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.