The torus centralizing subalgebra of 𝐷𝑖𝑠𝑡(𝐺ᵣ)

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