Maass–Selberg relations

In mathematics, the Maass–Selberg relations are some relations describing the inner products of truncated real analytic Eisenstein series, that in some sense say that distinct Eisenstein series are orthogonal. Maass (1949, p.169–170, 1964, p. 195–215) introduced the Maass–Selberg relations for the case of real analytic Eisenstein series on the upper half plane. Selberg (1963, p.183–184) extended the relations to symmetric spaces of rank 1. Harish-Chandra (1968, p.75) generalized the Maass–Selberg relations to Eisenstein series of higher rank semisimple group (and named the relations after Maass and Selberg). Harish-Chandra (1972, 1976) found some analogous relations between Eisenstein integrals, that he also called Maass–Selberg relations.

Informally, the Maass–Selberg relations say that the inner product of two distinct Eisenstein series is zero. However the integral defining the inner product does not converge, so the Eisenstein series first have to be truncated. The Maass–Selberg relations then say that the inner product of two truncated Eisenstein series is given by a finite sum of elementary factors that depend on the truncation chosen, whose finite part tends to zero as the truncation is removed.

References

Harish-Chandra (1968), Mars, J. G. M., ed., Automorphic forms on semisimple Lie groups, Lecture Notes in Mathematics, 62, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0098434, ISBN 978-3-540-04232-7, MR 0232893
Harish-Chandra (1972), "On the theory of the Eisenstein integral", in Gulick, Denny; Lipsman, Ronald L., Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Lecture Notes in Mathematics, 266, Berlin, New York: Springer-Verlag, pp. 123–149, doi:10.1007/BFb0059640, ISBN 978-3-540-05856-4, MR 0399355
Harish-Chandra (1976), "Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula", Annals of Mathematics. Second Series, 104 (1): 117–201, doi:10.2307/1971058, ISSN 0003-486X, JSTOR 1971058, MR 0439994
Kubota, Tomio (1973), Elementary theory of Eisenstein series, Tokyo: Kodansha Ltd., ISBN 978-0-470-50920-3, MR 0429749
Maass, Hans (1949), "Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen, 121: 141–183, doi:10.1007/BF01329622, ISSN 0025-5831, MR 0031519
Maass, Hans (1964), Lal, Sunder, ed., Lectures on modular functions of one complex variable (PDF), Tata Institute of Fundamental Research Lectures on Mathematics, 29, Bombay: Tata Institute of Fundamental Research, ISBN 978-3-540-12874-8, MR 0218305
Selberg, Atle (1963), "Discontinuous groups and harmonic analysis", Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Djursholm: Inst. Mittag-Leffler, pp. 177–189, MR 0176097

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