The microlocal–Whittaker equivalence for semistable Higgs bundles

Let $C$ be a smooth projective curve, let $G$ be a reductive group with Langlands dual group $G^{\vee}$, and let $w$ denote the relevant stratum or weight. Write $\mathsf{W}_{{\mathcal{C}\mkern-3mu\mathit{Conn}\mkern1mu}_G, \, \mathcal{N}}(-w)$ for the corresponding Whittaker category, and let $\mathsf{D}^{\mathrm{micro}}({\mathcal{H}\mkern-4mu\mathit{iggs}\mkern1mu}_{G^{\vee}}(w)^{\mathrm{ss}})^{\mathrm{cp}}$ be the compact objects in the microlocal category of the semistable $w$-stratum of the Higgs moduli stack. Microlocal–Whittaker equivalence. There is an equivalence

$$\mathsf{W}_{{\mathcal{C}\mkern-3mu\mathit{Conn}\mkern1mu}_G, \, \mathcal{N}}(-w) \simeq \mathsf{D}^{\mathrm{micro}}({\mathcal{H}\mkern-4mu\mathit{iggs}\mkern1mu}_{G^{\vee}}(w)^{\mathrm{ss}})^{\mathrm{cp}}.$$

This is expected to identify the semiorthogonal decomposition on the Whittaker side with the compact microlocal semiorthogonal decomposition for the Higgs moduli stack; the statement is presented as a conjectural compatibility under the cited equivalence, and its general status is not established in the supplied text.