Metaplectic Casselman–Shalika conjecture

Let $G$ be the reductive group with coweight lattice $\Lambda$, let $\Lambda^{\sharp,+}$ be the dominant metaplectic coweights, and let $\gamma\in\Lambda^{\sharp,+}$ and $\mu,\nu\in\Lambda$ satisfy $\mu+\nu\in\Lambda^+$. Let ${\mathcal C}$ be the category of finite-dimensional representations of the corresponding big quantum group, let $W^{\lambda,!}$ and $W^{\lambda,*}$ be its standard and costandard objects, and let $\operatorname{Fr}:\operatorname{Rep}(\check G_\zeta)\to{\mathcal C}$ be the quantum Frobenius functor. The complex specified in the source by the referenced formula vanishes unless $\mu\in\Lambda^+$. In the latter case, Metaplectic Casselman–Shalika conjecture. there is an isomorphism

$$\operatorname{R\Gamma}_c(\operatorname{Gr}_B^\nu\cap\overline{\operatorname{Gr}}_G^\gamma,(\chi_\mu^\nu)^*\mathcal L_\psi\otimes(s_B^\nu)^*\mathcal A_{\mathcal E}^\gamma)[\langle\nu,2\check\rho\rangle]\simeq\mathbb D\operatorname{RHom}_{\operatorname{D}({\mathcal C})}(W^{\mu+\nu,!},W^{\mu,*}\otimes\operatorname{Fr}(V(\gamma))).$$

This identifies the relevant metaplectic Whittaker cohomology with a derived Hom expression in the quantum-group category and is stated in the Casselman–Shalika section; the supplied text gives no resolution status.