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Irreducible characters are orthonormal class functions

Statement

Assume the Axiom of Choice. Let $G$ be a compact Lie group with normalized Haar measure $μ$ . The characters of pairwise inequivalent irreducible unitary finite-dimensional complex representations of $G$ are orthonormal in $L^{2} (G, μ)$ , and every such character is a conjugation-invariant class function.

Facts & Assumptions

Given: Assume the Axiom of Choice, a compact Lie group $G$ with normalized Haar measure $μ$ , and irreducible unitary representations $π, σ$ chosen from a family containing one representative of each equivalence class, with characters $χ_{π}, χ_{σ}$ . Thus either $π = σ$ with the same chosen model and basis, or $π$ and $σ$ are inequivalent.

[A1]

The Axiom of Choice is The Axiom of Choice; it enters through the normalized Haar measure and the Schur-orthogonality supplier [L1]. The finite-sum linearity statement [L3] is choice-free.

[L1]

Schur orthogonality with respect to the matrix coefficient functions fixed on this page gives $\int_{G} π_{i j} (g) \overline{σ_{k l} (g)} d μ (g) = 0$ for inequivalent irreducible $π, σ$ . For equivalent models and an intertwining isomorphism $S : V_{σ} \to V_{π}$ , the integral is $S_{i k} (S^{- 1})_{l j} / d_{π}$ ; in the equal-model, equal-basis case used below, $S = I$ and this specializes to $δ_{i k} δ_{j l} / d_{π}$ (Schur orthogonality).

[L2]

The character is $χ_{π} (g) = tr π (g) = \sum_{i = 1}^{d_{π}} π_{i i} (g)$ in an orthonormal basis, is independent of that basis, and is a class function: $χ_{π} (g h g^{- 1}) = χ_{π} (h)$ for all $g, h$ (Matrix coefficients and characters).

[L3]

The integral of a finite sum of integrable functions is the sum of the integrals (The Lebesgue integral is linear on $L^{1} (μ)$ ).

Proof

technique · direct

1.1

Expanding both characters by [L2] and using linearity of the integral [L3], $⟨ χ_{π}, χ_{σ} ⟩ = \int_{G} χ_{π} \overline{χ_{σ}} d μ = \sum_{i = 1}^{d_{π}} \sum_{k = 1}^{d_{σ}} \int_{G} π_{i i} (g) \overline{σ_{k k} (g)} d μ (g)$ , a finite sum of the orthogonality integrals of [L1].

L1L2L3

2.1

If $π$ and $σ$ are inequivalent, every term in step 1.1 vanishes by the first case of [L1], so $⟨ χ_{π}, χ_{σ} ⟩ = 0$ . If $π = σ$ , then the terms equal $δ_{i k} δ_{i k} / d_{π} = δ_{i k} / d_{π}$ by the second case of [L1] with $j = i$ and $l = k$ , so $⟨ χ_{π}, χ_{π} ⟩ = \sum_{i = 1}^{d_{π}} 1 / d_{π} = 1$ .

L1step 1.1

3.1

Conjugation invariance is [L2], so every character is a class function; combining with step 2.1, the characters of pairwise inequivalent irreducible unitary representations are orthonormal in $L^{2} (G, μ)$ . The Axiom of Choice entered only through the Haar-based supplier [L1].

A1L2step 2.1∎

Depends on

Used by

Dominant characters form the representation-ring basis Corollary
A compact group can have infinite-dimensional unitary representations False statement
Orthogonality identifies the Weyl numerator Lemma

Dependency tree · two levels

17 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)