Invariant holonomic systems on symmetric spaces and other polar representations

Abstract

Let $V$ be a symmetric space over a connected reductive Lie algebra $G$, with Lie algebra $\mathfrak{g}$ and discriminant $\delta\in \mathbb{C}[V]$. A fundamental object is the invariant holonomic system $\mathcal{G} =\mathcal{D}(V)\Big/ \Bigl(\mathcal{D}(V)\mathfrak{g}+ \mathcal{D}(V)(\mathrm{Sym}\, V)^G_+ \Bigr) $ over the ring of differential operators $\mathcal{D}(V)$. Jointly with Levasseur we have shown that there exists a surjective radial parts map $\mathrm{rad}$ from $ \mathcal{D}(V)^G$ to the spherical subalgebra $A_{\kappa}$ of a Cherednik algebra. When $A_{\kappa}$ is simple we show that $\mathcal{G}$ has no $\delta$-torsion submodule nor factor module and we determine when $\mathcal{G}$ is semisimple, thereby answering questions of Sekiguchi, respectively Levasseur-Stafford. In the diagonal case when $V=\mathfrak{g}$, these results reduce to fundamental theorems of Harish-Chandra and Hotta-Kashiwara. We generalise these results to polar representations $V$ satisfying natural conditions. By twisting the radial parts map, we obtain families of invariant holonomic systems. We introduce shift functors between the different twists. We show that the image of the simple summands of $\mathcal{G} $ under these functors is described by Opdam's KZ-twist.