[2110.06722] On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Representation Theory arXiv:2110.06722 (math) [Submitted on 13 Oct 2021 ( v1 ), last revised 29 Apr 2026 (this version, v4)] Title: On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures Authors: Bin Shu , Yunpeng Xue , Yufeng Yao View a PDF of the paper titled On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures, by Bin Shu and 1 other authors View PDF HTML (experimental) 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. Comments: Journal of Algebra (2026), in press Subjects: Representation Theory (math.RT) MSC classes: 20E45, 17B10, 05E10, 05E18 Cite as: arXiv:2110.06722 [math.RT] (or arXiv:2110.06722v4 [math.RT] for this version) https://doi.org/10.48550/arXiv.2110.06722 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Bin Shu [ view email ] [v1] Wed, 13 Oct 2021 13:50:45 UTC (38 KB) [v2] Thu, 14 Oct 2021 02:43:48 UTC (38 KB) [v3] Sun, 20 Feb 2022 04:35:02 UTC (38 KB) [v4] Wed, 29 Apr 2026 23:52:17 UTC (43 KB) Full-text links: Access Paper: View a PDF of the paper titled On enhanced reductive groups (II): Finiteness of nilpotent orbits under enhanced group action and their closures, by Bin Shu and 1 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.RT < prev | next > new | recent | 2021-10 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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