On enhanced general linear groups: nilpotent orbits and support variety for Weyl module

Yunpeng Xue; Yunpeng Xue

doi:10.3934/math.2023765

AIMS Mathematics

2023, Volume 8, Issue 7: 14997-15007. doi: 10.3934/math.2023765

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Research article

On enhanced general linear groups: nilpotent orbits and support variety for Weyl module

Yunpeng Xue ^,

School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

Received: 05 February 2023 Revised: 08 April 2023 Accepted: 13 April 2023 Published: 23 April 2023
MSC : 20E45, 17B10, 05E10, 05E18

Associated with a reductive algebraic group $ G $ and its rational representation $ (\rho, M) $ over an algebraically closed filed $ {\bf{k}} $, the authors define the enhanced reductive algebraic group $ {\underline{G}}: = G\ltimes_\rho M $, which is a product variety $ G\times M $ and endowed with an enhanced cross product in ^[5]. If $ {\underline{G}} = GL(V)\ltimes_{\eta} V $ with the natural representation $ (\eta, V) $ of $ {\text{GL}}(V) $, it is called an enhanced general linear algebraic group. And the authors give a precise classification of finite nilpotent orbits via a finite set of so-called enhanced partitions of $ n = \dim V $ for the enhanced group $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $ in [6, Theorem 3.5]. We will give another way to prove this classification theorem in this paper. Then we focus on the support variety of the Weyl module for $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $ in characteristic $ p $, and obtain that it coinsides with the closure of an enhanced nilpotent orbit under some mild condition.
- enhanced nilpotent orbits,
- enhanced general linear algebric groups,
- support variety,
- Weyl module
Citation: Yunpeng Xue. On enhanced general linear groups: nilpotent orbits and support variety for Weyl module[J]. AIMS Mathematics, 2023, 8(7): 14997-15007. doi: 10.3934/math.2023765

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Abstract

Associated with a reductive algebraic group $ G $ and its rational representation $ (\rho, M) $ over an algebraically closed filed $ {\bf{k}} $, the authors define the enhanced reductive algebraic group $ {\underline{G}}: = G\ltimes_\rho M $, which is a product variety $ G\times M $ and endowed with an enhanced cross product in ^[5]. If $ {\underline{G}} = GL(V)\ltimes_{\eta} V $ with the natural representation $ (\eta, V) $ of $ {\text{GL}}(V) $, it is called an enhanced general linear algebraic group. And the authors give a precise classification of finite nilpotent orbits via a finite set of so-called enhanced partitions of $ n = \dim V $ for the enhanced group $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $ in [6, Theorem 3.5]. We will give another way to prove this classification theorem in this paper. Then we focus on the support variety of the Weyl module for $ {\underline{G}} = {\text{GL}}(V)\ltimes_{\eta}V $ in characteristic $ p $, and obtain that it coinsides with the closure of an enhanced nilpotent orbit under some mild condition.

References

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