Complexe canonique de deuxième espèce, variété commutante et bicône nilpotent d'une algèbre de Lie réductive

Details Complexe canonique de deuxième espèce, variété commutante et bicône nilpotent d'une algèbre de Lie réductive Complexe canonique de deuxième espèce, variété commutante et bicône nilpotent d'une algèbre de Lie réductive

2005-09-13

arxiv.org/abs/math/0509272v1

Mathematics

Representation Theory

51 pages

Scientific paper

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 $g$.

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