Proof of a conjecture of Kostant

Abstract

Let g 0 = k 0 + p 0 {\mathfrak {g}_0} = {\mathfrak {k}_0} + {\mathfrak {p}_0} be a Cartan decomposition of a semisimple real Lie algebra and g = k + p \mathfrak {g} = \mathfrak {k} + \mathfrak {p} its complexification. Denote by G G the adjoint group of g \mathfrak {g} and by G 0 , K , K 0 {G_0},K,{K_0} the connected subgroups of G G with respective Lie algebras g 0 , k , k 0 {\mathfrak {g}_0},\mathfrak {k},{\mathfrak {k}_0} . A conjecture of Kostant asserts that there is a bijection between the G 0 {G_0} -conjugacy classes of nilpotent elements in g 0 {\mathfrak {g}_0} and the K K -orbits of nilpotent elements in p \mathfrak {p} which is given explicitly by the so-called Cayley transformation. This conjecture is proved in the paper.