On parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical.

*(English. Abridged French version)*Zbl 0922.20049Let \(G\) be a reductive algebraic group defined over an algebraically closed field \(K\) of characteristic \(0\). Let \(P\) be a parabolic subgroup of \(G\) with unipotent radical \(P_u\). The general problem of determining each parabolic subgroup \(P\) of a reductive algebraic group with a finite number of orbits on \(P_u\) was posed by V. Popov and G. Röhrle [Aust. Math. Soc. Lect. Ser. 9, 297-320 (1997; Zbl 0887.14020)]. G. Röhrle [Geom. Dedicata 60, No. 2, 163-186 (1996; Zbl 0853.20031)] has shown that when the simple factors of \(G\) are of type \(A_r\), \(B_r\) or \(C_r\) the number of orbits of \(P\) on \(P_u\) is infinite provided the class of nilpotency of \(P_u\) is at least 5.

In this paper, the authors obtain the following converse of the above result which was conjectured by Röhrle. Let \(l(P_u)\) denote the nilpotency class of \(P_u\). If \(G\) is a simple classical algebraic group and \(P\) a parabolic subgroup of \(G\), then the number of \(P\) orbits on \(P_u\) is finite provided: (i) \(G\) is of type \(A_r\), \(B_r\) or \(C_r\) and \(l(P_u)\leq 4\), (ii) \(G\) is of type \(D_r\) and \(l(P_u)\leq 3\), (iii) \(G\) is of type \(D_r\), \(l(P_u)=4\) and \(P\) is stable under the automorphism of \(G\) of order 2 stemming from the interchange of the simple roots \(\alpha_{r-1}\) and \(\alpha_r\). (iv) When \(G\) is of type \(D_r\), assume that \(P\) is invariant under the group automorphism of \(G\) of order 2. Then the number of \(P\) orbits on \(P_u\) is finite if and only if \(l(P_u)\leq 4\).

In this paper, the authors obtain the following converse of the above result which was conjectured by Röhrle. Let \(l(P_u)\) denote the nilpotency class of \(P_u\). If \(G\) is a simple classical algebraic group and \(P\) a parabolic subgroup of \(G\), then the number of \(P\) orbits on \(P_u\) is finite provided: (i) \(G\) is of type \(A_r\), \(B_r\) or \(C_r\) and \(l(P_u)\leq 4\), (ii) \(G\) is of type \(D_r\) and \(l(P_u)\leq 3\), (iii) \(G\) is of type \(D_r\), \(l(P_u)=4\) and \(P\) is stable under the automorphism of \(G\) of order 2 stemming from the interchange of the simple roots \(\alpha_{r-1}\) and \(\alpha_r\). (iv) When \(G\) is of type \(D_r\), assume that \(P\) is invariant under the group automorphism of \(G\) of order 2. Then the number of \(P\) orbits on \(P_u\) is finite if and only if \(l(P_u)\leq 4\).

Reviewer: Madathum K.Viswanath (Tambaram)

##### MSC:

20G15 | Linear algebraic groups over arbitrary fields |

14L35 | Classical groups (algebro-geometric aspects) |

20G05 | Representation theory for linear algebraic groups |