Remark

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

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

Let $x\in \mathfrak{g}$ . There exists a strictly positive real number A such that, for every irreducible representation $\tau : G\rightarrow \mathbf{G}\mathbf{L}(V)$ and every Hilbert structure on V invariant under G,

\|L(\tau )(x)\|^2\leq A.\widetilde{\Gamma}(\tau )

Indeed, with the notations in the preceding proof, we can choose the basis $(e_i)$ of $\mathfrak{g}$ so that $x=ae_1,a\in \mathbf{R}$ . Then, for $v\in V$ , we have

\langle x_Vv, x_Vv\rangle =|a|^2\langle e_1v, e_1v\rangle \leq |a|^2\widetilde{\Gamma}(\tau )\langle v, v\rangle

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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