Proposition 8 (Orthogonality relations)

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

Théories spectrales, chapter V, § 2. Représentations des groupes localement compacts

Let $\pi$ be an irreducible unitary representation of G in a Hilbert space E which is square-integrable modulo the center. One has

c_Z(\pi )\int_{G/Z}\langle x|\pi (g)x'\rangle \langle y|\pi (g)y'\rangle d\nu_Z(g) =\langle x|y\rangle \langle x'|y'\rangle

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

Let $w$ denote the morphism of Prop. 7. One has

\int_{G/Z}\langle x|\pi (g)x'\rangle \langle y|\pi (g)y'\rangle d\nu_Z(g) =\langle f_{x,x'}|f_{y,y'}\rangle

and by loc. cit., it follows that

\langle f_{x,x'}|f_{y,y'}\rangle =\frac{1}{c_Z(\pi)}\langle w(x\otimes x')|w(y\otimes y')\rangle

=\frac{1}{c_Z(\pi)}\langle x\otimes x'|y\otimes y'\rangle =\frac{1}{c_Z(\pi)}\langle x|y\rangle_{\overline{E}}\langle x'|y'\rangle_E

whence the result.

As a complement to the preceding proposition, one also has the following relations for nonisomorphic irreducible square-integrable representations.

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.

Cited by

Nothing.

Cites

Nothing.