Two problems in the representation theory of reduced enveloping algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912157758652416 |
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| author | Westaway, Matthew |
| author_facet | Westaway, Matthew |
| contents | In this paper we consider two problems relating to the representation theory of Lie algebras ${\mathfrak g}$ of reductive algebraic groups $G$ over algebraically closed fields ${\mathbb K}$ of positive characteristic $p>0$. First, we consider the tensor product of two baby Verma modules $Z_χ(λ)\otimes Z_{χ'}(μ)$ and show that it has a filtration of baby Verma modules of a particular form. Secondly, we consider the minimal-dimension representations of a reduced enveloping algebra $U_χ({\mathfrak g})$ for a nilpotent $χ\in{\mathfrak g}^{*}$. We show that under certain assumptions in type $A$ we can obtain the minimal-dimensional modules as quotients of certain modules obtained by base change from simple highest weight modules over ${\mathbb C}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two problems in the representation theory of reduced enveloping algebras Westaway, Matthew Representation Theory Rings and Algebras 17B10, 17B50 (Primary), 17B08, 17B20 (Secondary) In this paper we consider two problems relating to the representation theory of Lie algebras ${\mathfrak g}$ of reductive algebraic groups $G$ over algebraically closed fields ${\mathbb K}$ of positive characteristic $p>0$. First, we consider the tensor product of two baby Verma modules $Z_χ(λ)\otimes Z_{χ'}(μ)$ and show that it has a filtration of baby Verma modules of a particular form. Secondly, we consider the minimal-dimension representations of a reduced enveloping algebra $U_χ({\mathfrak g})$ for a nilpotent $χ\in{\mathfrak g}^{*}$. We show that under certain assumptions in type $A$ we can obtain the minimal-dimensional modules as quotients of certain modules obtained by base change from simple highest weight modules over ${\mathbb C}$. |
| title | Two problems in the representation theory of reduced enveloping algebras |
| topic | Representation Theory Rings and Algebras 17B10, 17B50 (Primary), 17B08, 17B20 (Secondary) |
| url | https://arxiv.org/abs/2409.17784 |