Wonderful compactification

In algebraic group theory, a wonderful compactification of a variety acted on by an algebraic group $$G$$ is a $$G$$-equivariant compactification such that the closure of each orbit is smooth. constructed a wonderful compactification of any symmetric variety given by a quotient $$G/G^{\sigma}$$ of an algebraic group $$G$$ by the subgroup $$G^{\sigma}$$ fixed by some involution $$\sigma$$ of $$G$$ over the complex numbers, sometimes called the De Concini–Procesi compactification, and  generalized this construction to arbitrary characteristic. In particular, by writing a group $$G$$ itself as a symmetric homogeneous space, $$G=(G \times G)/G$$ (modulo the diagonal subgroup), this gives a wonderful compactification of the group $$G$$ itself.