Holonomic modules over Cherednik algebras, I

TL;DR

This paper extends key concepts of $\

Contribution

It introduces a framework for holonomic modules over Cherednik algebras, connecting $\

Findings

Holonomic modules characterized by singular support and Gelfand-Kirillov dimension.

Pushforward preserves holonomicity for generic parameters.

Develops functorial operations on Cherednik modules.

Abstract

The goal of this paper is to generalize several basic results from the theory of $\cal{D}$-modules to the representation theory of rational Cherednik algebras. We relate characterizations of holonomic modules in terms of singular support and Gelfand-Kirillov dimension. We study pullback, pushforward, and dual on the derived category of (holonomic) Cherednik modules for certain classes of maps between varieties. We prove, in the case of generic parameters for the rational Cherednik algebra, that pushforward with respect to an open affine inclusion preserves holonomicity.