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.1. - Hypotheses and general notation
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
Hypotheses and general notation
Definitions
Let k be a global field and be the ring of adeles of k. For a finite place v of k, we write for the ring of integers. Let be the ring of finite adeles of k and, the product being over the archimedean places. If k is a function field, let q be the number of elements of its field of constants.
Let G be a connected reductive algebraic group defined over k. Fix an embedding into a linear group as follows. First choose an embedding, defined over k, with closed image. Then iG: G ↪ GL2n is defined by
There exists a finite set S of places of k, containing the archimedean places, such that the image of iG is defined and smooth over (see [Sp] §4.9). For v ∉ S, this allows us to define the group of points with values in. For almost all v ∉ S, this is a maximal compact subgroup of G(kv) (see [Sp] p.18, 1.3 and what follows). We fix a compact maximal subgroup K of G() such that K = ΠυKν product over all places of k, where Kν is a maximal compact subgroup of G(kυ). We suppose, as we may, that for almost all finite places. We will impose further properties on K in I.1.4.
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- Spectral Decomposition and Eisenstein SeriesA Paraphrase of the Scriptures, pp. 1 - 18Publisher: Cambridge University PressPrint publication year: 1995