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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

References

1. Borel, A., Seminar on algebraic groups and related finite groups, Springer Lecture notes, 131.Google Scholar
2. Borel, A. and Tits, J., Groupes réductifs, Publ. I.H.E.S. 27 (1965), 55–151.Google Scholar
3. Bourbaki, N., Groupes et algèbres de Lie IV, V, VI (Hermann, Paris, 1968).Google Scholar
4. Demazure, M. and Grothendieck, A., Schémas en groupes III, Springer lecture notes 153 ( = S.G.A.D. in this paper).Google Scholar
5. Godement, R., Domaines fondamentaux des groupes arithmétiques, Séminaire Bourbaki 257 (1962/63).Google Scholar
6. Godement, R., Introduction à la théorie deLanglands, Séminaire Bourbaki 321 (1966/67).Google Scholar
7. Godement, R. and Jacquet, H., Zêta functions of simple algebras, Springer lecture notes 260.Google Scholar
8. Harder, G., Halbeinfache Gruppenschemata ùber vollstdndigen Kurven, Inv. Math. 6 (1968), 107–149.Google Scholar
9. Harder, G., Minkowskische Reduktionsthéorie ùber Funktionenkôrpern, Inv. Math. 7 (1969), 33–54.Google Scholar
10. Harder, G., Chevalley groups over function fields and automorphic forms, Ann. of Math. 100 (1974), 249–306.Google Scholar
11. Harish-Chandra, , Automorphic forms on semi-simple Lie groups, Springer Lecture Notes 62. Google Scholar
12. Langlands, R. P., Eisenstein series, Proc. Symp. Pure Math. (A.M.S., Providence 9, 1966).Google Scholar
13. Langlands, R. P., On the functional equations satisfied by Eisenstein series, Springer lecture notes 544- Google Scholar
14. Mahler, K., An analogue to Minkowski1 s geometry of numbers in afield of series, Ann. of Math. 42 (1941).Google Scholar
15. Stone, M. H., Linear transformations in Hilbert space, and their applications to analysis, A.M.S. Coll. Publ. (1932), N.Y. Google Scholar
16. Tits, J., Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, J. reine angew. Math. 247 (1971), 196–220.Google Scholar