Wonderful compactification of a cartan subalgebra of a semisimple lie algebra

Abstract Let $\operatorname{\mathfrak h}$ be a Cartan subalgebra of a complex semisimple Lie algebra $\operatorname{\mathfrak g}.$ We define a compactification $\bar{\operatorname{\mathfrak h}}$ of $\operatorname{\mathfrak h}$, which is analogous to the closure $\bar{H}$ of the corresponding maximal torus $H$ in the adjoint group of $\operatorname{\mathfrak g}$ in its wonderful compactification, which was introduced and studied by De Concini and Procesi [11]. We observe that $\bar{\operatorname{\mathfrak h}}$ is a matroid Schubert variety and prove that the irreducible components of the boundary $\bar{\operatorname{\mathfrak h}} - \operatorname{\mathfrak h}$ of $\operatorname{\mathfrak h}$ are divisors indexed by root system data. We prove that $\bar{\operatorname{\mathfrak h}}$ is a normal variety and find an affine paving of $\bar{\operatorname{\mathfrak h}},$ where the strata are given by the orbits of $\operatorname{\mathfrak h}.$ We show that the strata of $\bar{\operatorname{\mathfrak h}}$ 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 $\bar{\operatorname{\mathfrak h}}$ in terms of well-known combinatorial invariants in the classical cases. We show that the Weyl group $W$ acts on $\bar{\operatorname{\mathfrak h}}$, and describe $H^{\bullet }(\bar{\operatorname{\mathfrak h}}, \operatorname{\mathbb C})$ as a representation of $W$, and compute the cup product for $H^{\bullet }(\bar{\operatorname{\mathfrak h}}, \operatorname{\mathbb Z})$.