Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T14:17:23.841Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Analytic Continuation of c-Functions

Published online by Cambridge University Press:  20 November 2018

Paul F. Ringseth
Paul F. Ringseth*
Affiliation:
University of Washington, Seattle, Washington
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let G be a reductive Lie group; Γ a nonuniform lattice in G. Then the theory of Eisenstein series plays a major role in the spectral decomposition of L2(G/Γ) (cf. [5]). One of the most difficult aspects of the subject is the analytic continuation of the Eisenstein series along with its associated c-function. This was originally done by Langlands using some very difficult analysis (cf. [5]). Later Harish-Chandra was able to simplify somewhat the most difficult part of the continuation, the continuation to zero, by the introduction of the Maas-Selberg relation. The purpose of this note is to give a simplified account of this particular part of the theory.

Our chief tool will be the truncation operator of Arthur (cf. [1] and [8]), the systematic utilization of which has the effect of streamlining the earlier accounts, especially in so far as continuation to zero is concerned, which is reduced to an elementary manipulation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 3 , 01 June 1988 , pp. 513 - 531
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Arthur, J., A trace formula for reductive groups II: Applications of a truncation operator, Compositio Mathematica 40 (1980), 87–121.Google Scholar
2. Arthur, J., On the inner product of truncated Eisenstein series, Duke Mathematical Journal 49 (1982), 35–70.Google Scholar
3. Verdière, Y. Colin de, Une nouvelle démonstration du prolongement méromorphe des séries d'Eisenstein, Lecture Notes in Math. 293 (1981), 361–363.Google Scholar
4. Harish-Chandra, , Automorphic forms on semisimple Lie groups, Lecture Notes in Math. 62 (Springer-Verlag, 1968).CrossRefGoogle Scholar
5. Langlands, R., On the functional equations satisfied by Eisenstein series, Lecture Notes in Math. 544 (Springer-Verlag, 1976).CrossRefGoogle Scholar
6. Langlands, R., Eisenstein series, Proc. Symp. Pure Math. 9, 235–252.Google Scholar
7. Osborne, M. S. and Warner, G., The theory of Eisenstein systems (Academic Press, 1981).Google Scholar
8. Osborne, M. S. and Warner, G., Selberg trace formula II, Pacific J. Math. 106 (1983), 307–496.Google Scholar
9. Osborne, M. S. and Warner, G., Selberg trace formula IV, Lecture Notes in Math. 1024 (Springer-Verlag, 1983), 112–263.Google Scholar
10. Selberg, A., Discontinuous groups and harmonic analysis, Proceedings of the International Congress of Mathematicians (1962), 177–189.Google Scholar
11. Yosida, K., Functional analysis, Grundlehren der mathematischen Wissenschaften 123 (Springer-Verlag, 1978).Google Scholar