The mixed Kazhdan–Lusztig equivalence for the Iwahori category

Let $G$ be the reductive group, let $I$ be an Iwahori subgroup, and let $\rep_{-\kappa}^I$ denote the corresponding category of Iwahori-equivariant modules at level $-\kappa$. Let $\Rep_q^{\operatorname{mxd}}(G)$ be the mixed representation category, and let $\KL(G,-\kappa):=\rep_{-\kappa}^{G({\mathcal O})}$ be the spherical Kazhdan–Lusztig category. For $\mu\in\Lambda$, write $J_\mu$ for the corresponding translation action, and write $k^\mu\in\Rep(\T)$ for the associated character. For $\clambda\in\Lambda$, let $\W_{-\kappa}^{\clambda}$ and $\M_{q,\operatorname{mxd}}^{\clambda}$ denote the affine Weyl and mixed quantum standard objects. The conjectural extension of the Kazhdan–Lusztig equivalence. There exists an equivalence

$$\F_{-\kappa}:\rep_{-\kappa}^I\simeq \Rep_q^{\operatorname{mxd}}(G)$$

such that the square relating the spherical equivalence $\KL(G,-\kappa)\simeq \Rep_q(G)_{\operatorname{ren}}$ to the forgetful functors commutes, the action of $J_\mu$ corresponds to the action of $k^\mu$, and

$$\F_{-\kappa}(\W_{-\kappa}^{\clambda})\simeq \M_{q,\operatorname{mxd}}^{\clambda}.$$

This is the paper’s proposed Iwahori-level extension of the Kazhdan–Lusztig equivalence; the source presents it as conjectural and uses it as the basis for subsequent comparisons.