Pairs of Short Root Subgroups in a Chevalley Group

The purpose of this paper is to describe subgroups of a Chevalley group G of type B l , C l or F 4 , generated by a pair of unipotent short root subgroups. In fact we do somewhat more: namely, we classify orbits of a Chevalley group acting by simultaneous conjugation on pairs of unipotent short root subgroups. In the sequel of this paper we consider analogous problems for the case of the Chevalley group of type G 2. For unipotent long root subgroups such a description is very well known, see M.Aschbacher and G.M.Seitz AS]. Namely, it turns out that any pair (X 1 ; X 2) of long root subgroups is simultaneously conjugate to a pair of elementary long roots subgroups (X ; X). The orbits of G on pairs of root subgroups are essentially determined by the angle between and , so that generically there are ve possible conngurations, some of which may give two or three orbits, compare Theorem A below. This result played an important role in understanding the irreducible subgroups generated by long root subgroups. Geometry of long root subgroups in Chevalley groups is now a well established eld, see, for example, J.McLaughlin M1], M2], B.