2005/09/13 by Jean-Yves Charbonnel, Charbonnel, Jean-Yves
Mathematics · #14A10 #18G05 #18G10 #22E46 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #math.AG #math.RT #msc:14A10 #msc:18G05 #msc:18G10 #msc:22E46
paper · pdf · doi:10.48550/arxiv.math/0509272
51 pages
arxiv created 2005/09/13 · arxiv updated 2009/12/01
Let g be a finite dimensional complex reductive Lie algebra and <.,.> an invariant non degenerated bilinear form on g× 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× 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× 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× 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× 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× g is the commuting variety of g.