Slices for biparabolics of index one

Abstract

Let $\mathfrak{a}$ be an algebraic Lie subalgebra of a simple Lie algebra $\mathfrak{g}$ with index $\mathfrak a \leq \rank \mathfrak g$. Let $Y(\mathfrak{a})$ denote the algebra of $\mathfrak{a}$ invariant polynomial functions on ${\mathfrak{a}}^{\ast }$. An algebraic slice for $\mathfrak{a}$ is an affine subspace $\eta +V$ with $\eta \in {\mathfrak{a}}^{\ast }$ and $V\subset {\mathfrak{a}}^{\ast }$ a subspace of dimension index $\mathfrak{a}$ such that restriction of function induces an isomorphism of $Y(\mathfrak{a})$ onto the algebra $R[\eta +V]$ of regular functions on $\eta +V$. Slices have been obtained in a number of cases through the construction of an adapted pair $(h,\eta )$ in which $h\in \mathfrak{a}$ is ad-semisimple, $\eta$ is a regular element of ${\mathfrak{a}}^{\ast }$ which is an eigenvector for $h$ of eigenvalue minus one and $V$ is an $h$ stable complement to $(\ad \mathfrak a)η$ in ${\mathfrak{a}}^{\ast }$. The classical case is for $\mathfrak{g}$ semisimple. 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" or a centralizer. In some of these cases (particular when the biparabolic is a Borel subalgebra) it was found that $\eta$ could be taken to be the restriction of a regular nilpotent element in $\mathfrak{g}$. Moreover this calculation suggested 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 (and then $\mathfrak{g}$ is of type $A$). In this case it is shown that the second member of an adapted pair $(h,\eta )$ for $\mathfrak{a}$ is the restriction of a particularly carefully chosen regular nilpotent element of $\mathfrak{g}$.

31 pages, 7 figures