Book contents
- Frontmatter
- Contents
- Foreword
- Conventional notations and terminology
- PART I REPRESENTATIONS OF COMPACT GROUPS
- PART II REPRESENTATIONS OF LOCALLY COMPACT GROUPS
- 11 Groups with few finite-dimensional representations
- 12 Invariant measures on locally compact groups and homogeneous spaces
- 13 Continuity properties of representations
- 14 Representations of G and of L1(G)
- 15 Schur's lemma: unbounded version
- 16 Discrete series of locally compact groups
- 17 The discrete series of S12(ℝ)
- 18 The principal series of S12(ℝ)
- 19 Decomposition along a commutative subgroup
- 20 Type I groups
- 21 Getting near an abstract Plancherel formula
- Epilogue
- References
- Index
17 - The discrete series of S12(ℝ)
Published online by Cambridge University Press: 20 March 2010
- Frontmatter
- Contents
- Foreword
- Conventional notations and terminology
- PART I REPRESENTATIONS OF COMPACT GROUPS
- PART II REPRESENTATIONS OF LOCALLY COMPACT GROUPS
- 11 Groups with few finite-dimensional representations
- 12 Invariant measures on locally compact groups and homogeneous spaces
- 13 Continuity properties of representations
- 14 Representations of G and of L1(G)
- 15 Schur's lemma: unbounded version
- 16 Discrete series of locally compact groups
- 17 The discrete series of S12(ℝ)
- 18 The principal series of S12(ℝ)
- 19 Decomposition along a commutative subgroup
- 20 Type I groups
- 21 Getting near an abstract Plancherel formula
- Epilogue
- References
- Index
Summary
Let (π,H) be a unitary irreducible representation in the discrete series of some group G. Fix u ∈H with ∥u∥2 = d and consider the coefficients cv = cuv of π. By definition of the discrete series, they are square summable on G and more precisely (16.3.a) shows that
is an isometry which embeds π in the right regular representation r of G. If a sequence vn → v in H, → cv uniformly on G
In the model of π consisting of right translations in {cv : v ∈ H}, all functions cv are continuous and
→ cv in L2 (G) ⇒ → cv uniformly
(because the assumption is equivalent to vn → v in H). This situation is very peculiar and reminds one of properties of holomorphic (or harmonic) functions. Indeed, the discrete series of Sl2(ℝ) will be constructed in spaces of holomorphic (or anti-holomorphic) functions.
The group Sl2(ℝ) acts on the upper half-plane Im(z) > 0, but it will be more convenient to let it act in the unit disc |w| < 1 (conformally equivalent to the upper half-plane) and thus, to use a conjugate SU(1,1) of S12(ℝ) in S12(ℂ). Here is a description of this new group.
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- Publisher: Cambridge University PressPrint publication year: 1983