Proposition 3

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

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

The elements $J(\lambda \rho )/\surd\overline{w(G)}$ , for $\lambda \in X_{++}$ , form an orthonormal basis of the subspace of anti-invariant elements of $L^2(T)$ , and an algebraic basis of the subspace of anti-invariant elements of $\Theta (T)$ .

The proof is identical to that of Chap. VI, §3, no. 3, Prop. 1.

By Chap. VI, loc. cit., Prop. 2,

J(\rho ) =\rho \prod_{\alpha >0}(1-\alpha^{-1}) =\rho^{-1}\prod_{\alpha >0}(\alpha -1) \tag{3}

J(\rho )J(\rho ) =\prod_{\alpha}(\alpha -1) \tag{4}

By Cor. 2 of Th. 1 (§6, no. 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.

Cited by

Theorem 2 (H. Weyl), § 7
Corollary 2, § 7

Cites

Cor. 2 of Th. 1 (§6, no. 2)
Chap. VI, §3, no. 3, Prop. 1 not in this corpus