Automorphisms of $\overline{T}$

TL;DR

This paper proves that for a certain class of algebraic groups, the maximal torus is exactly the identity component of the automorphism group of its closure in the wonderful compactification, revealing structural symmetries.

Contribution

It establishes a precise characterization of the automorphism group of the torus closure within the wonderful compactification for a broad class of algebraic groups.

Findings

The maximal torus equals the identity component of the automorphism group.

The result applies to simple affine algebraic groups with trivial center, excluding PSL(2,C).

Provides insight into the symmetry structure of compactified algebraic groups.

Abstract

Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb C$ such that its center is trivial and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and denote by $\overline T$ its closure in $\overline G$. We prove that $T$ coincides with the connected component, containing the identity element, of the group of automorphisms of the variety $\overline T$.