Sums of orbits of algebraic groups I

Let be a simple algebraic group which is defined and split over a field K and let be a corresponding Lie algebra. Further, let R be the corresponding root system. We prove here that if charK=0 or a very good prime for R and |K|>|R+|, then there exists an orbit with respect to the adjoint action of such that L=O+O. This is an analogue of the corresponding result for Chevalley groups (the Thompson problem; see [E.W. Ellers, N. Gordeev, On the conjecture of J. Thompson and O. Ore, Trans. Amer. Math. Soc. 350 (1998) 3657–3671]). We also prove that if charK≠2 for R=Br,Cr,F4 and charK≠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) -orbits of elements of coincides with . 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 .