The Torus Centralizing Subalgebra of $\text{Dist}(G_r)$

TL;DR

We study Dist(G_r)^T, the subalgebra ofDist(G_r) fixed by the adjoint action of a maximal torus in characteristic $p>0$, and show it has a PBW-like basis and is augmented over Dist$(T_r)$, yet is noncommutative and not a Hopf subalgebra in general. We classify simple Dist(G_r)^T-modules by restricting weight spaces of simple $G_rT$-modules, establishing a precise equivalence of simples via weight data modulo $p^r$. This framework connects the representation theory of Dist$(G_r)^T$ to that of $G_rT$, enabling transfer of composition multiplicities and offering a path to computing characters of simple $G_rT$-modules and, ultimately, simple $G$-modules. The paper also discusses automorphisms, generators, and open problems, including the minimal generating set for Dist$(G_r)^T$ and the broader applicability to multiplicities in $G_rT$-modules.

Abstract

Let $G$ be a simple and simply connected algebraic group over a field of characteristic $p>0$, and $G_r$ its $r$-th Frobenius kernel. In this paper, we initiate a general study of $\text{Dist}(G_r)^T$, the subalgebra of $\text{Dist}(G_r)$ consisting of fixed points for the adjoint action of a maximal torus $T$ of $G$. We analyze the structure of this algebra, and classify its simple modules, which essentially are just the non-zero weight spaces of the simple $G_rT$-modules of $p^r$-restricted highest weight. Further connections between the representations of $\text{Dist}(G_r)^T$ and $G_rT$ are shown, demonstrating the potential usefulness of this algebra.

Theorems & Definitions (18)

• Proposition 3.1.1
• Lemma 3.2.1
• proof
• Theorem 3.2.1
• proof
• Proposition 3.4.1
• proof
• Proposition 4.2.1
• proof
• Proposition 4.2.2