Alexander Premet

NILPOTENT ORBITS IN GOOD CHARACTERISTICS: A NEW APPROACH

The aim of my talk is to outline a new approach to the classification of nilpotent orbits in a reductive Lie algebra 𝔤 = Lie G over a field of good characteristic, and to give a unified description of the component group Z_{G}(e)/Z_{G}(e)^{∘} of the centraliser of a nilpotent element e ∈ 𝔤. The latter is based on a generalisation of a recent result of Sommers to fields of good characteristics, while the former combines my earlier computation of dimensions of standard nilpotent orbits with an idea of Spaltenstein and provides an alternative an shorter proof of Pommerening’s theorem (which states that all nilpotent orbits in 𝔤 are standard). The approach I use is based on the Kempf–Rousseau theory of optimal tori for unstable vectors. This allows one to eliminate 𝔰𝔩(2)-triples from the theory.

References

[1] R.W. Carter, Finite groups of Lie type, Pure Appl. Math., John Wiley & Sons, New York, 1985.

[2] K. Pommerening, Über die unipotenten Klassen reductiver Gruppen, II J. Algebra, 1980, 65, 373–399.

[3] A. Premet, An analogue of the Jacobson–Morozov theorem for Lie algebras of reductive groups of good characteristics, Tran. Amer. Math. Soc., 1995, 347, 2961–2988.

[4] E. Sommers, A generalization of the Bala–Carter theorem for nilpotent orbits, Internat. Math. Res. Notices, 1998, no. 11, 539–562.