On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures

TL;DR

This paper classifies when the nilpotent cone of an enhanced algebraic group has finitely many orbits, explores the geometry of these orbits, and describes their closures and intersection cohomology.

Contribution

It provides a complete classification of finite nilpotent orbits under enhanced group actions and describes their geometric and cohomological properties.

Findings

Finite nilpotent orbits occur only for specific modules: 1D, natural, dual (for n>2), or small irreducible modules.

Classifies finite nilpotent orbits via enhanced partitions.

Describes orbit closures using enhanced flag varieties and establishes intersection cohomology decomposition.

Abstract

This is a sequel to \cite{osy} and \cite{sxy}. Associated with $G:=\GL_n$ and its rational representation $(\rho, M)$ over an algebraically closed filed $\bk$, we define an enhanced algebraic group $\uG:=G\ltimes_\rho M$ which is a product variety $\GL_n\times M$, endowed with an enhanced cross product. In this paper, we first show that the nilpotent cone $\ucaln:=\caln(\ugg)$ of the enhanced Lie algebra $\ugg:=\Lie(\uG)$ has finite nilpotent orbits under adjoint $\uG$-action if and only if up to tensors with one-dimensional modules, $M$ is isomorphic to one of the three kinds of modules: (i) a one-dimensional module, (ii) the natural module $\bk^n$, (iii) the linear dual of $\bk^n$ when $n>2$; and $M$ is an irreducible module of dimension not bigger than $3$ when $n=2$. We then investigate the geometry of enhanced nilpotent orbits when the finiteness occurs. Our focus is on the enhanced group $\uG=\GL(V)\ltimes_{\eta}V$ with the natural representation $(\eta, V)$ of $\GL(V)$, for which we give a precise classification of finite nilpotent orbits via a finite set $\scrpe$ of so-called enhanced partitions of $n=\dim V$, then give a precise description of the closures of enhanced nilpotent orbits via constructing so-called enhanced flag varieties. Finally, the $\uG$-equivariant intersection cohomology decomposition on the nilpotent cone of $\ugg$ along the closures of nilpotent orbits is established.