Spectral decomposition and Eisenstein series.
(Décomposition spectrale et séries d’Eisenstein. Une paraphrase de l’écriture.)

*(French)*Zbl 0794.11022
Progress in Mathematics (Boston, Mass.). 113. Basel: Birkhäuser Verlag. xxix, 341 p. (1994).

Let \(\Delta\) be the Laplace-Beltrami operator on the upper half plane \(H= \{z= x+iy\mid \text{Im }z>0\}\); with respect to the hyperbolic metric it is given as \(\Delta= y^ 2 (\partial^ 2/ \partial x^ 2+ \partial^ 2/ \partial y^ 2)\). The arithmetic interest in the eigenfunctions of \(\Delta\) invariant under the modular group \(SL_ 2(\mathbb{Z})\) and its congruence subgroups \(\Gamma\) as shown in the work of E. Hecke and H. Maass, is still there. In particular, the study of the so called Hecke operators (they commute with \(\Delta\) and act on its eigenspaces) plays an important role in dealing with diophantine problems. For example, recall the link between zeta functions of Shimura varieties and automorphic \(L\)-functions as exhibited in the work of M. Eichler and G. Shimura in the case \(GL_ 2\). The congruence relation, in its generalization of symplectic groups, establishes a close relation between the eigenvalues of Frobenius and the eigenvalues of the Hecke operators.

The spectral theory of the Laplace-Beltrami operator on \(\Gamma\)- invariant functions is an analytic problem of interest in its own right for any discrete subgroup \(\Gamma\) of the special linear group \(SL_ 2(\mathbb{R})\) whose fundamental domain has finite volume. If the quotient \(H\) by \(\Gamma\) is compact the spectrum is discrete, but otherwise there is a continuous spectrum built up by so called Eisenstein series. The construction of the eigenfunctions for the continuous spectrum via an analytic continuation of a given series attached to a cusp, was given by W. Roelcke for congruence subgroups, in which case these Eisenstein series reduce to familiar series. Then a complete solution of the problem was indicated by A. Selberg in 1956. The essential tool for the analytic continuation is provided by certain inequalities for the coefficients occurring in the constant term at the cusp attached to the Eisenstein series in question. However, Selberg never published a complete proof.

Beyond the spectral theory, Selberg derived, among other things, a finite closed formula for the trace of the Hecke operators on the space of modular forms of weight \(2k\), for \(k>1\), and given level. Later on, Selberg outlined an argument for establishing a trace formula for a non- compact quotient (but of finite volume) when the underlying group \(G\) has real rank one. At that time, by various reasons, it turned out that the proper setting for the theory of automorphic forms is a given reductive group and an arithmetically defined subgroup \(\Gamma\). Many aspects of this theory are nothing but a detailed analysis of the (infinite- dimensional) regular representation of \(G\) on the Hilbert space \(L^ 2(\Gamma \setminus G)\). During the International Congress, Stockholm 1962, both I. M. Gelfand and A. Selberg stressed the general problems posed in the theory. The latter emphasized the importance of establishing a trace formula in general, and of applying it to the theory of Hecke operators. The former discussed the spectral aspects, i.e. that of explicitly describing the decomposition of the regular representation of \(G\) on \(L^ 2(\Gamma \setminus G)\) into a direct integral of irreducible representations. A decisive step in developing the theory of automorphic forms on groups of higher rank was the introduction by Gelfand in 1962 of the notion of cusp forms. As a result to be mentioned, the representation on the space of cusp forms is a discrete sum of irreducible representations when \(G\) is semisimple.

In 1963/64, R. P. Langlands wrote his treatise “On the functional equations satisfied by Eisenstein series” (published in 1976, Lect. Notes Math. 544, Springer (see Zbl 0332.10018)] where he gave an account of the spectral theory of Eisenstein series, viewing his study as a preliminary to the development of a trace formula. There he carried out the spectral decomposition and the theory of the general Eisenstein series simultaneously, by an inductive procedure.

The book under review by C. Moeglin and J. L. Waldspurger is claimed to be an exposition of Langlands’ book alluded to above, ‘une paraphrase de l’écriture’, as the authors have put it. First, the book does what it says it will do, namely to give a coherent presentation of the theory of Eisenstein series, its constant terms, their analytic continuation, its functional equations, the intertwining operators involved. It contains a complete treatment of the spectral decomposition, i.e., a construction of the discrete spectrum by means of residues of Eisenstein series resp. a description of the continuous spectrum by means of the discrete spectrum attached to Levi subgroups of \(G\). It is a demanding but rewarding book on a subject playing a central role in an important and even spreading part of mathematics. But second, it is more: The book of Moeglin and Waldspurger is more comprehensive and general in its outlet as Langlands’ book; the case of non-connected groups is dealt with, and groups defined over a function field is treated as well. It makes the subject (i.e. one of the central themes in the current theory of automorphic forms) accessible to mathematicians with even little background in the automorphic theory. However, a solid knowledge of algebraic group theory is required.

The book is divided into six chapters. Chapter I mainly deals with the general theory of automorphic forms and their constant terms and the way in which an automorphic form is determined by its constant terms. In chapter II, Eisenstein series and so called pseudo-Eisenstein series, i.e. certain integrals of Eisenstein series, are introduced and their basic properties are discussed. The computation of the scalar product of two pseudo-Eisenstein series is given at the end. As a consequence, a first orthogonal decomposition of \(L^ 2(G(k)\setminus G(\mathbb{A}_ k))\) is given. Chapter IV deals with the analytic continuation of Eisenstein series resp. related intertwining operators. The proof is based on an approach due to H. Jacquet. By use of results in III on Hilbert operators, various properties of singularities of Eisenstein series and intertwining operators are derived. The functional equation is proved. Finally, Chapters V and VI give a spectral decomposition. An elaborate induction is required.

