense of Legendre polynomials. For example in ( 11 ), § 17.4, the following result is attributed to Heine: such limiting formulae are also known for certain other special functions (cf. (1), (5)). Apart from their intrinsic interest, these formulae have been used in the theory of special functions to obtain product formulae, etc. for the limit function from those of the approximating sequence.

Topologies on the group of homeomorphisms of a Cantor set

Let

Non-Bernoulli systems with completely positive entropy

\text{\rm Homeo}(\Omega)

Topologies on the group of Borel automorphisms of a standard Borel space

be the group of all homeomorphisms of a Cantor set

Nonsingular dynamical systems, Bratteli diagrams and Markov odometers

\Omega