An algebraic slice in the coadjoint space of the Borel and the Coxeter element
Let \(\mathfrak{g}\) be a complex simple Lie algebra and \(\mathfrak{b}\) a Borel subalgebra. There is a canonically determined ideal \(\mathfrak{b}_E\) of \(\mathfrak{b}\) such that the algebra generated by semi-invariants on \(\mathfrak{b}^*\) coincides with the algebra \(Y(\mathfrak{b}_E)\) of invariants on \(\mathfrak{b}^*_E\). If \(\mathfrak{g}\) is of type \(A\) there exists an algebraic slice to coadjoint action defined to be an affine translate \(y+V\subset\mathfrak{b}^*_E\) of a vector subspace \(V\) of \(\mathfrak{b}^*_E\) such that the restriction map induces an isomorphism of \(Y(\mathfrak{b}^*_E)\) onto the algebra \(R[y+V]\) of regular functions on \(y+V\). This result also holds to all biparabolic subalgebras in type \(A\). The author proves that an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of type \(B(2m)\), \(C(n)\), 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 the previous construction.
scientific article; zbMATH DE number 5880833
scientific article