Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra

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 $H^{\bullet}(\bar {\mathfrak h}, \mathbb Z)$.