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More a comment than an answer:

I think the lagrangian-to-sheaf dictionary is more “immediately applicable” than is commonly supposed. In particular, it’s possible to immediately apply standard theorems to construct some (plausibly Hecke eigen) sheaves from smooth Hitchin fibers. See the following note:

https://arxiv.org/abs/2108.13571

More a comment than an answer:

I think the lagrangian-to-sheaf dictionary is more “immediately applicable” than is commonly supposed. In particular, it’s possible to immediately apply the dictionary to construct some (plausibly Hecke eigen) sheaves from smooth Hitchin fibers. See the following note:

https://arxiv.org/abs/2108.13571

I think the lagrangian-to-sheaf dictionary is more “immediately applicable” than is commonly supposed: in particular, it’s possible to immediately apply standard theorems to construct some (plausibly Hecke eigen) sheaves from Hitchin fibers. See the following note:

https://arxiv.org/abs/2108.13571

More a comment than an answer:

I think the lagrangian-to-sheaf dictionary is more “immediately applicable” than is commonly supposed. In particular, it’s possible to immediately apply standard theorems to construct some (plausibly Hecke eigen) sheaves from smooth Hitchin fibers. See the following note:

https://arxiv.org/abs/2108.13571

I am less pessimistic than the other commenters about the extent to which the lagrangian-to-sheaf dictionary is “immediately applicable”: in particular, it’s possible to immediately apply standard theorems to construct some objects which are plausibly Hecke eigensheaves in some situations, see the following note:

https://arxiv.org/abs/2108.13571

I think the lagrangian-to-sheaf dictionary is more “immediately applicable” than is commonly supposed: in particular, it’s possible to immediately apply standard theorems to construct some (plausibly Hecke eigen) sheaves from Hitchin fibers. See the following note:

https://arxiv.org/abs/2108.13571