The parabolic-Whittaker geometric equivalence

Let $P$ be a parabolic subgroup of $G$, let $Q$ be another parabolic subgroup, and let $D_{IW_Q}({{\mathcal F}\ell}_P)$ be the corresponding category of partial Whittaker sheaves on the partial affine flag variety. Let ${Q\check{\ }}$ and ${P\check{\ }}$ be the corresponding parabolic subgroups of ${ G\check{\ }}$, and define

$$\tilde{{\frak g}}_{Q\check{\ }}=\{({\frak q},x)\mid x\in{\frak q}\},\qquad {\tilde{\mathcal N}}_{P\check{\ }}=\{({\frak p},x)\mid x\in\operatorname{rad}({\frak p})\}.$$

Then the parabolic-Whittaker equivalence conjecture. There is a canonical equivalence

$$D_{IW_Q}({{\mathcal F}\ell}_P)\cong D^b\left(\operatorname{Coh}^{ G\check{\ }}(\tilde{{\frak g}}_{Q\check{\ }}\times_{ {\frak g\check{\ }}}{\tilde{\mathcal N}}_{P\check{\ }})\right).$$

This conjecture proposes a geometric description of partial Whittaker categories in terms of coherent sheaves on a fiber product of parabolic and nilpotent resolutions, generalizing the equivalences discussed earlier in the paper.