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Finite-dimensional compact-group representations are unitarizable

Statement

Assume the Axiom of Choice. Every finite-dimensional continuous complex representation of a compact Lie group preserves some positive-definite Hermitian inner product.

Facts & Assumptions

Given: Assume the Axiom of Choice, a compact Lie group $G$ with normalized Haar measure $μ$ , a finite-dimensional complex representation $π : G \to GL (V)$ as in Continuous and unitary representations, and a positive-definite Hermitian inner product $(\cdot, \cdot)_{0}$ on $V$ .

[A1]

The Axiom of Choice is The Axiom of Choice; it enters here through the existence of $μ$ and its invariance [L2].

[L1]

A representation is a continuous homomorphism $π : G \to GL (V)$ with $π (g h) = π (g) π (h)$ and $π (e) = {i d}_{V}$ ; its matrix entries in any basis are continuous functions on $G$ ; an inner product is linear in the first argument, conjugate-symmetric and positive definite (Continuous and unitary representations, Real and complex inner-product spaces and their induced length).

[L2]

For the normalized Haar measure $μ$ and every integrable $f$ , $\int_{G} f (g h) d μ (g) = \int_{G} f (g) d μ (g)$ for every $h \in G$ (Haar integration is translation and conjugation invariant).

[L3]

Every left Haar integral is strictly positive on every nonzero nonnegative continuous compactly supported function; in particular, if a continuous $f \geq 0$ on $G$ satisfies $f (g_{0}) > 0$ for some $g_{0}$ , then $\int_{G} f d μ > 0$ (Haar measure is positive on nonempty open sets and finite on compact sets).

[L4]

A finite-dimensional complex vector space admits a positive-definite Hermitian inner product: choose a basis (Every vector space has a basis) and transport the standard inner product of $C^{n}$ (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[L5]

The Lebesgue integral is complex-linear on integrable functions, and for nonnegative Borel functions it is monotone and scalar-homogeneous; consequently, if a measurable $h$ satisfies $∣ h ∣ \leq c$ everywhere with $c \geq 0$ and $μ$ is a finite measure, then $\int ∣ h ∣ d μ \leq c μ (G) < + \infty$ , so $h$ is integrable with $∣ \int h d μ ∣ \leq \int ∣ h ∣ d μ$ (The Lebesgue integral is linear on $L^{1} (μ)$ , Monotonicity and nonnegative homogeneity of the nonnegative integral, Integrable real and complex functions, and their integrals).

Proof

technique · direct

1.1L1L4L5

Fix a positive-definite Hermitian inner product $(\cdot, \cdot)_{0}$ on $V$ and a basis of $V$ , and for $v, w \in V$ define $⟨ v, w ⟩ : = \int_{G} (π (g) v, π (g) w)_{0} d μ (g)$ . The integrand is a finite sum of products of the continuous matrix entries of $π$ with the coordinates of $w$ , hence continuous on $G$ , and it is bounded because $G$ is compact; so by [L5] the integral exists in $C$ and the definition is unambiguous.

2.1L5step 1.1

The form $⟨ \cdot, \cdot ⟩$ is sesquilinear: for $a, b \in C$ the identity $(π (g) (a v + b v^{'}), π (g) w)_{0} = a (π (g) v, π (g) w)_{0} + b (π (g) v^{'}, π (g) w)_{0}$ holds pointwise, and complex linearity of the integral [L5] turns it into $⟨ a v + b v^{'}, w ⟩ = a ⟨ v, w ⟩ + b ⟨ v^{'}, w ⟩$ ; conjugate symmetry follows pointwise from conjugate symmetry of $(\cdot, \cdot)_{0}$ together with reality of the integral of a real-valued function, and positive semidefiniteness follows because each integrand $(π (g) v, π (g) v)_{0} \geq 0$ and the integral of a nonnegative function is nonnegative.

2.2L1L3step 1.1

If $v \neq 0$ , the continuous function $ϕ (g) : = (π (g) v, π (g) v)_{0}$ is nonnegative and satisfies $ϕ (e) = (v, v)_{0} > 0$ , so [L3] gives $⟨ v, v ⟩ = \int_{G} ϕ d μ > 0$ ; the form is therefore positive definite.

3.1L1L2step 2.2

For every $h \in G$ and all $v, w \in V$ one has $⟨ π (h) v, π (h) w ⟩ = \int_{G} (π (g) π (h) v, π (g) π (h) w)_{0} d μ (g) = \int_{G} (π (g h) v, π (g h) w)_{0} d μ (g) = ⟨ v, w ⟩$ , where the middle equality is the homomorphism property $π (g) π (h) = π (g h)$ of [L1] and the last equality is the right-translation invariance [L2] applied to $f (g) = (π (g) v, π (g) w)_{0}$ .

4.1A1step 2.1step 2.2step 3.1∎

By steps 2.1, 2.2 and 3.1 the form $⟨ \cdot, \cdot ⟩$ is a positive-definite Hermitian inner product preserved by every $π (h)$ , so $π$ is unitary for it; the Axiom of Choice was used only through the existence and translation invariance of $μ$ in [L2].

Depends on

Used by

Complete reducibility for compact Lie groups Corollary
Dominant characters form the representation-ring basis Corollary
Matrix coefficients and characters Definition
Roots of a compact connected Lie group Definition
Peter–Weyl decomposition of L2(SU(2)) Example
Peter–Weyl gives density, not finite equality False statement
Orthogonality identifies the Weyl numerator Lemma
The Chevalley–Eilenberg Laplacian is scalar on weight components Lemma
Differentiation and integration of highest weights Proposition

Dependency tree · two levels

39 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)