Bourbaki

Corollary

Tag 0162, permanent. Cite it as [Bourbaki, Tag 0162].

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

If λ\lambda is a weight of E(ω)E(\omega ) distinct from ω\omega,

mλ=−∑w∈W,w≠1ε(w)mλ+ρ−wρ m_{\lambda}=-\sum_{w\in W,w\not=1}\varepsilon (w)m_{\lambda+\rho-w\rho}

Apply Prop. 1 with ω=0\omega = 0. If μ∈P\mu\in P-- {0}\{0\}, we find that

0=∑w∈Wε(w)P(wρ+μ−ρ) 0 =\sum_{w\in W}\varepsilon (w)\mathfrak{P}(w\rho +\mu-\rho )

hence

P(μ)=−∑w∈W,w≠1ε(w)P(μ+wρ−ρ)(1) \mathfrak{P}(\mu) =-\sum_{w\in W,w\not=1}\varepsilon (w)\mathfrak{P}(\mu+w\rho -\rho ) \tag{1}

Prop. 1 also gives

mλ=−∑w∈Wε(w)∑w′∈W,w′≠1ε(w′)P(w(ω+ρ)−(λ+ρ)+w′ρ−ρ) m_{\lambda}=-\sum_{w\in W}\varepsilon (w)\sum_{w'\in W,w'\not=1}\varepsilon (w')\mathfrak{P}(w(\omega +\rho )-(\lambda +\rho ) +w'\rho -\rho )

since w(ω+ρ)≠λ+ρw(\omega +\rho )\not=\lambda +\rho for all w∈W(w\in W (§7, Prop. 5 (iii)). Hence,

mλ=−∑w′∈W,w′≠1ε(w′)∑w∈Wε(w)P(w(ω+ρ)−(λ+ρ−w′ρ+ρ)) m_{\lambda}=-\sum_{w'\in W,w'\not=1}\varepsilon (w')\sum_{w\in W}\varepsilon (w)\mathfrak{P}(w(\omega +\rho )-(\lambda +\rho -w'\rho +\rho ))

=−∑w′∈W,w′≠1ε(w′)mλ+ρ−w′ρ=-\sum_{w'\in W,w'\not=1}\varepsilon (w')m_{\lambda+\rho-w'\rho} (Prop.1.1).

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

Nothing.

Cites

Transcribed from the printed volume. Not translated.