Equation of some wonderful compactifications

TL;DR

This paper investigates the defining equations of the wonderful compactification of certain symmetric spaces, showing that under specific rank conditions, it is characterized by linear equations related to invariant forms.

Contribution

It proves that when the rank of the symmetric space equals the rank of the group, the compactification is defined by linear equations involving invariant trilinear forms.

Findings

The variety is defined by linear equations when ranks are equal.

Invariant alternate trilinear form vanishes on the (-1)-eigenspace.

Provides explicit equations for the compactification in this case.

Abstract

De Concini and Procesi have defined the wonderful compactification of a symmetric space X=G/H with G a semisimple adjoint group and H the subgroup of fixed points of G by an involution s. It is a closed subvariety of a grassmannian of the Lie algebra L of G. In this paper, we prove that, when the rank of X is equal to the rank of G, the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form w on L vanishes on the (-1)-eigenspace of s.