Complexe canonique d'une alg\`ebre de Lie r\'eductive

Let $\text{goth}g$ be a finite dimensional complex reductive Lie algebra and $\text{dv}\mathrm{.}\mathrm{.}$ an invariant non degenerated bilinear form on ${\goth g}\times {\goth g}$whichextendstheKillingformof$[{\goth g},{\goth g}]$. We define the homology complex $C_{\bullet }(\text{goth}g)$. Its space is the algebra $\tk {{\Bbb C}}{\e Sg}\tk {{\Bbb C}}{\e Sg}\ex {}{{\goth g}}$where$\e Sg$and$\ex {}{{\goth g}}$arethesymmetricandexterioralgebrasof${\goth g}$. The differential of $C_{\bullet }(\text{goth}g)$ is the $\tk {{\Bbb C}}{\e Sg}\e Sg$-derivationwhichassociatestotheelement$v$of${\goth g}$thefunction$(x,y)\mapsto \dv v{[x,y]}$on${\goth g}\times {\goth g}$\mathrm{.}Thenthecomplex$C_{\bullet}({\goth g})$hasnohomologyindegreestrictlybiggerthan$\rk {\goth g}$.