Corollary 2

Tag 03DT, permanent. Cite it as [Bourbaki, Tag 03DT].

Théories spectrales, chapter V, § 4. Représentations des groupes compacts

a) Let $\pi$ be an irreducible unitary representation of G in a Hilbert space E. One has

$\int'$ 1 $''$

\langle x|\pi (g)x'\rangle \langle y|\pi (g)y\rangle d\mu(g) =\langle x|y\rangle \langle x|y\rangle

$_G$ dim(E)

for every $(x, y, x', y')\in E^4$ ;

b) Let $\pi_1($ resp. $\pi_2)$ be an irreducible unitary representation of G in a Hilbert space $E_1($ resp. $E_2)$ . If $\pi_1$ and $\pi_2$ are not isomorphic, one has

\int_G\langle x|\pi_1(g)x'\rangle \langle y|\pi_2(g)y'\rangle d\mu(g) = 0

for every $(x, x', y, y')\in E^2_1\times E^2_2$ .

This follows at once from Prop. 8 of V, p. 424 and Prop. 9 of V, p. 424, taking into account Lemma 1 and the formula $c_\mu(\pi ) =$ dim(E) (Prop. 2).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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