Representations of the rational Cherednik algebra $H_{t,c}(S_3,\h)$ in positive characteristic

TL;DR

This paper investigates the structure of rational Cherednik algebra representations for type A2 in positive characteristic, providing explicit calculations of Hilbert polynomials, characters, and submodules across all parameters.

Contribution

It offers a comprehensive analysis of irreducible category O representations of the rational Cherednik algebra in positive characteristic, including explicit formulas and generators.

Findings

Calculated Hilbert polynomials for all parameters.

Determined characters of irreducible modules.

Explicit generators of maximal proper submodules.

Abstract

We study the rational Cherednik algebra $H_{t,c}(S_3,\h)$ of type $A_2$ in positive characteristic $p$, and its irreducible category $\mathcal{O}$ representations $L_{t,c}(\tau)$. For every possible value of $p,t,c$, and $\tau$ we calculate the Hilbert polynomial and the character of $L_{t,c}(\tau)$, and give explicit generators of the maximal proper graded submodule of the Verma module.