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We consider Lie algebra monomorphisms ip : ^ —> £2 between various noncompact real forms 01, 02 of complex simple Lie algebras gf, #2I* 1 ^ cases that we consider, 0f or £^ is of exceptional type, with one exception. For each adjoint nilpotent orbit O of 01 we determine the adjoint nilpotent orbit of 02 which contains the image ^(O). The adjoint nilpotent orbits of ^ and ^2 are themselves parametrized by using the Kostant-Sekiguchi correspondence. 1. Preliminaries. Let g be a semisimple real Lie algebra and g its complexification. Let 0 be a Cartan involution of g and g = E+p the corresponding Z2-gradation (a Cartan decomposition). By complexifying, we obtain the Z2-gradation 0 = 6-fp and we extend 0 to a complex linear automorphism 6 of 0. Denote by G (resp. G) the adjoint group of 0 (resp. g). Thus G is the connected Lie subgroup of G with Lie algebra g. The group G (resp. G) acts on g (resp. g) via the adjoint action: (a, x) —> a x = Ad(a)(x) where a G G (resp. G) and x € g (resp. g). A G (resp. G)-orbit is nilpotent if it consists of nilpotent elements of g (resp g). There are only finitely many nilpotent G (resp. G)-orbits in g (resp. g). Let K (resp. K) be the connected' Lie subgroup of G (resp. G) whose Lie algebra is t (resp. t). By restricting the adjoint action of G, we obtain an action of K on p. The number of nilpotent K-oibits in p is also finite [4]. Let O be a nilpotent G-orbit in g. The intersection O fl g consists of finitely many connected components Ai, i = l,...,k. Moreover, each of these connected components is a single nilpotent G-orbit, and dimj^(^) = dim.Q(0) for each i. The intersection Onp also consists of k connected components, say 23$, i = 1,...,fc, each of them is a single nilpotent K-orbit and dimQ(Bz) = |dim(2)(0) for each i. The Kostant-Sekiguchi correspondence (see [10, 4]) establishes a bijection from {Ai} to {Bi}. If E,H,F € g are nonzero elements satisfying [H,E] = 2E, [H,F] = —2F, and [F, E] = H, then we say that (E, H, F) is a standard triple. If moreover E, F G p and H £ %, then we say that (JE, iJ, F) is a normal triple. Let us fix a Cartan subalgebra \) of t and f} of g such that ^25Let f) and ^ be their respective complexifications. Let R be the root system of (£,f)), and R that of (g, fj ). Finally, let W be the Weyl group of i?, 11 = {Pi^2,.. •} a base of R (a system of fundamental roots), and define W, R and ft = {ai,a2,...} similarly. If R (resp. R) is irreducible, we denote its highest root by a (resp. /?). We say that the Z2-graded Lie algebra g is of inner type if E and g have the same rank, i.e., l) = ^ , and otherwise of cmter iy^e. In the former case we may view R as a root subsystem of R, and we say that the roots in R are compact and the other roots in R are noncompact. In the root diagrams, we shall represent compact (resp. noncompact) roots by black (resp. white) nodes. * Received January 20, 2002; accepted for publication July 4, 2002. "^Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada (djokovic@uwaterloo.ca). The author was supported in part by the NSERC Grant A-5285. * Department of Mathematics, Tokyo University of Agriculture and Technology, Koganei, Tokyo 184-8588, Japan (sekiguti@cc.tuat.ac.jp). The author was supported by Grant-in-Aid for Scientific Research (No. 13640039), Ministry of Education, Culture, Sports, Science and Technology, Japan.

Mapping of nilpotent orbits under embeddings of real forms of exceptional complex Lie algebras