Separation of variables for the moduli of Higgs bundles and connections

For a fixed Riemann surface X, the Hitchin moduli space of stable G-Higgs bundles can be regarded as a compactification of the cotangent of the moduli space of stable G-bundles. It admits a Białynicki-Birula stratification induced by a C*-action.

For G=SL_2, I will explain by picking subbundles of the underlying bundles of Higgs bundles, one can assign points in the symmetric product of T*X that essentially encodes this stratification. The degeneration of these points corresponds to a limit in a lower stratum. A parallel phenomenon occurs in the moduli space of holomorphic connections: picking sub-bundles induces projective connections with apparent singularities, the degeneration of which corresponds to limiting to lower strata. I will sketch how this work prepares for an explicit construction of Hecke-eigensheaves in the geometric Langlands correspondence.