Theorem 3

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

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

The dimension of the space of an irreducible representation of G of highest weight $\lambda$ is given by

\chi_{\lambda}(e) =\prod_{\alpha\in R_+}\frac{\langle\lambda + \rho , K_{\alpha}\rangle}{\langle\rho , K_{\alpha}\rangle}

Put $\gamma =\frac{1}{2}\sum_{\alpha >0}K_{\alpha}$ , so $\delta (\alpha )(\gamma ) = 2\pi i$ for every simple root $\alpha$ (Chap. VI, §1, no. 10, Prop. 29). The line $\mathbf{R}\gamma$ is not contained in any of the hyperplanes Ker $\delta (\alpha )$ , so exp( $z\gamma$ ) is a regular element of G for all sufficiently small $z\in \mathbf{R}^*$ ; for all $\mu\in X(T)$ and all $z\in \mathbf{R}$ , we have

$J(\mu$ )(exp( $z\gamma$ )) $=\sum_{w\in W}\varepsilon (w)e^{z\delta(\mu)(w^{-1}\gamma)}$ .

Let us accept provisionally the following lemma:

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

Cites

Chap. VI, §1, no. 10, Prop. 29 not in this corpus