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Induction from the trivial subgroup

Statement

Assume AC. For $H = {e}$ with its standard Haar measure of mass $1$ and the one-dimensional trivial representation, choose $ρ = 1$ . Then $G / H = G$ , $μ_{ρ}$ is left Haar measure, and ${Ind}_{{e}}^{G} 1$ is the left regular representation on $L^{2} (G)$ , $(λ (g) f) (x) = f (g^{- 1} x)$ .

Facts & Assumptions

Given: AC and an LCH group $G$ with a left Haar measure.

[F1]

The closed-subgroup induction construction and action (Unitary induction from a closed subgroup).

[F2]

The Weil formula and uniqueness of its quotient measure (Weil formula with a rho-function).

[F3]

For the trivial subgroup the left regular representation acts by $f (x) \mapsto f (g^{- 1} x)$ (Left and right regular unitary representations of an LCH group).

[A1]

AC is the choice-function principle required by the stated hypothesis (The Axiom of Choice).

Proof

technique · direct

1.1F2givenA1

When $H = {e}$ , the rho covariance imposes no restriction, and $ρ = 1$ is valid. The averaging map $T_{H}$ is identity and the Weil formula becomes $\int_{G} f (x) d x = \int_{G} f (x) d μ_{ρ} (x)$ . Radon uniqueness [F2] identifies $μ_{ρ}$ with left Haar measure.

2.1A1F1F2F3step 1.1

Covariance is empty for the trivial subgroup, so the dense model is $C_{c} (G)$ and its completion is $L^{2} (G)$ . The density cocycle is $D_{g} (x) = 1$ , hence the induced action is $F (x) \mapsto F (g^{- 1} x)$ , exactly [F3]. The construction theorem [F1] supplies strong continuity and unitarity. ∎

Sources

Bekka–de la Harpe–Valette, Kazhdan’s Property (T), Appendix E §E.1, Example E.1.8(i), PDF p. 414; Vogan, On the Definition of Induced Representations, §2. All relevant lines were inspected.

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Sources

Bekka–de la Harpe–Valette, Kazhdan’s Property (T), Appendices B and E (standard reference, not scraped)
Bruhat, Lectures on Lie Groups and Representations of Locally Compact Groups, Chapters 1 and 7 (standard reference, not scraped)
David Vogan, Unitary Representations of Locally Compact Groups and Induced Representations (standard reference, not scraped)