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Eisenstein Series for Reductive Groups over Global Function Fields I.

The Cusp Form Case

Published online by Cambridge University Press:  20 November 2018

L. E. Morris
L. E. Morris*
Affiliation:
Clark University, Worcester, Massachusetts
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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.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 1 , 01 February 1982 , pp. 91 - 168
Copyright
Copyright © Canadian Mathematical Society 1982

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