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.3 - Cuspidal components
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
AM-finite functions
Let M be a standard Levi subgroup of G.
Remark
Let Hom(AM,) be the set of characters of AM, i.e. of continuous homomorphisms of AM into. The restriction map of M to AM defines a map
(1) We can use this to prove surjectivity by giving a description of Hom(AM) analogous to the description of XM (see I.1.4). As AMM1 is of finite index in M, the kernel of (1) is finite. It contains since AM⊂ZM. We will call it XM(AM). We obtain an isomorphism
Suppose first that k is a number field. Let (AM) be the enveloping algebra of the (complex) Lie algebra of the real group AM. We have
Thus (4M) is identified, by a map which we will denote by z ↦, with the polynomial algebra over the complex space, itself isomorphic to XM and even to XM/XM(AM), since XM(AM) = {0}. Suppose now that k is a function field. Let (AM) be the convolution algebra of functions with compact support on AM. We associate with z ∊ (AM) its Fourier-Mellin transform, which is the function on XM/XM(AM) defined by
Fix a basis (ai=1,…,n of the ℤ-module ZM. Then (AM) can be identified via z ↦ with the space of polynomials in the variables for i = 1,…,n.
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- Spectral Decomposition and Eisenstein SeriesA Paraphrase of the Scriptures, pp. 40 - 48Publisher: Cambridge University PressPrint publication year: 1995