Book contents
- Frontmatter
- Contents
- Preface
- 0 Introduction
- I Basic material on SL2(ℝ), discrete subgroups, and the upper half-plane
- II Automorphic forms and cusp forms
- III Eisenstein series
- IV Spectral decomposition and representations
- 13 Spectral decomposition of L2(Γ\G)m with respect to C
- 14 Generalities on representations of G
- 15 Representations of G
- 16 Spectral decomposition of L2(Γ\G): the discrete spectrum
- 17 Spectral decomposition of L2(Γ\G): the continuous spectrum
- 18 Concluding remarks
- References
- Notation index
- Subject index
14 - Generalities on representations of G
from IV - Spectral decomposition and representations
Published online by Cambridge University Press: 07 October 2011
- Frontmatter
- Contents
- Preface
- 0 Introduction
- I Basic material on SL2(ℝ), discrete subgroups, and the upper half-plane
- II Automorphic forms and cusp forms
- III Eisenstein series
- IV Spectral decomposition and representations
- 13 Spectral decomposition of L2(Γ\G)m with respect to C
- 14 Generalities on representations of G
- 15 Representations of G
- 16 Spectral decomposition of L2(Γ\G): the discrete spectrum
- 17 Spectral decomposition of L2(Γ\G): the continuous spectrum
- 18 Concluding remarks
- References
- Notation index
- Subject index
Summary
We first review some notions and facts about continuous representations of
in a locally complete topological vector space V. In fact, all we need is the representation by right translations of G in C∞(G), L2(G), C∞(H\G), or L2(H\G) (H closed unimodular subgroup, mainly F). Much of this has been encountered earlier, implicitly or explicitly. All this is valid in a much more general framework (see e.g. [7]).
A continuous representation (π, V) of G into V is a homomorphism of G into the group of automorphisms of V that is continuous; in other words, the map (g, v) ↦ π(g).v is a continuous map of G × V into V. If V is a Hilbert space then it is said to be i>unitary if π (g) (g ∈ G) leaves the scalar product of V invariant. Then the operator norm ∥π(g)∥ is uniformly bounded (by 1), and it is known that continuity already follows from separate continuity:
(1) for every v ∈ V, the map g ↦ π(g).v of G into V is continuous (see e.g. [7, §3] or [10, §VIII.1]).
The assumption “locally complete” is made to ensure that if α ∈ Cc(G) then ∫G α(x)π(x).v dx converges to an element of V. More generally, ∫ µ(x)π(x).v converges if µ is a compactly supported measure on G (see an important example in 14.4).
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- Automorphic Forms on SL2 (R) , pp. 153 - 157Publisher: Cambridge University PressPrint publication year: 1997