## Holomorphy and convexity in Lie theory.(English)Zbl 0936.22001

de Gruyter Expositions in Mathematics. 28. Berlin: de Gruyter. xxi, 778 p. (1999).
The main purpose of the author of this book is the introduction into the “circle of ideas connecting holomorphic and unitary representations with invariant convexity in Lie algebras”. The author is a well-known German mathematician, his area of interests contains the Lie theory of semigroups and representation theory. This monograph is a remarkable collection of facts, theorems and theories concerning such subjects as holomorphy and convexity in Lie theory. The book is divided into five parts A-E. In Part A, consisting of Chapters I–IV, the abstract representation theory of involutive semigroups is given. In Chapter I the author develops the theory of positive kernels and reproducing kernel Hilbert spaces. Positive definite kernels are a tool to deal with Hilbert spaces of functions on a set $$X$$ for which the evaluations in points of $$X$$ are continuous. In Chapter II the representation theory of involutive semigroups, i.e. semigroups $$S$$ endowed with an involutive antiautomorphism $$s\mapsto s^*$$, is considered. In Chapter IV the author studies the relation between several continuity concepts for representations of topological involutive semigroups, whereas in Chapters I–III any topology is not taken into account. Part B consists of Chapters V and VI, where Chapter V provides the background on convex sets and convex functions in finite-dimensional vector spaces. Here the author develops results and facts on convex functions that will be needed in the remainder of the book. Chapter VI deals with the representation theory of cones and tubes. The representation theory of open cones $$W$$ in real vector spaces $$V$$ and the associated tube domains $$\Gamma_V(W):=V+iW$$ is given. Since tubes are the simplest examples of Ol’shanskii semigroups, this chapter can be viewed as a discussion on holomorphic representation theory for the simple class of vector groups. Using the enveloping $$C^*$$-algebras of an open cone, the author obtains a generalization of Nussbaum’s theorem on the representation of positive definite functions as Laplace transforms of positive measures, which is a direct link to the theory of convex functions. Part C contains the theory of invariant convex sets and cones in a Lie algebra $$\mathfrak g$$, and invariant convex sets and coadjoint orbits in the dual space $$\mathfrak g^*$$ (Chapters VII and VIII). In Chapter VII convexity in Lie algebras is investigated. The notion of invariance of objects related to a Lie algebra $$\mathfrak g$$ (resp., $$\mathfrak g^*$$) refers to the adjoint (resp., coadjoint) action. The reduction of questions concerning invariant convex sets directly leads to the class of admissible Lie algebras: a Lie algebra $$\mathfrak g$$ is admissible if it contains a generating invariant convex set $$C$$ not containing affine lines. The existence of such a set has remarkable consequences for the structure of $$\mathfrak g$$. For example, it implies the existence of a Cartan subalgebra $$\mathfrak t$$ which is compactly embedded, i.e. the closure of the group $$e^{\text{ ad } \mathfrak t}\subseteq \operatorname{Aut}(\mathfrak g)$$ is compact. The first step of analysis of the fine points concerning convexity in Lie algebras can be found in Chapter VIII. After infinitesimal level in Part C, the author passes on to the global level in Part D (Chapters IX–XII). In Chapters IX–XI the theory of highest weight representations on the three levels of Lie algebras, Lie groups and complex semigroups is given. In Chapter IX the algebraic side of the theory of unitary highest weight representations is investigated. Section IX.I contains general facts, and in Section IX.2 necessary conditions for the existence of faithful unitary highest weight representations are discussed. In particular, it is shown that the Lie algebra under consideration must be admissible. In Section IX.5 the qualitative picture of the classification for simple Lie algebras is described. In Chapters X, XI, XII the methods of Chapter IX are applied to several other situations. For holomorphic representations in Chapter XI a global representation theory is developed and the decomposition of arbitrary representations into irreducible ones which are highest weight representations in some sense is shown. Chapter XII is devoted to convenient realizations of highest weight representations in spaces of holomorphic functions on domains in complex vector spaces. In Part E (Chapters XIII-XV) the close relation between complex geometry and representation theory which is well-visible for complex Ol’shanskii semigroups and highest weight representations is considered. In Chapter XIII the holomorphic representation theory is used to explore the complex geometry of Ol’shanskii semigroups, and in Chapter XIV the complex analysis of these semigroups is studied by investigating two paradigmatic classes of Hardy spaces. Since the complex manifold $$S=\Gamma_G(W):=G\text{ Exp}(iW)$$ is a semigroup, the connected Lie group $$G$$ also acts from the right, so that we obtain a natural action of $$G\times G$$ on this space. Objects invariant under this action are called biinvariant. Important information on the complex geometry of this action is contained in the knowledge of the biinvariant domains of holomorphy and the biinvariant plurisubharmonic functions. A biinvariant domain $$G\text{ Exp}(iB) \subseteq S$$ is Stein if and only if $$B$$ is convex, and a biinvariant function $$\phi$$ on such a domain is plurisubharmonic if and only if the corresponding invariant function $$X\mapsto \phi(\text{Exp }iX)$$ on $$B\subseteq \mathfrak g$$ is locally convex. The subject of Chapter XIV is the turn from complex geometry to complex analysis. It is proved that each biinvariant Hilbert space of holomorphic functions on a biinvariant domain $$D$$ is a direct integral of irreducible representations of $$G\times G$$ which are highest weight representations. The end of this chapter is dedicated to two types of Hardy spaces on complex Ol’shanskii semigroups. The author discusses analogs of the classical $$H^{\infty}$$-spaces, and the analog of the classical $$H^2$$-spaces. The main result of Chapter XV is the characterization of unitary highest weight representations establishing an additional link between representation theory and complex geometry. It is shown that if $$(\pi, {\mathcal H})$$ is an irreducible unitary highest weight representation of a connected admissible Lie group $$G$$ with discrete kernel, then $$\pi$$ is a unitary highest weight representation if and only if some $$G$$-orbit $$G.[v]$$ in the projective space $$P({\mathcal H}^\infty)$$ is a complex manifold. Such orbits are called coherent state orbits. Note that in the classical setup of the oscillator representation of the oscillator group on $$L^2(\mathbb R)$$, i.e. the representation corresponding to the quantum mechanical harmonic oscillator with a quadratic potential, the coherent states can be characterized as Gaussian functions, hence as functions minimizing the product of the “uncertainties” of the position and momentum operators. Finally, in Section XV.3 an abstract version of Heisenberg’s Uncertainty Principle in the context of coherent state orbits is discussed.
Reviewer: A.K.Guts (Omsk)

### MSC:

 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 22E15 General properties and structure of real Lie groups 22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 17B05 Structure theory for Lie algebras and superalgebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 32E10 Stein spaces 32U05 Plurisubharmonic functions and generalizations 43A35 Positive definite functions on groups, semigroups, etc. 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 81R30 Coherent states