Proof of a conjecture of Kostant

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.