Lemma 1

Tag 015T, permanent. Cite it as [Bourbaki, Tag 015T].

Lie Groups and Lie Algebras, chapter VIII, § 9. The formula of Hermann Weyl

In the ring $\mathbf{Z}\langle P\rangle$ , we have $K.\prod_{\alpha\in R_+}(1-e^{-\alpha}) = Ke^{-\rho}d= 1$ .

Indeed,

K =\prod_{\alpha\in R_+}(e^0+e^{-\alpha}+e^{-2\alpha}+\cdots )

Ke^{-\rho}d=\prod_{\alpha\in R_+}(1 +e^{-\alpha}+e^{-2\alpha}+\cdots )\prod_{\alpha\in R_+}(1-e^{-\alpha}) = 1

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

Lemma 2, § 9
Proposition 1, § 9

Cites

Nothing.