Plectic Galois action on the cohomology of Hilbert modular varieties

A result of Jan Nekovář says that the Galois action on p-adic intersection cohomology of Hilbert modular varieties with coefficients in automorphic local systems is semisimple.  We will explain a new proof of this result for the non-CM part of the cohomology.  Interestingly, Nekovář’s approach is based on the construction of partial Frobenii at places away from p, while our method uses partial Sen operators at p. This is joint work with Yuanyang Jiang.