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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 L2(Γ\G), on which G acts by right translations. If Γ\G is 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 L2(Γ\G) splits into a discrete spectrum Ld2 which behaves as if Γ\G were compact, and a continuous spectrum Lc2 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 Lc2 however, one analytically continues them to meromorphic functions over all of C, and shows them to satisfy functional equations.

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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