The proposed structure of the irreducible standard representation for p/3 < c < p/2

Let $H_{1,c}(S_3,\mathfrak{h})$ be the rational Cherednik algebra over an algebraically closed field of characteristic $p>3$, with $c\in\mathbb{F}_p$. Let $M_{1,c}(\mathrm{Stand})$ be the Verma module for the standard representation, and let $L_{1,c}(\mathrm{Stand})$ be its irreducible quotient. For $p/3<c<p/2$, the proposed quotient is obtained by factoring out the submodule generated by one-dimensional vectors of types $\mathrm{Sign}$, $\mathrm{Triv}$, $\mathrm{Sign}$, and $\mathrm{Triv}$ in degrees $-p+3c$, $3p-3c$, $p+3c$, and $5p-3c$, respectively.

Proposed structure. The quotient and its character and Hilbert polynomial are

$$L_{1,c}(\mathrm{Stand})=M_{1,c}(\mathrm{Stand})/\langle\text{the four specified generating spaces}\rangle,$$ $$\chi_{L_{1,c}(\mathrm{Stand})}(z)=\chi_{S\mathfrak{h}^*}(z)\left([\mathrm{Stand}]-z^{-p+3c}[\mathrm{Sign}]-z^{3p-3c}[\mathrm{Triv}]-z^{p+3c}[\mathrm{Sign}]-z^{5p-3c}[\mathrm{Triv}]+z^{4p}[\mathrm{Stand}]\right),$$ $$h_{L_{1,c}(\mathrm{Stand})}(z)=\frac{2-z^{-p+3c}-z^{3p-3c}-z^{p+3c}-z^{5p-3c}+2z^{4p}}{(1-z)^2}.$$

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