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A. Moreau Departement Mathematik, ETH Zurich Zurich, Switzerland anne.moreau@math.ethz.ch

Nilp otent bicone of a reductive Lie algebra
Let g be a finite dimensional complex reductive Lie algebra and Ng its nilpotent cone. We denote by Ng the subset of elements (x, y ) in g в g such that the subspace generated by x and y is contained in Ng . This subset is called the nilpotent bicone of g. It was first introduced to study the commuting variety of g. It is naturally endowed with a structure of subscheme of g в g. We prove, in joint work with J-Y Charbonnel [1], that Ng , as subscheme of g в g, is a complete intersection (non reduced) of dimension 3(bg - rkg), where bg and rkg are respectively the dimension of a Borel subalgebra of g and the rank of g. In this talk I will present the main steps of our proof (which uses in particular some arguments of motivic integration [2] [5]) and I will give some applications of this result in invariant theory. References [1] J-Y Charbonnel and A. Moreau. Nilpotent bicone and characteristic submodule of a reductive Lie algebra. arXiv:math.RT/0705.2685, 2007. [2] J. Denef and F. Loeser. Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math., 135(1):201­232, 1999. [3] H. Kraft and N. R. Wallach. On the nullcone of representations of reductive groups. Pacific J. Math., 224(1):119­139, 2006. [4] M. Losik, P. W. Michor, and V. L. Popov. On polarizations in invariant theory. J. Algebra, 301(1):406­424, 2006. [5] M. Mustat Jet schemes of locally complete intersection canonical singularities. Invent. ёa. Math., 145(3):397­424, 2001. With an appendix by David Eisenbud and Edward Frenkel.

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