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

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.