Unitary representation theory of exponential Lie groups
A finite dimensional real Lie group \(G\) is called exponential if the exponential function \(\text{exp}: {\mathfrak g} \to G\) is a diffeomorphism from the Lie algebra \(\mathfrak g\) of \(G\) onto \(G\). It is called solvable if the commutator series \({\mathfrak g},{\mathfrak g}_1 = [{\mathfrak g},{\mathfrak g}]\), \({\mathfrak g}_2 = [{\mathfrak g}_1,{\mathfrak g}_1], \dots\) of \(\mathfrak g\) reaches zero. Bernat showed, generalizing a fundamental result of Kirillov, that the irreducible unitary representations of an exponential solvable Lie group can be parametrized by the \(G\)-orbits in the dual space \({\mathfrak g}^*\) of \(\mathfrak g\) under the coadjoint action. The map \(K : {\mathfrak g}^*/ G \to \widehat {G}\) which associates to a coadjoint orbit the corresponding representation is called the Kirillov map. Both domain and range of the Kirillov map carry natural topologies: On \({\mathfrak g}^*/G\) one has the quotient topology of the Euclidean topology and on \(\widehat{G}\) the induced topology from the projection onto the primitive ideal space of the group \(C^*\)-algebra with its Jacobson topology. The book under review is devoted to the proof of the fact that the Kirillov map is a homeomorphism with respect to these topologies. Kirillov had conjectured this in his 1962 paper for the nilpotent case and I. D. Brown proved it in that case in 1973. The exponential solvable case was settled by J. Ludwig in the late eighties. A crucial ingredient in the proof of Kirillov's conjecture in the exponential case is the possibility to modify the group structure along the way. This allows to write certain positive definite functions in terms of positive definite functions of lower dimensional modified subgroups and use induction to prove the convergence of orbits from the convergence of positive definite functions. Ludwig introduced the concept of variable structures such as \textit{Variable Lie Groups} (VLG) and \textit{Variable Banach Algebras} (VLA) to give the term \textit{modify} a precise meaning. They essentially consist of a family of algebraic structures (with additional structure elements like topologies, norms etc.) together with a parameter space and certain compatibility conditions between the single structures. It is then easy to formulate a parametrized version of Kirillovs conjecture and it is this parametrized version that is proved in the book. Apart from the idea of variable structures the proof of Kirillov's conjecture is a blend of non-commutative abstract harmonic analysis and quite concrete structure theory for exponential solvable Lie groups which makes heavy use of special bases for the Lie algebras and induction on the dimension. Since there is not much textbook literature on solvable Lie groups and their unitary representations available, the authors devote about one half of the book to the exposition of background material such as the structure of induced representations, Mackey's imprimitivity theorem, polarization and the construction of the Kirillov map. Nevertheless this is not an introductory text but a book for specialists which wants to be reasonably selfcontained. In particular the proof of the continuity of the inverse Kirillov map with its many different cases to consider is by no means easy reading.
scientific article; zbMATH DE number 715335