An algebraic slice in the coadjoint space of the Borel and the Coxeter element

Abstract

Let g be a complex simple Lie algebra and b a Borel subalgebra. The algebra Y of polynomial semi-invariants on the dual b ⁎ of b is a polynomial algebra on rank g generators (Grothendieck and Dieudonné (1965–1967)) [16]. The analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action, that is an affine translate y + V of a vector subspace of b ⁎ such that the restriction map induces an isomorphism of Y onto the algebra R [ y + V ] of regular functions on y + V . This holds in type A and even extends to all biparabolic subalgebras (Joseph (2007)) [20]; but the construction fails in general even with respect to the Borel. Moreover already in type C ( 2 ) no algebraic slice exists. Very surprisingly the exception of type C ( 2 ) is itself an exception. Indeed an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of types B ( 2 m ) , C ( n ) and F ( 4 ) . Outside type A, the slice obtained meets an open dense subset of regular orbits, even though the special point y of the slice is not itself regular. This explains the failure of our previous construction.