Principal-filtration conjecture for short Kostka–Foulkes analogues

Let $\lambda$ be dominant, let $V_\lambda$ be the irreducible $G$-module of highest weight $\lambda$, and let $V_\lambda^\mu$ be its $\mu$-weight space. Let $H$ be the connected semisimple subgroup with root system $\Delta_l$, let $\mathfrak X_{+,H}$ denote the $H$-dominant weights, and let $V_\lambda^{U(H),\mu}$ be the subspace of $H$-highest vectors in $V_\lambda^\mu$. For nonzero root vectors $e_\alpha$ with $\alpha\in\Delta^+$, set

$$e_s=\sum_{\alpha\in\Pi_s}e_\alpha,$$

and define

$$J^p_{e_s}(V_\lambda^{U(H),\mu})=\{v\in V_\lambda^{U(H),\mu}\mid e_s^{p+1}\cdot v=0\}.$$

The associated jump polynomial is

$$\overline r_\lambda^\mu(q)=\sum_{p\geqslant0}\dim\left(J^p_{e_s}(V_\lambda^{U(H),\mu})/J^{p-1}_{e_s}(V_\lambda^{U(H),\mu})\right)q^p.$$

Principal-filtration conjecture. If $\mu\in\mathfrak X_{+,H}$ satisfies the vanishing conditions of Theorem~, then

$$\overline r_\lambda^\mu(q)=\overline{{\mathfrak m}}_\lambda^\mu(q).$$

This is the proposed analogue, for short $q$-analogues, of Brylinski's principal-filtration description of Lusztig's $q$-analogues. The stated cohomological vanishing is the condition under which the equality is conjectured.