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**Invariant function spaces on homogeneous manifolds of Lie groups and applications. Translated by A. I. Zaslavsky from an original Russian manuscript.**
*(English)*
Zbl 0791.43012

Translations of Mathematical Monographs. 126. Providence, RI: American Mathematical Society. x, 131 p. (1993).

This book consists of an updating of the author’s dissertation written to obtain a Doctorate in Mathematics in 1987 at the Siberian Branch of the Russian Academy of Sciences. Before starting the review of its contents, I would like to say that it is very well written and nicely translated (there are even reasonably few errata).

The object of study on this volume are invariant spaces or algebras on homogeneous manifolds. The purpose is to find out the connection between the geometric properties of the manifold and the structure (or classification) of these invariant subspaces. An example of application of this theory (in the last chapter of the book) is a characterization of those functions \(f \in L^ 2 (B,dxdy)\) (where \(B\) is the unit disk) which are holomorphic by means of a Morera like theorem; e.g. \(f\in{\mathcal H}(B)\) if (and only if) there is a piecewise smooth Jordan curve \(\gamma\) in \(B\) such that for almost every Moebius transform \(\omega\), \(\int_{\omega (\gamma)} f(z)dz=0\). In fact, the original impulse for this work lies in the Morera theorems proved in the early 70s by the author and R. Val’skij [Sib. Mat. Zh. 12, 3-12 (1971; Zbl 0221.46052)].

The first chapter considers mainly general properties of symmetric spaces of noncompact type. A typical theorem from this chapter is the following Theorem: Let \(X\) be a connected irreducible symmetric space of the noncompact type and \(I_ 0(X)\) the connected component of the isometry group of \(X\) containing the identity: Then the following conditions are equivalent: (1) Every closed \(I_ 0(X)\)-invariant subspace of \(L^ 2(X)\) is invariant under complex conjugation. (2) All positive definite spherical functions in \(X\) are real valued. (3) The root system associated to \(X\) is neither \(E_ 6\) nor \(A_ n\), \(D_{2_{n - 1}}(n>1)\).

The second chapter is mainly about closed invariant subspaces and subalgebras of \(C_ 0(\mathbb{H}^ n)\), \(\mathbb{H}^ n\) denotes the Heisenberg group. The third chapter considers invariant spaces and algebras on the Shilov boundaries of the classical bounded hermitian domains. Finally, the fourth chapter studies several Morera type theorems as the one mentioned earlier.

The object of study on this volume are invariant spaces or algebras on homogeneous manifolds. The purpose is to find out the connection between the geometric properties of the manifold and the structure (or classification) of these invariant subspaces. An example of application of this theory (in the last chapter of the book) is a characterization of those functions \(f \in L^ 2 (B,dxdy)\) (where \(B\) is the unit disk) which are holomorphic by means of a Morera like theorem; e.g. \(f\in{\mathcal H}(B)\) if (and only if) there is a piecewise smooth Jordan curve \(\gamma\) in \(B\) such that for almost every Moebius transform \(\omega\), \(\int_{\omega (\gamma)} f(z)dz=0\). In fact, the original impulse for this work lies in the Morera theorems proved in the early 70s by the author and R. Val’skij [Sib. Mat. Zh. 12, 3-12 (1971; Zbl 0221.46052)].

The first chapter considers mainly general properties of symmetric spaces of noncompact type. A typical theorem from this chapter is the following Theorem: Let \(X\) be a connected irreducible symmetric space of the noncompact type and \(I_ 0(X)\) the connected component of the isometry group of \(X\) containing the identity: Then the following conditions are equivalent: (1) Every closed \(I_ 0(X)\)-invariant subspace of \(L^ 2(X)\) is invariant under complex conjugation. (2) All positive definite spherical functions in \(X\) are real valued. (3) The root system associated to \(X\) is neither \(E_ 6\) nor \(A_ n\), \(D_{2_{n - 1}}(n>1)\).

The second chapter is mainly about closed invariant subspaces and subalgebras of \(C_ 0(\mathbb{H}^ n)\), \(\mathbb{H}^ n\) denotes the Heisenberg group. The third chapter considers invariant spaces and algebras on the Shilov boundaries of the classical bounded hermitian domains. Finally, the fourth chapter studies several Morera type theorems as the one mentioned earlier.

Reviewer: C.A.Berenstein (College Park)

### MSC:

43A85 | Harmonic analysis on homogeneous spaces |

53C35 | Differential geometry of symmetric spaces |

32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |

22E46 | Semisimple Lie groups and their representations |

43A90 | Harmonic analysis and spherical functions |

17B66 | Lie algebras of vector fields and related (super) algebras |