Slices for biparabolics of index one

Details Slices for biparabolics of index one Slices for biparabolics of index one

2010-11-03

arxiv.org/abs/1011.0928v1

Mathematics

Representation Theory

31 pages, 7 figures

Scientific paper

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 $\mathfrak a^*$. An algebraic slice for $\mathfrak a$ is an affine subspace $\eta+V$ with $\eta \in \mathfrak a^*$ and $V \subset \mathfrak a^*$ 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 $\mathfrak a^*$ which is an eigenvector for $h$ of eigenvalue minus one and $V$ is an $h$ stable complement to $(\ad \mathfrak a)\eta$ in $\mathfrak a^*$. 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 $\mathfrak g$.

Affiliated with

Also associated with

No associations

Rating

Slices for biparabolics of index one does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Slices for biparabolics of index one, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Slices for biparabolics of index one will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-603687