Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra

TL;DR

This paper introduces a new compactification of a Cartan subalgebra in a semisimple Lie algebra, analyzing its geometric, combinatorial, and topological properties, and relating it to known structures like matroid Schubert varieties and hyperplane arrangements.

Contribution

It constructs and studies the properties of a compactification of a Cartan subalgebra, linking it to matroid Schubert varieties, hyperplane arrangements, and Weyl group actions.

Findings

The boundary components are divisors indexed by root system data.

The variety is normal and admits an affine paving with strata from orbits.

Betti numbers are expressed via combinatorial invariants.

Abstract

Let $\mathfrak h$ be a Cartan subalgebra of a complex semisimple Lie algebra $\mathfrak g.$ We define a compactification $\bar {\mathfrak h}$ of $\mathfrak h$, which is analogous to the closure $\bar H$ of the corresponding maximal torus $H$ in the adjoint group of $\mathfrak g$ in its wonderful compactification, which was introduced and studied by De Concini and Procesi \cite{DCP}. We observe that $\bar {\mathfrak h}$ is a matroid Schubert variety and prove that the irreducible components of the boundary $\bar {\mathfrak h} - \mathfrak h$ of $\mathfrak h$ are divisors indexed by root system data. We prove that $\bar {\mathfrak h}$ is a normal variety and find an affine paving of $\bar {\mathfrak h},$ where the strata are given by the orbits of $\mathfrak h.$ We show that the strata of $\bar {\mathfrak h}$ correspond bijectively to subspaces of the corresponding Coxeter hyperplane arrangement studied by Orlik and Solomon, and prove that the associated posets are isomorphic. As a consequence, we express the Betti numbers of $\bar {\mathfrak h}$ in terms of well-known combinatorial invariants in the classical cases. We show that the Weyl group $W$ acts on $\bar {\mathfrak h}$, and describe $H^{\bullet}(\bar {\mathfrak h}, \mathbb C)$ as a representation of $W$, and compute the cup product for $H^{\bullet}(\bar {\mathfrak h}, \mathbb Z)$.