Lemma 4

Tag 01GC, permanent. Cite it as [Bourbaki, Tag 01GC].

Lie Groups and Lie Algebras, chapter IX, § 7. Irreducible representations of connected compact Lie groups

If $\varphi$ and $\psi$ are two continuous central functions on G,

\int_G\overline{\varphi(g)}\psi (g)dg=\frac{1}{w(G)}\int_T\overline{(\varphi(t)J(\rho)(t))}.(\psi (t)J(\rho )(t))dt

For all $\lambda \in X_{++}$ , denote by $\chi_{\lambda}$ the character of an irreducible representation of G of highest weight $\lambda$ .

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

Theorem 2 (H. Weyl), § 7
Corollary 4, § 7

Cites

Nothing.