##
**Harmonic analysis on symmetric spaces and applications. II.**
*(English)*
Zbl 0668.10033

Berlin etc.: Springer-Verlag. x, 385 p. DM 128.00 (1988).

The book under review is the second volume of a two-volume introduction to harmonic analysis on symmetric spaces. In the following we continue the review of volume I [Springer 1985; Zbl 0574.10029].

Volume II splits into two chapters of rather different length: Chapter IV (257 pp.) deals with harmonic analysis on the space \({\mathcal P}_ n\) of positive definite (n\(\times n)\)-matrices whereas Chapter V (93 pp.) is devoted to the general noncompact symmetric space.

The aim of Chapter IV is to extend to \({\mathcal P}_ n\) as many of the results of Chapter III (Vol. I). The author starts with the necessary geometric and analytic prerequisites: Iwasawa decompositions, invariant Riemannian metric, geodesics, invariant integration, use of polar coordinates, convolution, point pair invariants, structure of the ring of invariant differential operators on \({\mathcal P}_ n\). Throughout the entire book fascinating use is made of Selberg’s theory of invariant differential and integral operators.

In Section 4.2 various special functions on \({\mathcal P}_ n\) are introduced: Power and gamma functions, K-Bessel and Whittaker functions, and spherical functions. This material turns out to be necessary e.g. when analogues of Hecke’s correspondence between modular forms and Dirichlet series are considered. The gamma function for \({\mathcal P}_ n\) comes up later in functional equations for L-functions and Eisenstein series for GL(n,\({\mathbb{Z}})\), and the K-Bessel function appears in Fourier expansions of automorphic forms for GL(n,\({\mathbb{Z}})\). Various interesting equivalent definitions of spherical functions are discussed.

Section 4.3 deals with harmonic analysis on \({\mathcal P}_ n\) in polar coordinates. Here, a highlight is the investigation of the properties of the Helgason Fourier transform and in particular the inversion formula.

Next the action of GL(n,\({\mathbb{Z}})\) on \({\mathcal P}_ n\) is studied and fundamental domains are determined in Section 4.4. First the author analyzes Minkowski’s fundamental domain and then Grenier’s fundamental domain is studied. Integration over fundamental domains is used e.g. to prove the Minkowski-Hlawka-Theorem, and it is also used later in order to prove the convergence of Eisenstein series.

Section 4.5 on automorphic forms for GL(n,\({\mathbb{Z}})\) deals mainly with various types of Eisenstein series. There is a detailed discussion of Eisenstein series with lower rank automorphic forms, Selberg’s Eisenstein series for parabolic subgroups, Koecher zeta functions, method of theta functions, analytic continuation and functional equations, Hecke operators for GL(n,\({\mathbb{Z}})\) and Fourier expansions of Eisenstein series. There are also some remarks and indications about the spectral theory of the invariant differential operators, on \(L^ 2({\mathcal P}_ n/GL(n,{\mathbb{Z}}))\) and its possible applications, but the author shrinks back from giving the long and tedious details.

The final Chapter V contains a rather sketchy discussion of analysis on a general noncompact symmetric space \(X=G/K\) and it contains in particular the classification of these spaces. The author’s goal is to lay the foundations for possible extensions of the preceding chapters to other symmetric spaces, e.g. the Siegel upper half space or the three- dimensional hyperbolic space. The emphasis is more on examples than on abstract theory; the main motivation comes from number theory.

The final Section 5.2 gives a very brief sketch of the theory of automorphic forms for certain subgroups \(\Gamma\) of G acting discontinuously on the symmetric space \(X=G/K\). Attention is paid mainly to the specific cases \(\Gamma =GL(n,{\mathfrak o})\), where \({\mathfrak o}\) is the ring of integers in an algebraic number field and \(\Gamma =Sp(n,{\mathbb{Z}})\), the Siegel modular group. In particular, the author considers holomorphic Hilbert modular forms and Siegel modular forms (Fourier expansion, Eisenstein series, Hecke operators, Relations with Dirichlet series), and there are applications to number theory based on Epstein’s zeta-functions.

The book is written in the author’s typical informal style. Details of most of the proofs are left to the reader, but the author helps the reader with this task by means of explanations and cupious hints to the vast literature. There are also many indications of open problems which may lead the reader to new research.

