Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra

Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $\bar H$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $\mathfrak h$ of $H$, we define an analogous compactification $\bar {\mathfrak h}$ of $\mathfrak h$, to be referred to as the wonderful compactification of $\mathfrak h$, as a subvariety of a variety of Lagrangian subalgebras.  We will describe various properties of the cohomology of $\bar {\mathfrak h}$. In particular, we will connect the Betti numbers of $\bar {\mathfrak h}$ with some classical combinatorial sequences (Stirling numbers, Whitney numbers of the Dowling lattice, etc.). The ring structure of the cohomology of $\bar {\mathfrak h}$ will be explained in terms of the intersection lattice of the Coxeter hyperplane arrangement.
This is joint work with Sam Evens.