On One Class of Unipotent Subgroups of Semisimple Algebraic Groups

This paper contains a complete proof of a fundamental theorem on the normalizers of unipotent subgroups in semisimple algebraic groups. A similar proof was given later by A. Borel and J. Tits (see also bibliographical remarks in the introduction to their paper). I.I. Pyatetskii–Shapiro suggested me to prove the following theorem which in his opinion would be of interest for the theory of discrete subgroups of semisimple Lie groups. Theorem. Let k be an arbitrary field, G a semisimple algebraic group defined over k, and H a unipotent subgroup of G. If the nilpotent radical of the normalizer NG(H) of H in G, coincides with H, then NG(H) is a parabolic subgroup of G. Lemma. Let N and H be nilpotent groups, H ⊂ N . If NN (H) = H, then N = H. Let N0 = N , Ns+1 = [Ns, N ] with Nq = 1. Then Nq ⊂ H . Further, if Ns+1 ⊂ H , then Ns ⊂ H since [Ns, H ] ⊂ [Ns, N ] = Ns+1 ⊂ H . Thus Ns ⊂ H for any s, which completes the proof. Proof of the Theorem. First we prove that NG(H) contains a maximal torus T of the group G. For, let NG(H) = SH where S is a reductive group. Let us choose two Borel subgroups B and B in G satisfying the following conditions: B ∩NG(H) and B ′ ∩NG(H) are Borel subgroups of NG(H), and B∩B ′ ∩NG(H) = T̃H , where T̃ is a maximal torus of S. Let N be the unipotent part of B ∩B. Then NN (H) = NG(H)∩N = H , and by lemma, N = H . But, it is well–known ([3], n 2.16) that B ∩ B = TN where T is a maximal torus of G and N is a normal subgroup of TN , whence T ⊂ NG(H). Translated from Russian, Uspehi Mat. Nauk, 21, 2(128) (1966), 222-223 A. Borel, J. Tits, Eléments unipotents et sous–groupes paraboliques de groupes réductifs I, Invent. Math. 12 (1971), 95–104. 1991 Mathematics Subject Classification. 20G15, 22E46.