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**Automorphic forms on GL(3,\({\mathbb{R}})\).**
*(English)*
Zbl 0543.22005

Lecture Notes in Mathematics. 1083. Berlin etc.: Springer-Verlag. XI, 184 p. DM 26, 50 (1984).

From the author’s preface: ”The theory of automorphic forms on GL(3) was greatly advanced by the work of Jacquet, Piatetski-Shapiro and Shalika, who proved the converse theorem in Hecke theory, and in the process developed much important machinery. Their work, as that of most experts, uses the full machinery of representation theory. In these notes, we shall attempt to lay a reasonable foundation, from a classical point of view, for the study of automorphic forms on \(GL(3,{\mathbb{R}})\). Here ”classical” means that we shall consider the forms as defined on the real group, rather than the adèle group, and that we shall avoid, as much as possible, the language of representation theory.

The main topics we shall consider are the theory of Whittaker functions, their differential equations, and their analytic continuation and functional equations; Fourier expansions on \(GL(3)\); the Fourier expansions of the Eisenstein series, and the theory of Ramanujan sums on \(GL(3)\), which arise in the Fourier expansions; the interpretation of the Fourier coefficients of the Eisenstein series as generalized divisor sums, expressed in terms of Schur polynomials; the L-series associated with an automorphic form, their analytic continuation and functional equations; Hecke operators, and the Euler product satisfied by the L- series associated with an automorphic form; double L-series, and the double Mellin transform of the Whittaker functions. Some of the results are new, but the general outline of this theory has been known to experts for several years now. - The biggest novelty of our approach is that it is very direct and explicit.-”

I should have added italics to this last sentence in order to emphasize that the author’s approach is indeed remarkably concrete. For example, in Chapter VII he describes the Fourier expansion of an Eisenstein series on \(GL_ 3({\mathbb{R}})\) explicitly in terms of generalized divisor sums, and then Chapter X he exploits this description in order to compute Mellin transforms of Whittaker functions on \(GL_ 3({\mathbb{R}})\). These results complement the (Archimedean) theory of Jacquet, Shalika and Piatetski- Shapiro, and should be useful in applications of Hecke theory to number theory. Though technical in nature, these notes are very clearly written; moreover, the author has included explanations of how his computations fit in with the existing (less classical) theory of automorphic forms and L-series.

The main topics we shall consider are the theory of Whittaker functions, their differential equations, and their analytic continuation and functional equations; Fourier expansions on \(GL(3)\); the Fourier expansions of the Eisenstein series, and the theory of Ramanujan sums on \(GL(3)\), which arise in the Fourier expansions; the interpretation of the Fourier coefficients of the Eisenstein series as generalized divisor sums, expressed in terms of Schur polynomials; the L-series associated with an automorphic form, their analytic continuation and functional equations; Hecke operators, and the Euler product satisfied by the L- series associated with an automorphic form; double L-series, and the double Mellin transform of the Whittaker functions. Some of the results are new, but the general outline of this theory has been known to experts for several years now. - The biggest novelty of our approach is that it is very direct and explicit.-”

I should have added italics to this last sentence in order to emphasize that the author’s approach is indeed remarkably concrete. For example, in Chapter VII he describes the Fourier expansion of an Eisenstein series on \(GL_ 3({\mathbb{R}})\) explicitly in terms of generalized divisor sums, and then Chapter X he exploits this description in order to compute Mellin transforms of Whittaker functions on \(GL_ 3({\mathbb{R}})\). These results complement the (Archimedean) theory of Jacquet, Shalika and Piatetski- Shapiro, and should be useful in applications of Hecke theory to number theory. Though technical in nature, these notes are very clearly written; moreover, the author has included explanations of how his computations fit in with the existing (less classical) theory of automorphic forms and L-series.

Reviewer: S.Gelbart

### MSC:

22E30 | Analysis on real and complex Lie groups |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

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

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

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11F12 | Automorphic forms, one variable |