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Characters of connected Lie groups. (English) Zbl 0934.22002
Mathematical Surveys and Monographs. 71. Providence, RI: American Mathematical Society (AMS). xvii, 128 p. (1999).
The present book is about the representation theory of general connected Lie groups. Its central result is the description of a bijection between the set $$\text{Prim}(G)$$ of primitive ideals of the group $$C^*$$-algebra $$C^*(G)$$ and the quasi-equivalence classes of normal representations of $$G$$. A continuous unitary representation $$\pi$$ of $$G$$ is called normal if the von Neumann algebra generated by the image of $$C^*(G)$$ under $$\pi$$ is a factor and there exists an element $$f \in C^*(G)$$ for which $$\pi(f)$$ is a trace class operator. According to a result of J. Dixmier, the kernel of such a representation is a primitive ideal, so that the main problem is to show that every primitive ideal of $$C^*(G)$$ arises that way, and that normal representations with the same kernel are quasi-equivalent. These results are made more explicit for connected solvable Lie groups in terms of a generalization of Kirillov’s orbit theory to arbitrary connected solvable Lie groups. Originally Pukanszky intended to write a six chapter book on this topic, but unfortunately he died before he had finished the last two chapters. The other four chapters are published in the present book which is structured as follows: We recall that a locally algebraic group is a Lie group which is locally isomorphic to some algebraic group. Chapter I is devoted to the proof of Dixmier’s theorem that these groups are type I. As an application, one finds in Section I.4 the fact that the left regular representation of any Lie group (not necessarily unimodular) is semifinite, which is not true for every locally compact group. Chapter II deals with representations of elementary groups, i.e., central $$\mathbb T$$-extensions of closed subgroups of some $$\mathbb R^n$$. This material prepares Chapter III, where the bijection between $$\text{Prim}(G)$$ and the quasi-equivalence classes of normal representations is established. Here Pukanszky notes that many of the results in this book can be derived from more general results of D. Poguntke [Ann. Sci. Éc. Norm. Supér. (4) 16, 151-172 (1983; Zbl 0523.22007)]. The goal of Chapter IV is to describe for a simply connected solvable Lie group $$G$$ the set $$\text{Prim}(G)$$ in terms of the coadjoint action of $$G$$ on $${\mathbf g}^*$$ and thus extending the work of Kirillov and Auslander/Kostant. This is done by constructing a $$G$$-equivariant fiber space $$q : {\mathcal R} \to {\mathbf g}^*$$ (which would be trivial for nilpotent $${\mathbf g}$$) and to each $$p \in {\mathcal R}$$ one associates a semifinite factor representation $$F(p)$$ in such a way that all elements in the same $$G$$-orbit lead to equivalent representations. One thus obtains a map $$J : {\mathcal R} \to \text{Prim}(G)$$, $$p \mapsto \ker F(p)$$ whose fibers are called generalized orbits (for nilpotent $${\mathbf g}$$ these are coadjoint orbits). It is shown that generalized orbits $$O$$ carry a natural manifold structure and that the corresponding normal representation can be constructed as a directly integral of the $$F(p)$$’s over $$O$$ with respect to an absolutely continuous measure. For each generalized orbit $$O \subseteq {\mathcal R}$$ the set $${\mathcal O} := q(O) \subseteq {\mathbf g}^*$$ is an invariant subset, hence a union of coadjoint orbits. The next step is a characterization of those generalized orbits corresponding to type I primitive ideals of $$G$$, and this specializes to the theorem of Auslander-Kostant that a solvable Lie group $$G$$ is type I if and only if all coadjoint orbits are locally closed and integral. There are not many recent books on the representation theory of non-semisimple Lie groups and most of them deal exclusively with nilpotent or solvable groups (see for instance the book of H. Leptin and J. Ludwig, “Unitary representation theory of exponential Lie groups” (Berlin 1994; Zbl 0833.22012)) which deals with the class of exponential Lie groups which are “most well behaved” solvable Lie groups, whereas the key ideas of Chapter IV of the present book are concerned with generalizing the theory of coadjoint orbits to arbitrary connected solvable Lie groups, where Pukanszky introduced several completely new concepts. Since the book uses the representation theory of $$C^*$$-algebras and quite a portion of the Auslander-Moore representation theory of solvable Lie groups, the background required for reading the book in detail is considerable. On the other hand every section starts with an explanation of its goals and main results, so one is also well guided if one skips some of the details. Apart from many important results which appear for the first time in book form, the present book is a very valuable source for many techniques in the representation theory of general Lie groups. These techniques are beautiful combinations of methods in abstract harmonic analysis and others which are more specific to Lie theory and related to coadjoint orbits. I can recommend the book to everyone interested in general and abstract aspects of the representation theory of Lie groups.

##### MSC:
 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 22E15 General properties and structure of real Lie groups 22D25 $$C^*$$-algebras and $$W^*$$-algebras in relation to group representations