Volume II splits into two chapters of rather different length: Chapter IV (257 pp.) deals with harmonic analysis on the space \({\mathcal P}_ n\) of positive definite (n\(\times n)\)-matrices whereas Chapter V (93 pp.) is devoted to the general noncompact symmetric space.

The aim of Chapter IV is to extend to \({\mathcal P}_ n\) as many of the results of Chapter III (Vol. I). The author starts with the necessary geometric and analytic prerequisites: Iwasawa decompositions, invariant Riemannian metric, geodesics, invariant integration, use of polar coordinates, convolution, point pair invariants, structure of the ring of invariant differential operators on \({\mathcal P}_ n\). Throughout the entire book fascinating use is made of Selberg’s theory of invariant differential and integral operators.

In Section 4.2 various special functions on \({\mathcal P}_ n\) are introduced: Power and gamma functions, K-Bessel and Whittaker functions, and spherical functions. This material turns out to be necessary e.g. when analogues of Hecke’s correspondence between modular forms and Dirichlet series are considered. The gamma function for \({\mathcal P}_ n\) comes up later in functional equations for L-functions and Eisenstein series for GL(n,\({\mathbb{Z}})\), and the K-Bessel function appears in Fourier expansions of automorphic forms for GL(n,\({\mathbb{Z}})\). Various interesting equivalent definitions of spherical functions are discussed.

Section 4.3 deals with harmonic analysis on \({\mathcal P}_ n\) in polar coordinates. Here, a highlight is the investigation of the properties of the Helgason Fourier transform and in particular the inversion formula.

Next the action of GL(n,\({\mathbb{Z}})\) on \({\mathcal P}_ n\) is studied and fundamental domains are determined in Section 4.4. First the author analyzes Minkowski’s fundamental domain and then Grenier’s fundamental domain is studied. Integration over fundamental domains is used e.g. to prove the Minkowski-Hlawka-Theorem, and it is also used later in order to prove the convergence of Eisenstein series.

Section 4.5 on automorphic forms for GL(n,\({\mathbb{Z}})\) deals mainly with various types of Eisenstein series. There is a detailed discussion of Eisenstein series with lower rank automorphic forms, Selberg’s Eisenstein series for parabolic subgroups, Koecher zeta functions, method of theta functions, analytic continuation and functional equations, Hecke operators for GL(n,\({\mathbb{Z}})\) and Fourier expansions of Eisenstein series. There are also some remarks and indications about the spectral theory of the invariant differential operators, on \(L^ 2({\mathcal P}_ n/GL(n,{\mathbb{Z}}))\) and its possible applications, but the author shrinks back from giving the long and tedious details.

The final Chapter V contains a rather sketchy discussion of analysis on a general noncompact symmetric space \(X=G/K\) and it contains in particular the classification of these spaces. The author’s goal is to lay the foundations for possible extensions of the preceding chapters to other symmetric spaces, e.g. the Siegel upper half space or the three- dimensional hyperbolic space. The emphasis is more on examples than on abstract theory; the main motivation comes from number theory.

The final Section 5.2 gives a very brief sketch of the theory of automorphic forms for certain subgroups \(\Gamma\) of G acting discontinuously on the symmetric space \(X=G/K\). Attention is paid mainly to the specific cases \(\Gamma =GL(n,{\mathfrak o})\), where \({\mathfrak o}\) is the ring of integers in an algebraic number field and \(\Gamma =Sp(n,{\mathbb{Z}})\), the Siegel modular group. In particular, the author considers holomorphic Hilbert modular forms and Siegel modular forms (Fourier expansion, Eisenstein series, Hecke operators, Relations with Dirichlet series), and there are applications to number theory based on Epstein’s zeta-functions.

The book is written in the author’s typical informal style. Details of most of the proofs are left to the reader, but the author helps the reader with this task by means of explanations and cupious hints to the vast literature. There are also many indications of open problems which may lead the reader to new research.

Reviewer: J.Elstrodt

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

11F27 | Theta series; Weil representation; theta correspondences |

43A85 | Harmonic analysis on homogeneous spaces |

11M35 | Hurwitz and Lerch zeta functions |

22E30 | Analysis on real and complex Lie groups |

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |

53C35 | Differential geometry of symmetric spaces |