Slices for biparabolics of index 1

Abstract

Let \( \mathfrak{a} \) be an algebraic Lie subalgebra of a simple Lie algebra \( \mathfrak{g} \) with index \( \mathfrak{a} \) ≤ rank \( \mathfrak{g} \). Let \( Y\left( \mathfrak{a} \right) \) denote the algebra of \( \mathfrak{a} \) invariant polynomial functions on \( {\mathfrak{a}^*} \). An algebraic slice for \( \mathfrak{a} \) is an affine subspace η + V with \( \eta \in {\mathfrak{a}^*} \) and \( V \subset {\mathfrak{a}^*} \) subspace of dimension index \( \mathfrak{a} \) such that restriction of function induces an isomorphism of \( Y\left( \mathfrak{a} \right) \) onto the algebra R[η + V] of regular functions on η + V. Slices have been obtained in a number of cases through the construction of an adapted pair (h, η) in which \( h \in \mathfrak{a} \) is ad-semisimple, η is a regular element of \( {\mathfrak{a}^*} \) which is an eigenvector for h of eigenvalue minus one and V is an h stable complement to \( \left( {{\text{ad}}\;\mathfrak{a}} \right)\eta \) in \( {\mathfrak{a}^*} \). The classical case is for \( \mathfrak{g} \) semisimple [16], [17]. Yet rather recently many other cases have been provided; for example, if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is a “truncated biparabolic” [12] or a centralizer [13]. In some of these cases (in particular when the biparabolic is a Borel subalgebra) it was found [13], [14], that η could be taken to be the restriction of a regular nilpotent element in \( \mathfrak{g} \). Moreover, this calculation suggested [13] how to construct slices outside type A when no adapted pair exists. This article makes a first step in taking these ideas further. Specifically, let \( \mathfrak{a} \) be a truncated biparabolic of index one. (This only arises if \( \mathfrak{g} \) is of type A and \( \mathfrak{a} \) is the derived algebra of a parabolic subalgebra whose Levi factor has just two blocks whose sizes are coprime.) In this case it is shown that the second member of an adapted pair (h, η) for \( \mathfrak{a} \) is the restriction of a particularly carefully chosen regular nilpotent element of \( \mathfrak{g} \). A by-product of our analysis is the construction of a map from the set of pairs of coprime integers to the set of all finite ordered sequences of ±1.