Summarizing of Orbits of Algebraic Groups I

  • Published 2004


Let G̃ be a simple algebraic group which is defined and split over a field K and let L̃ be a corresponding Lie algebra. Further, let R be the corresponding root system. We prove here that if char K = 0 or a very good prime for R and | K |>| R |, then there exists an orbit O ⊂ L = L̃(K) with respect to the adjoint action of G = G̃(K) such that L = O + O. This is an analogue of the corresponding result for Chevalley groups (the Thompson problem; see [EG2]). We also prove that if char K 6= 2 for R = Br, Cr, F4 and char K 6= 3 for R = G2, then the Zariski closure of a sum of any r (R = Br (r > 3), Dr, Er, F4), r + 1 (R = Ar, B3 , G2), 2r (R = Cr) G̃-orbits of elements of L̃ \ Z(L̃) coincides with L̃. For the group G = SL2(K) where K is an algebraically closed field of characteristic zero we list all cases of rational K[G]-modules V which have an orbit O ⊂ V such that O + O = V .

Cite this paper

@inproceedings{GORDEEV2004SummarizingOO, title={Summarizing of Orbits of Algebraic Groups I}, author={NIKOLAI GORDEEV}, year={2004} }