There are four appendices, the first one of a more technical nature, the second one dealing with a new result on automorphic forms and Eisenstein series over a function field, namely, that the automorphic form may be written as a linear combination of derivatives of Eisenstein series.

Appendix III gives some interesting computations concerning the discrete spectrum for \(G_ 2\); this complements the appendix III in Langlands’ book. Finally, appendix IV gives the necessary modifications for extending the theory to the case of non-connected groups \(G\).

P.S.: There is an English edition to come.

The spectral theory of the Laplace-Beltrami operator on \(\Gamma\)- invariant functions is an analytic problem of interest in its own right for any discrete subgroup \(\Gamma\) of the special linear group \(SL_ 2(\mathbb{R})\) whose fundamental domain has finite volume. If the quotient \(H\) by \(\Gamma\) is compact the spectrum is discrete, but otherwise there is a continuous spectrum built up by so called Eisenstein series. The construction of the eigenfunctions for the continuous spectrum via an analytic continuation of a given series attached to a cusp, was given by W. Roelcke for congruence subgroups, in which case these Eisenstein series reduce to familiar series. Then a complete solution of the problem was indicated by A. Selberg in 1956. The essential tool for the analytic continuation is provided by certain inequalities for the coefficients occurring in the constant term at the cusp attached to the Eisenstein series in question. However, Selberg never published a complete proof.

Beyond the spectral theory, Selberg derived, among other things, a finite closed formula for the trace of the Hecke operators on the space of modular forms of weight \(2k\), for \(k>1\), and given level. Later on, Selberg outlined an argument for establishing a trace formula for a non- compact quotient (but of finite volume) when the underlying group \(G\) has real rank one. At that time, by various reasons, it turned out that the proper setting for the theory of automorphic forms is a given reductive group and an arithmetically defined subgroup \(\Gamma\). Many aspects of this theory are nothing but a detailed analysis of the (infinite- dimensional) regular representation of \(G\) on the Hilbert space \(L^ 2(\Gamma \setminus G)\). During the International Congress, Stockholm 1962, both I. M. Gelfand and A. Selberg stressed the general problems posed in the theory. The latter emphasized the importance of establishing a trace formula in general, and of applying it to the theory of Hecke operators. The former discussed the spectral aspects, i.e. that of explicitly describing the decomposition of the regular representation of \(G\) on \(L^ 2(\Gamma \setminus G)\) into a direct integral of irreducible representations. A decisive step in developing the theory of automorphic forms on groups of higher rank was the introduction by Gelfand in 1962 of the notion of cusp forms. As a result to be mentioned, the representation on the space of cusp forms is a discrete sum of irreducible representations when \(G\) is semisimple.

In 1963/64, R. P. Langlands wrote his treatise “On the functional equations satisfied by Eisenstein series” (published in 1976, Lect. Notes Math. 544, Springer (see Zbl 0332.10018)] where he gave an account of the spectral theory of Eisenstein series, viewing his study as a preliminary to the development of a trace formula. There he carried out the spectral decomposition and the theory of the general Eisenstein series simultaneously, by an inductive procedure.

The book under review by C. Moeglin and J. L. Waldspurger is claimed to be an exposition of Langlands’ book alluded to above, ‘une paraphrase de l’écriture’, as the authors have put it. First, the book does what it says it will do, namely to give a coherent presentation of the theory of Eisenstein series, its constant terms, their analytic continuation, its functional equations, the intertwining operators involved. It contains a complete treatment of the spectral decomposition, i.e., a construction of the discrete spectrum by means of residues of Eisenstein series resp. a description of the continuous spectrum by means of the discrete spectrum attached to Levi subgroups of \(G\). It is a demanding but rewarding book on a subject playing a central role in an important and even spreading part of mathematics. But second, it is more: The book of Moeglin and Waldspurger is more comprehensive and general in its outlet as Langlands’ book; the case of non-connected groups is dealt with, and groups defined over a function field is treated as well. It makes the subject (i.e. one of the central themes in the current theory of automorphic forms) accessible to mathematicians with even little background in the automorphic theory. However, a solid knowledge of algebraic group theory is required.

The book is divided into six chapters. Chapter I mainly deals with the general theory of automorphic forms and their constant terms and the way in which an automorphic form is determined by its constant terms. In chapter II, Eisenstein series and so called pseudo-Eisenstein series, i.e. certain integrals of Eisenstein series, are introduced and their basic properties are discussed. The computation of the scalar product of two pseudo-Eisenstein series is given at the end. As a consequence, a first orthogonal decomposition of \(L^ 2(G(k)\setminus G(\mathbb{A}_ k))\) is given. Chapter IV deals with the analytic continuation of Eisenstein series resp. related intertwining operators. The proof is based on an approach due to H. Jacquet. By use of results in III on Hilbert operators, various properties of singularities of Eisenstein series and intertwining operators are derived. The functional equation is proved. Finally, Chapters V and VI give a spectral decomposition. An elaborate induction is required.

There are four appendices, the first one of a more technical nature, the second one dealing with a new result on automorphic forms and Eisenstein series over a function field, namely, that the automorphic form may be written as a linear combination of derivatives of Eisenstein series.

Appendix III gives some interesting computations concerning the discrete spectrum for \(G_ 2\); this complements the appendix III in Langlands’ book. Finally, appendix IV gives the necessary modifications for extending the theory to the case of non-connected groups \(G\).

P.S.: There is an English edition to come.

Reviewer: J.Schwermer (Eichstätt)

##### MSC:

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |

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