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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Proof of a conjecture of Kostant


Author: Dragomir Ž. Đoković
Journal: Trans. Amer. Math. Soc. 302 (1987), 577-585
MSC: Primary 17B20; Secondary 17B45, 22E60
MathSciNet review: 891636
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Abstract: Let $ {\mathfrak{g}_0} = {\mathfrak{k}_0} + {\mathfrak{p}_0}$ be a Cartan decomposition of a semisimple real Lie algebra and $ \mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ its complexification. Denote by $ G$ the adjoint group of $ \mathfrak{g}$ and by $ {G_0},K,{K_0}$ the connected subgroups of $ G$ with respective Lie algebras $ {\mathfrak{g}_0},\mathfrak{k},{\mathfrak{k}_0}$. A conjecture of Kostant asserts that there is a bijection between the $ {G_0}$-conjugacy classes of nilpotent elements in $ {\mathfrak{g}_0}$ and the $ K$-orbits of nilpotent elements in $ \mathfrak{p}$ which is given explicitly by the so-called Cayley transformation. This conjecture is proved in the paper.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0891636-0
PII: S 0002-9947(1987)0891636-0
Article copyright: © Copyright 1987 American Mathematical Society