The full automorphism group of $\overline{T}$

Abstract

Let $\overline{G}$ be the wonderful compactification of a simple affine algebraic group $G$ of adjoint type defined over $\mathbb{C}.$ Let $\overline{T}\subset \overline{G}$ be the closure of a maximal torus $T\subset G.$ We prove that the group of all automorphisms of the variety $\overline{T}$ is the semi-direct product $N_G(T)⋊D,$ where $N_G(T)$ is the normalizer of $T$ in $G$ and $D$ is the group of all automorphisms of the Dynkin diagram, if $G\ne \mathrm{PSL}(2,\mathbb{C})$. Note that if $G=\mathrm{PSL}(2,\mathbb{C})$, then $\overline{T}=\mathbb{C}{\mathbb{P}}^1$ and so in this case $\text{Aut}(\overline{T})=\mathrm{PSL}(2,\mathbb{C})$.

Final version