DefinitionDefinition: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Matrix coefficient of a unitary representation

Definition

Let $π : G \to U (H)$ be a group homomorphism and let $ξ, η \in H$ . Its matrix coefficient associated with $(ξ, η)$ is $c_{ξ, η} : G \to C, c_{ξ, η} (g) = ⟨ π (g) ξ, η ⟩ .$ We use the convention that the Hilbert pairing is linear in its first argument and conjugate-linear in its second. Thus $c_{ξ, η}$ is linear in $ξ$ and conjugate-linear in $η$ .

When $G$ is a topological group and $π$ is strongly continuous, every such coefficient is continuous:

Facts & Assumptions

[A1]

If $G$ is a topological group and $π$ is strongly continuous, then for every $v \in H$ the orbit map $g \mapsto π (g) v$ is norm-continuous (Strongly continuous unitary representations, invariant linear subspaces and intertwiners).

[A2]

For vectors $x, y$ in a complex inner-product space, $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[A3]

Continuity of a map into $C$ is measured with the metric $d_{C} (z, w) = ∣ z - w ∣$ (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane).

Proof

Given: A group homomorphism $π : G \to U (H)$ and vectors $ξ, η \in H$ ; for the continuity assertion, assume that $G$ is a topological group and $π$ is strongly continuous.

Proof technique: direct.

1.1A1

Fix $g_{0} \in G$ . By strong continuity, the orbit map for $ξ$ is continuous at $g_{0}$ , so $∥ π (g) ξ - π (g_{0}) ξ ∥ \to 0$ as $g \to g_{0}$ .

2.1A2A3step 1.1∎

Linearity in the first argument and Cauchy–Schwarz give $∣ c_{ξ, η} (g) - c_{ξ, η} (g_{0}) ∣ = ∣ ⟨ π (g) ξ - π (g_{0}) ξ, η ⟩ ∣ \leq ∥ π (g) ξ - π (g_{0}) ξ ∥ ∥ η ∥ \to 0$ by step 1.1. By [A3] this is continuity of the coefficient at the arbitrary point $g_{0}$ , hence on $G$ . This also holds when $η = 0$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Bekka, de la Harpe and Valette, Kazhdan's Property (T), Definition A.1.1, Appendix A, printed p. 305 (standard reference, not scraped)
Emmanuel Kowalski, An Introduction to the Representation Theory of Groups, §3.4, printed pp. 106–107 (standard reference, not scraped)