Mirror-support and multiplicity conjectures for reductive Hitchin systems

Let $c\in C$ and let $\mathrm G$ be a reductive group. For $\mu\in X_+(\mathrm G^\vee)$, let $\rho_\mu$ be the corresponding irreducible representation of the Langlands dual group, let ${\mathbb E}_c$ be the universal Higgs bundle restricted to $c$, and let $\delta_c^\mu$ denote the associated divisor. Let $W^+_{\delta_c^\mu}$ be the corresponding upward flow, let $h_{\mathrm G}:\overline{\overline{W^+_{\delta_c^\mu}}}\to{\mathcal A}$ be the restricted Hitchin map, and let $\overline{\operatorname{Gr}}^\mu$ be the corresponding affine Grassmannian Schubert variety. Mirror-support and multiplicity conjectures. For every $\mu\in X_+(\mathrm G^\vee)$, the support of the mirror of $\rho_\mu({\mathbb E}_c)$ is $\overline{\overline{W^+_{\delta_c^\mu}}}$. If $\rho_\mu$ is multiplicity free, then the mirror is ${\mathcal O}_{\overline{\overline{W^+_{\delta_c^\mu}}}}$, and the multiplicity algebra of $h_{\mathrm G}$ is isomorphic to the cohomology ring of $\overline{\operatorname{Gr}}^\mu$. These expectations extend the proposed mirror description for very stable upward flows and relate mirror sheaves to geometric representation theory.