Maass–Selberg relations
Encyclopedia
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. introduced the Maass–Selberg relations for the case of real analytic Eisenstein series on the upper half plane. extended the relations to symmetric spaces of rank 1. generalized the Maass–Selberg relations to Eisenstein series of higher rank semisimple group (and named the relations after Maass and Selberg). found some analogous relations between Eisenstein integral
s, 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.
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 and in analytic number theory...
, that in some sense say that distinct Eisenstein series are orthogonal. introduced the Maass–Selberg relations for the case of real analytic Eisenstein series on the upper half plane. extended the relations to symmetric spaces of rank 1. generalized the Maass–Selberg relations to Eisenstein series of higher rank semisimple group (and named the relations after Maass and Selberg). found some analogous relations between Eisenstein integral
Eisenstein integral
In mathematical representation theory, the Eisenstein integral is an integral introduced by in the representation theory of semisimple Lie groups, analogous to Eisenstein series in the theory of automorphic forms....
s, 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.