Abstract
Steinberg's tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.
| Original language | English |
|---|---|
| Pages (from-to) | 232-255 |
| Journal | Nagoya Mathematical Journal |
| Volume | 244 |
| Early online date | 2 Jun 2020 |
| DOIs | |
| Publication status | Published - Dec 2021 |
Bibliographical note
Funding Information:Proof. ΩP : Z[r](G) → Z(g) is a homomorphism of commutative algebras, so it induces a morphism Inna Capdeboscq for their continued assistance with this project. I would also like to thank Lewis Topley for some useful discussions regarding this subject. Finally, I want to thank Alexander Premet, Adam Thomas, and the referee for their useful comments, which have helped improve the paper. I was supported during this research by a PhD studentship from the Engineering and Physical Sciences Research Council.
Keywords
- Frobenius kernel
- higher universal enveloping algebra
- representation
- Steinberg decomposition
ASJC Scopus subject areas
- General Mathematics