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 $N_G(T)\rtimes D,$ where $N_G(T)$ is the normalizer of $T$ in $G$ and $D$ is the group of all automorphisms of the Dynkin diagram, if $G\not= {\rm PSL}(2,\mathbb{C})$. Note that if $G = {\rm PSL}(2,\mathbb{C})$, then $\overline{T} = {\mathbb C}{\mathbb P}^1$ and so in this case $\text{Aut}(\overline T)= {\rm PSL}(2,\mathbb{C})$.