On differential operators on complete symmetric varieties of type $A_1$ and $A_2$

TL;DR

This paper studies the algebra of global differential operators on certain symmetric varieties, revealing new operators, finite type structure, and simplicity of modules, with implications for cohomology.

Contribution

It constructs a new global differential operator not from the Lie algebra action and analyzes the structure of differential operator algebras on symmetric varieties of types A1 and A2.

Findings

Existence of a new global differential operator outside the Lie algebra action

Finite type algebra of differential operators for type A2 varieties

Cohomology groups are either zero or simple modules over differential operators

Abstract

In this paper, we will look at the algebra of global differential operators $D_X$ on wonderful compactifications $X$ of symmetric spaces $G/H$ of type $A_1$ and $A_2$. We will first construct a global differential operator on these varieties that does not come from the infinitesimal action of $\mathfrak{g}$. We will then focus on type $A_2$, where we will show that $D_X$ is an algebra of finite type, and that for any invertible sheaf ${\cal L}$ on $X$, $H^{0}(X,{\cal L})$ is either 0 or a simple left $D_{X,{\cal L}}$-module. Finally, we will show with the help of local cohomology that this is still true for higher cohomology groups $H^{i}(X,{\cal L})$.