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

Published version