Bourbaki

Proposition 1

Tag 0161, permanent. Cite it as [Bourbaki, Tag 0161].

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

Let ω∈P++\omega \in P_{++}. For all λ∈P\lambda \in P, the multiplicity of λ\lambda in E(ω)E(\omega ) is

mλ=∑w∈Wε(w)P(w(ω+ρ)−(λ+ρ)) m_{\lambda}=\sum_{w\in W}\varepsilon (w)\mathfrak{P}(w(\omega +\rho )-(\lambda +\rho ))

By Th. 1 and Lemma 1,

ch E(ω)=Ke−ρdE(\omega ) = Ke^{-\rho}d ch E(ω)=Ke−ρ∑w∈Wε(w)ew(ω+ρ)E(\omega ) = Ke^{-\rho}\sum_{w\in W}\varepsilon (w)e^{w(\omega+\rho)}

so

ch E(ω)=∑w∈W,γ∈Q+ε(w)P(γ)e−ρ+w(ω+ρ)−γE(\omega ) =\sum_{w\in W,\gamma\in Q_+}\varepsilon (w)\mathfrak{P}(\gamma )e^{-\rho+w(\omega+\rho)-\gamma}

and

mλ=∑ε(w)P(γ) m_{\lambda}=\sum\varepsilon (w)\mathfrak{P}(\gamma )
w∈W,γ∈Q+,γ=−λ−ρ+w(ω+ρ)w\in W,\gamma \in Q_+,\gamma =-\lambda -\rho +w(\omega +\rho )

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.

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Cites

Transcribed from the printed volume. Not translated.