Closures of conjugacy classes in classical real linear Lie groups

Abstract

By a classical group we mean one of the groups GLJ(R), GLn(C), GL,(H), U(p, q), OQ(C), O(p, q), SO*(2n), Sp21(C), Sp2n(R), or Sp(p, q). Let G be a classical group and L its Lie algebra. For each x E L we determine the closure of the orbit G x (for the adjoint action of G on L). The problem is first reduced to the case when x is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of G. 0. Introduction. Let G be a classical real linear Lie group (see ?1 for precise definition) and L its Lie algebra. There are ten infinite series of such groups and we use indexj (1 s j s 10) to label these series. G acts on L via the adjoint representation and on itself by conjugation. The classification of orbits of G in L and the classification of conjugacy classes of G are now well known; see for instance [1] and the references mentioned there. In this paper we determine the closure of an arbitrary orbit 0 and the closure of an arbitrary conjugacy class C(. (The topological terms refer to the ordinary topology of G and L.) For complex groups this problem was solved by M. Gerstenhaber [5,6]. An independent proof for complex special linear groups was given by J. Dixmier [2], and for complex orthogonal and symplectic groups by W. Hesselink [7] and the author [3]. In fact the results of Gerstenhaber and Hesselink are more general since they consider classical groups over any algebraically closed field. In our previous paper [4] on the same subject we have announced the main result of the present paper. For the sake of convenience we repeat all the necessary definitions so that the reader does not need to consult [4] except for the proofs of Theorem 1 (?2), Theorem 2 (?3), and the necessity part of Theorem 5 (?7). The two problems, closures of orbits and closures of conjugacy classes, are closely related. We give a full treatment to the first problem while the second is treated in a cavalier fashion in ?13. In this introduction we shall comment only on the first problem. In ?2 we state the Centralizer Theorem (Theorem 1) which asserts that the centralizer CG(x) in G of a semisimple element x E L is a direct product of classical groups. The direct factors of CG(x) may belong to different series of classical groups. Received by the editors May 5, 1980 and, in revised form, January 22, 1981. 1980 Mathematics Subject Classification. Primary 20G20; Secondary 22E46, 22E60.