Microlocal sheaves on certain affine Springer fibers

For homogeneous affine Springer fibers (those with GmGm symmetry), we realize them as Lagrangian cycles inside ambient symplectic varieties, and make sense of microlocal sheaves supported on these affine Springer fibers. We also propose a conjectural equivalence between the microlocal sheaf category and a coherent sheaf category in terms of Springer fibers for the Langlands dual group, and are able to prove the conjecture in some cases. This is joint work with Roman Bezrukavnikov, Pablo Boixeda Alvarez and Michael McBreen.