Properties of the quantum Langlands image of automorphic sheaves

Let $X$ be a smooth projective curve and let $E$ be an irreducible $\operatorname{SL}_2$-local system on $X$. Let $\operatorname{Aut}_E\in\operatorname{D}(\operatorname{Bun}_H)$ be the corresponding automorphic sheaf, let $\mathcal K_E:=QL(\operatorname{Aut}_E)$, and retain the functors $F_G$ and $F_H$ and the Fourier transform appearing in the source.

Whittaker and theta-lifting conjecture. (i) $\mathcal K_E$ is a perverse sheaf, each of whose $\mathbb Z/2\mathbb Z$-parity pieces is irreducible, and

$$\mathbb D(\mathcal K_E)\\,\widetilde\to\\,\mathcal K_E.$$

(ii) The complex $F_HF_G(\operatorname{Aut}_E)$ is isomorphic to $\operatorname{Aut}_E$ tensored by

$$\bigoplus_{\theta\geq 0}\operatorname{R\Gamma}(X^{(\theta)},E^{(\theta)})[\theta].$$

(iii) Assuming the square-root conjecture at ${\cal E}={\cal O}$, with $SQ_E$ the corresponding fibre, there is, for a suitable map $\underline p:\operatorname{RCov}^d\to B(\mu_2)$, an isomorphism over $\operatorname{RCov}^d$

$$(\operatorname{Four}_\psi\tilde\nu_B^*F_G(\operatorname{Aut}_E))|_{\operatorname{RCov}^d}\\,\widetilde\to\\,\underline p^*SQ_E\otimes(\operatorname{Four}_\psi\tilde\nu_B^*\mathcal K_E)|_{\operatorname{RCov}^d}.$$

These assertions connect quantum Langlands, theta lifting, and geometric Whittaker coefficients. They are stated conjecturally in the source; part (iii) is explicitly conditional on the square-root conjecture.