Geodesic
Eisenstein Series for Reductive Groups over Global Function Fields I.
Canadian journal of mathematics, Tome 34 (1982) no. 1, pp. 91-168

Voir la notice de l'article provenant de la source Cambridge University Press

Let G be the Lie group SL(2, R) and Γ a discrete subgroup of arithmetic type. The homogeneous space Γ\G can be equipped with an invariant measure so that there is a Hilbert space of square integrable functions, denoted L 2(Γ\G), on which G acts by right translations. If Γ\Gis compact then this Hilbert space breaks up into a countable direct sum of irreducible representations of G, each occurring with finite multiplicity. Quite often however Γ\G is not compact, but of finite volume; in this case L 2(Γ\G) splits into a discrete spectrum L d 2 which behaves as if Γ\G were compact, and a continuous spectrum L c 2 which is described by the so called theory of Eisenstein series. These are generalized eigenfunctions of the Casimir operator of G, which are parametrized by a right half plane in C, and as such are analytic functions on this half-plane; in the course of describing the continuous spectrum L c 2 however, one analytically continues them to meromorphic functions over all of C, and shows them to satisfy functional equations. 
Morris, L. E. Eisenstein Series for Reductive Groups over Global Function Fields I.. Canadian journal of mathematics, Tome 34 (1982) no. 1, pp. 91-168. doi: 10.4153/CJM-1982-009-2
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