Invariant holonomic systems on symmetric spaces and other polar representations

TL;DR

This paper studies invariant holonomic systems on symmetric spaces and polar representations, establishing their structure, simplicity, and semisimplicity, and connecting them to Cherednik algebras and KZ-twists.

Contribution

It generalizes fundamental theorems of Harish-Chandra and Hotta-Kashiwara to polar representations and introduces a framework linking invariant systems with Cherednik algebras.

Findings

Existence of a surjective radial parts map to Cherednik algebra

Conditions for holonomic systems to be semisimple

Description of simple summands via Opdam's KZ-twist

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.