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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
DOI: https://doi.org/10.1090/S0002-9947-1987-0891636-0
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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Article copyright: © Copyright 1987 American Mathematical Society