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Let 8 be a semisimple Lie algebra over C and let 0 be a complex linear involution of g. Then we obtain a direct sum decomposition g=!+F, where I-{Xeg; 0(X)=X} and V={XtE§6 ( X ) = -X}. In this paper the pair (9, I) is called a symmetric pair and V is the vector space associated to it. Our main concern is the nilpotent subvariety Jf (V) of V. An element X of V is contained in Jf (V) if and only if adz(X) is nilpotent. We first describe the motivation of the study. Let us assume that 9 is a direct sum of another semisimple Lie algebra g', that is, g=g'©g' and the involution 0 is defined by 0(X, Y) = (Y, X) (X, Yea')Then we obtain a symmetric pair (8, I) and the vector space V for 0, In this case, I and V are obviously isomorphic to a' itself. Furthermore, the nilpotent subvariety Jf(V) of V is identified with that of 8'Then due to the celebrated result of Brieskorn [Br], we find that if g' is simple of type A/, D/ or Eh the generic singularity cf <A(V) is smoothly equivalent to the rational double point of the corresponding type. In the case when g' is simple of type BI, Ci, F^ or G2? a similar result is obtaind by Slodowy [SI]. The results of Brieskorn and Slodowy naturally lead us to the problem of the determination of the generic singularities of J f ( V } for a general symmetric pair (g, !). When (g, I) is of the normal type in the sense of [S-S], we obtain a result similar to that of Brieskorn. But in general, it seems to be hard to determine the generic singularities. To explain the reasons, we mention some

The nilpotent subvariety of the vector space associated to a symmetric pair