Complexe canonique de deuxi\`{e}me esp\`{e}ce, vari\'{e}t\'{e} commutante et bic\^{o}ne nilpotent d'une alg\`{e}bre de Lie r\'{e}ductive

Abstract

Let $g$ be a finite dimensional complex reductive Lie algebra and <.,.> an invariant non degenerated bilinear form on $g\times g$ which extends the Killing form of $[g,g]$. We define a subcomplex $E\_{\bullet}(g)$ of the canonical complex $C\_{\bullet}(g)$ of $g$. There exists a well defined sub-module $B\_{g}$ of the module of polynomial maps from $g\times g$ to $g$ which is free of rank equal to the dimension b of the borel subalgebras of $g$. Moreover, $B\_{g}$ is contained in the space of cycles of the canonical complex of $g$. The complex $E\_{\bullet}(g)$ is the ideal of $C\_{\bullet}(g)$ generated the exterior power of degree b of the module $B\_{g}$. We denote by ${\cal N}\_{g}$ the set of elements in $g\times g$ whose components generate a subsbspace contained in the nilpotent cone of $g$ and we say that $g$ has property (N) if the codimension of ${\cal N}\_{g}$ in $g\times g$ is strictly bigger than the dimension of the space of nilpotent elements in a borel subalgebra of $g$. Let $I\_{g}$ be the ideal of polynomial functions on $g\times g$ generated by the functions whose value in $(x,y)$ is the scalar product of $v$ and $[x,y]$ where $v$ is in $g$. The main result is the theorem: Let us suppose that for any semi-simple element in $g$, the simple factors of its centralizer in $g$ have the property (N). Then the complex $E\_{\bullet}(g)$ has no homology in degree different from b and its homology in degree b is the reduced algebra of regular functions on the commuting variety. In particular, $I\_{g}$ is a prime ideal whose set of zeros in $g\times g$ is the commuting variety of $g$.