Categorical Heisenberg action I: rational Cherednik algebras

TL;DR

This paper develops a categorical Heisenberg algebra action on modules over rational Cherednik algebras linked to symmetric groups, demonstrating exactness, equivalence with prior constructions, and behavior in modular reduction.

Contribution

It introduces a new categorical Heisenberg action on rational Cherednik algebra modules and relates it to existing frameworks, including modular reduction analysis.

Findings

The functor for the Heisenberg action is exact.

The categorical action on category O matches Shan and Vasserot's construction.

Simple objects become semisimple under the action in large prime characteristic.

Abstract

In this paper we introduce and study a categorical action of the positive part of the Heisenberg Lie algebra on categories of modules over rational Cherednik algebras associated to symmetric groups. We show that the generating functor for this action is exact. We then produce a categorical Heisenberg action on the categories $\mathcal{O}$ and show it is the same as one constructed by Shan and Vasserot. Finally, we reduce modulo a large prime $p$. We show that the functors constituting the action of the positive half of the Heisenberg algebra send simple objects to semisimple ones, and we describe these semisimple objects.