Book contents
- Front Matter
- Contents
- Preamble
- Notation
- I Hypotheses, automorphic forms, constant terms
- I.1. Hypotheses and general notation
- I.2. Automorphic forms: growth, constant terms
- I.3 Cuspidal components
- I.4. Upper bounds as functions of the constant term
- II Decomposition according to cuspidal data
- III Hilbertian operators and automorphic forms
- IV Continuation of Eisenstein series
- V Construction of the discrete spectrum via residues
- Appendix I Lifting of unipotent subgroups into a central extension
- Appendix II Automorphic forms and Eisenstein series over a function field
- Appendix III On the discrete spectrum of G2
- Appendix IV Non-connected groups
- Bibliography
- Index
I.4. - Upper bounds as functions of the constant term
from I - Hypotheses, automorphic forms, constant terms
Published online by Cambridge University Press: 22 September 2009
- Front Matter
- Contents
- Preamble
- Notation
- I Hypotheses, automorphic forms, constant terms
- I.1. Hypotheses and general notation
- I.2. Automorphic forms: growth, constant terms
- I.3 Cuspidal components
- I.4. Upper bounds as functions of the constant term
- II Decomposition according to cuspidal data
- III Hilbertian operators and automorphic forms
- IV Continuation of Eisenstein series
- V Construction of the discrete spectrum via residues
- Appendix I Lifting of unipotent subgroups into a central extension
- Appendix II Automorphic forms and Eisenstein series over a function field
- Appendix III On the discrete spectrum of G2
- Appendix IV Non-connected groups
- Bibliography
- Index
Summary
Lemma
Let ϕ be an automorphic form on G(k)\G. For every standard parabolic subgroup P = MU of G, let us take a set
of cuspidal data for (see 1.3.3). Then there exists c > 0 such that for all g є S, we have the upper bound
see 1.3.3 for the definition of Reл; deg(Q) is the total degree of Q. More generally for all, there exists such that for all g є S, we have the upper bound
where µM is the projection of onto (see 1.1.6 (9)).
Proof (a) We proceed by induction on the semi-simple rank of G. Suppose the lemma is proved for every proper standard Levi subgroup M of G. We immediately deduce a similar lemma concerning automorphic forms on M(k)U(A)\G for every proper standard parabolic P = MU of G. Note that if are two such subgroups, we have the equality of cuspidal components
We deduce from this that for all and all, we have an upper bound
A fortiori, we can replace the sum over P′ by the sum over all P′ G and restrict ourselves to g ε S.
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- Spectral Decomposition and Eisenstein SeriesA Paraphrase of the Scriptures, pp. 49 - 77Publisher: Cambridge University PressPrint publication year: 1995