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• W2007409105 abstract "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 h∈g. 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,y∈s are nilpotent, and h∈k semisimple. In addition, we assume x¯=y, where x¯ denotes the complex conjugation which commutes with θ. Then a=−1(x−y) 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)x∩s¯, if x is even nilpotent." @default.
• W2007409105 created "2016-06-24" @default.
• W2007409105 creator A5032414388 @default.
• W2007409105 date "2011-03-01" @default.
• W2007409105 modified "2023-10-16" @default.
• W2007409105 title "Asymptotic cone of semisimple orbits for symmetric pairs" @default.
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• W2007409105 doi "https://doi.org/10.1016/j.aim.2010.12.004" @default.
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