Corollary 2

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

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

For all $\lambda \in X_{++}$ and every regular element $t$ of T,

\chi_{\lambda}(t) =\sum\sum^w\varepsilon_w(w\varepsilon )(\lambda w()wt\rho ()wt\rho ()wt) \tag{5}

where the two sums are over the elements $w$ of W.

Indeed, $J(\rho )(t)$ is non-zero, since $t$ is regular (formula (4)).

If $\varphi$ is a central function on G, the restriction of $\varphi$ to T is invariant under W, so $J(\rho ).\varphi |T$ is anti-invariant under W. Further, by Spectral Theories and Th. 1, the family $(\chi_{\lambda})_{\lambda\in X_{++}}$ is an algebraic basis of the space of central representative functions on G and an orthonormal basis of the space ZL $^2(G)$ of classes of square-integrable central functions on G.

Thus, from Prop. 3 and Th. 2 we deduce:

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.

Corollary 2

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