, This implies that, for each sheet S G and sl 2 -triple S = (e, h, f ) as in §1.4, the set e + X(S G , S ) = e + X(S ) contains a semisimple element. For symmetric pairs of type AI or AII, the K-sheets are all of the
Theorems 3.2.1 and 4.1.2); thus, in these cases, any K-sheet is a Dixmier K-sheet ,
, In type AIII there exist K-sheets containing no semisimple element and one can characterize them in terms of the partition ? associated to the nilpotent element e ? S G ? p as follows
, In type AIII, a K-sheet is Dixmier if and only if the partition ? satisfies: ? i ? ? i+1 is odd for at most one i ?
, Corollary 1.6.4 and a study of semisimple elements in e + c. Observe that the condition for a K
, When g is of type A the only rigid nilpotent orbit is {0}. In other cases it may happen that a rigid orbit O 1 contains a non-rigid orbit O 2 in its closure (see the classification of rigid nilpotent orbits in [CM]). Observe that, since the nilpotent cone is closed, a sheet containing O 2 cannot be contained in the closure of O 1 . One gets in this way some sheets whose closure is not a union of sheets. One can ask if similar facts occur for symmetric pairs (g, k), in particular when g is of type A. Let (g, k, p) be a symmetric Lie algebra; a nilpotent K-orbit in p which is a K-sheet in p ?
, Assume that (g, k, p) is of type AIII, z(g) ? k, and recall from the proof of Proposition 4.1.3 (using Remark 1.4.8) that S K (K.e) = K.e if and only if dim c = 0. The arguments given in (2) about K-sheets can be, {0} is the only rigid nilpotent K-orbit in these cases
, Note that the previous result depends only on the partition ? and not on the ab-diagram of e. In particular, K.e is rigid if and only if each K-orbit
This orbit contains in its closure a nilpotent K-orbit O 2 with partition (3, 1, 1, 1), cf. [Oh2], but O 2 is not rigid ,
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