Categorical Gelfand–Kazhdan conjecture for cuspidal representations

Let $G=GL_n(\a0\mathcal{K})$ and let $P$ be its mirabolic subgroup, with unipotent subgroup $U$ and character $\theta$ as above. Let $\a0\mathcal{C}$ be an irreducible cuspidal categorical representation of $G$, meaning a category equipped with a strong action of $G$ such that the Whittaker invariants functor establishes an equivalence

$$\mathcal{C}\simeq \operatorname{Vect}.$$

For the functor

$$\Phi:\mathcal{C}\to D(P)_{/(U,\theta)}$$

constructed from categorical matrix coefficients, the categorical Gelfand–Kazhdan conjecture. The functor $\Phi$ is an equivalence of categories. This is proposed as a categorical analogue of the Gelfand–Kazhdan theorem, which identifies the restriction of a cuspidal irreducible representation of $GL_n(\mathcal{K})$ to the mirabolic subgroup with its standard irreducible representation. The conjecture concerns the extension of this phenomenon to strong categorical representations; no resolution is supplied here.