In mathematics, the Maass–Selberg relations are some relations describing the inner products of truncated

real analytic Eisenstein series In mathematics, the simplest real analytic Eisenstein series is a special function of two variables. It is used in the representation theory of SL(2,R) and in analytic number theory. It is closely related to the Epstein zeta function. There are ma ...

, that in some sense say that distinct Eisenstein series are orthogonal.

Hans Maass Hans Maass (german: Hans Maaß; June 17, 1911, Hamburg – April 15, 1992) was a German mathematician who introduced Maass wave forms and Koecher–Maass series and Maass–Selberg relations and who proved most of the Saito–Kurokawa conjec ...

introduced the Maass–Selberg relations for the case of real analytic Eisenstein series on the upper half plane.

Atle Selberg Atle Selberg (14 June 1917 – 6 August 2007) was a Norwegian mathematician known for his work in analytic number theory and the theory of automorphic forms, and in particular for bringing them into relation with spectral theory. He was awarded t ...

extended the relations to symmetric spaces of rank 1.

Harish-Chandra Harish-Chandra Fellow of the Royal Society, FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups. ...

generalized the Maass–Selberg relations to Eisenstein series of higher rank semisimple group (and named the relations after Maass and Selberg) and 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.

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* * * * * * * Modular forms Representation theory {{Math-stub