Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6426172 | Advances in Mathematics | 2011 | 14 Pages |
Let G be a reductive algebraic group over C and denote its Lie algebra by g. Let Oh be a closed G-orbit through a semisimple element hâg. By a result of Borho and Kraft (1979) [4], it is known that the asymptotic cone of the orbit Oh is the closure of a Richardson nilpotent orbit corresponding to a parabolic subgroup whose Levi component is the centralizer ZG(h) in G. In this paper, we prove an analogue on a semisimple orbit for a symmetric pair.More precisely, let θ be an involution of G, and K=Gθ a fixed point subgroup of θ. Then we have a Cartan decomposition g=k+s of the Lie algebra g=Lie(G) which is the eigenspace decomposition of θ on g. Let {x,h,y} be a normal sl2 triple, where x,yâs are nilpotent, and hâk semisimple. In addition, we assume x¯=y, where x¯ denotes the complex conjugation which commutes with θ. Then a=â1(xây) is a semisimple element in s, and we can consider a semisimple orbit Ad(K)a in s, which is closed. Our main result asserts that the asymptotic cone of Ad(K)a in s coincides with Ad(G)xâ©s¯, if x is even nilpotent.