Logarithmic Differential Operators on the Wonderful Compactification

If $G$ is a simply-connected semisimple complex algebraic group, its adjoint form admits a particularly nice equivariant completion called the De Concini--Procesi or wonderful compactification. The complement of the adjoint group in its wonderful compactification $\mathbf{X}$ is a divisor $\mathbf{Y}$ with normal crossings and smooth irreducible components, so it makes sense to consider the sheaf of logarithmic differential operators $\mathcal{D}_{\mathbf{X}, \mathbf{Y}}$ on the pair $(\mathbf{X}, \mathbf{Y})$. After reviewing the construction of $\mathbf{X}$ and $\mathcal{D}_{\mathbf{X}, \mathbf{Y}}$, we relate the latter object to a canonical $(G \times G)$-action on $\mathbf{X}$. In particular, this gives rise to a homomorphism from the Lie algebra $\mathfrak{g} \oplus \mathfrak{g}$ to the global logarithmic vector fields on $\mathbf{X}$, which extends to a homomorphism from the universal enveloping algebra to the global logarithmic differential operators on $\mathbf{X}$. We show that the latter homomorphism is surjective, compute its kernel, and relate the result to global differential operators on the adjoint group. We also demonstrate analogous results in the setting of logarithmic differential operators twisted by an invertible sheaf on $\mathbf{X}$. We end with a short application of the results to certain modules over $\mathcal{D}_{\mathbf{X}, \mathbf{Y}}$ with support on the closed orbit in $\mathbf{X}$, relating them to the now classical Beilinson-Bernstein theory of differential operators on flag varieties.