The semi-infinite Finkelberg–Mirković conjecture

Let $\ell\geq h$. The category $\mathrm{Perv}_{I_\mathrm{u}}(\mathcal{F}l^{\frac{\infty}{2}})$ has simple objects $\mathrm{IC}_x^{\frac{\infty}{2}}$, standard objects $\nabla_x^{\frac{\infty}{2}}$ and costandard objects $\Delta_x^{\frac{\infty}{2}}$ indexed by $x\in W_{\mathrm{ext}}$. The convolution functor $\widetilde{\mathrm{For}}_0$ and the forgetful functor $\mathrm{For}_0$ are as defined in the source, and $\mathrm{FM}$, $\widehat{\mathrm{L}}$, $\widehat{\mathrm{Z}}'$ and $\widehat{\mathrm{Z}}$ denote the corresponding representation-theoretic functors and modules. Semi-infinite Finkelberg–Mirković conjecture. There exists an equivalence of categories

$$\widetilde{\mathrm{FM}}:\mathrm{Perv}_{I_\mathrm{u}}(\mathcal{F}l^{\frac{\infty}{2}})\xrightarrow{\sim}\mathrm{Rep}_{[0]}(\check{\mathbf{G}}_1\check{\mathbf{T}})$$

such that $\widetilde{\mathrm{FM}}\circ\widetilde{\mathrm{For}}_0\simeq \mathrm{For}_0\circ\mathrm{FM}$, with

$$\widetilde{\mathrm{FM}}(\mathrm{IC}_x^{\frac{\infty}{2}})\simeq \widehat{\mathrm{L}}(x^{-1}\cdot_\ell0),\quad \widetilde{\mathrm{FM}}(\nabla_x^{\frac{\infty}{2}})\simeq \widehat{\mathrm{Z}}'(x^{-1}\cdot_\ell0),\quad \widetilde{\mathrm{FM}}(\Delta_x^{\frac{\infty}{2}})\simeq \widehat{\mathrm{Z}}(x^{-1}\cdot_\ell0),$$

for all $x\in W_{\mathrm{ext}}$, and

$$\widetilde{\mathrm{FM}}(\mathcal{G}\star\mathcal{F})\simeq \widetilde{\mathrm{FM}}(\mathcal{F})\otimes\mathrm{Fr}^*(\mathrm{Sat}(\mathrm{sw}^*\mathcal{G}))$$

functorially in $\mathcal{F}\in\mathrm{Perv}_{I_\mathrm{u}}(\mathcal{F}l^{\frac{\infty}{2}})$ and $\mathcal{G}\in\mathrm{Perv}_{G[[t]]}(\mathrm{Gr})$. This is the paper's proposed geometric–representation-theoretic correspondence, rooted in earlier work of Finkelberg and Mirković; the source provides no resolution status.