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Let p denote a maximal (truncated) parabolic subalgebra of a simple Lie algebra g$$ \mathfrak{g} $$. It is known that the Poisson centre Y (p$$ \mathfrak{p} $$) is a polynomial algebra in many cases. We construct a slice for the coadjoint action of p, thus extending a theorem of Kostant. The role of the principal sl$$ \mathfrak{s}\mathfrak{l} $$2-triple is played by an adapted pair.

SLICES FOR MAXIMAL PARABOLIC SUBALGEBRAS OF A SEMISIMPLE LIE ALGEBRA