Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2021
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2110.06722 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915968248184832 |
|---|---|
| author | Shu, Bin Xue, Yunpeng Yao, Yufeng |
| author_facet | Shu, Bin Xue, Yunpeng Yao, Yufeng |
| contents | This is a sequel to \cite{osy} and \cite{sxy}. Associated with $G:=\GL_n$ and its rational representation $(ρ, M)$ over an algebraically closed filed $\bk$, we define an enhanced algebraic group $\uG:=G\ltimes_ρM$ 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 $\uG=\GL(V)\ltimes_ηV$ with the natural representation $(η, V)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_06722 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures Shu, Bin Xue, Yunpeng Yao, Yufeng Representation Theory 20E45, 17B10, 05E10, 05E18 This is a sequel to \cite{osy} and \cite{sxy}. Associated with $G:=\GL_n$ and its rational representation $(ρ, M)$ over an algebraically closed filed $\bk$, we define an enhanced algebraic group $\uG:=G\ltimes_ρM$ 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 $\uG=\GL(V)\ltimes_ηV$ with the natural representation $(η, V)$ 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. |
| title | On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures |
| topic | Representation Theory 20E45, 17B10, 05E10, 05E18 |
| url | https://arxiv.org/abs/2110.06722 |