Remark 1

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

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

For all $w\in W$ , put $\rho_w=^w\rho /\rho$ ; then

\sum_w\varepsilon (w)\rho_w=\prod_{\alpha >0}(1-\alpha^{-1}) =\rho^{-2}\prod_{\alpha >0}(\alpha -1) \tag{6}

If $t$ is a regular element of T, we deduce from (5) that

\chi_{\lambda}(t) =\sum\sum^w\varepsilon_w(w\varepsilon )(\overset{w}{w}\lambda )\rho (t_w)(\rho t^w)(t)=\sum\prod^w_{\alpha >}\varepsilon (_0w(1)\overset{w}{-}\lambda \alpha (t()t\rho )^{w-}(_1t)) \tag{7}

Note that $\rho_w$ is a linear combination of roots with integer coefficients, and hence belongs to X(T) even if we do not assume that $\rho \in X(T)$ . It follows that formula (7) is valid without the assumption that $\rho \in X(T):$ indeed, to prove this we replace G by a suitable connected covering, and are then reduced to Cor. 2.

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

Nothing.

Cites

Cor. 2