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.

中文翻译：

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李代数表示的高阶变形 II

Steinberg 的张量积定理表明，对于半单代数群，对更高 Frobenius 核的不可约表示的研究归结为对第一个 Frobenius 核的不可约表示的研究。在本系列的前一篇论文中，对更高 Frobenius 核的分布代数进行变形会产生一系列变形，称为更高约简包络代数。在本文中，我们证明了 Steinberg 分解可以类似地变形，从而使我们能够将关于这些代数的表示理论问题简化为关于约简包络代数的问题。我们使用它来推导出关于这些代数上的模块的结构结果。另外，我们还表明，在没有还原性假设的情况下，前面论文中的许多结果都成立。