Bourbaki

Corollary 4

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

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

Let φ\varphi be a continuous central function on G. Then,

∫Gχλ(g)‾φ(g)dg=∫Tλ(t)‾∏α>0(1−α(t)−1)φ(t)dt=∫Tλρ(t)‾.φ(t)J(ρ)(t)dt \int_G\overline{\chi_{\lambda}(g)}\varphi (g)dg=\int_T\overline{\lambda(t)}\prod_{\alpha >0}(1-\alpha (t)^{-1})\varphi (t)dt=\int_T\overline{\lambda \rho(t)}.\varphi (t)J(\rho )(t)dt

Indeed, by Lemma 4 and Th. 2,

∫Gχλ(g)‾φ(g)dg=1w(G)∫Tχλ(t)J(ρ)(t)(φ(t)J(ρ)(t))dt \int_G\overline{\chi_{\lambda}(g)}\varphi (g)dg=\frac{1}{w(G)}\int_T\chi_{\lambda}(t)J(\rho )(t)(\varphi (t)J(\rho )(t))dt
=1w(G)∫TJ(λρ)(t)‾φ(t)J(ρ)(t)dt =\frac{1}{w(G)}\int_T\overline{J(\lambda \rho)(t)}\varphi (t)J(\rho )(t)dt

But the function t→φ(t)J(ρ)(t)t \rightarrow \varphi (t)J(\rho )(t) is anti-invariant and 1w(G)J(λρ)\frac{1}{w(G)}J(\lambda \rho ) is the orthogonal projection of λρ\lambda \rho onto the subspace of anti-invariant elements of L2L^2(T), so

1w(G)∫TJ(λρ)(t)‾φ(t)J(ρ)(t)dt=∫Tλρ(t)‾φ(t)J(ρ)(t)dt \frac{1}{w(G)}\int_T\overline{J(\lambda \rho)(t)}\varphi (t)J(\rho )(t)dt=\int_T\overline{\lambda \rho(t)}\varphi (t)J(\rho )(t)dt

finally, by formula (3), we have ρ(t)J(ρ)(t)=∏α>0(1−α(t)−1)\rho (t)J(\rho )(t) =\prod_{\alpha >0}(1-\alpha (t)^{-1}), hence the corollary.

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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Transcribed from the printed volume. Not translated.