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 and extend it to the locally analytic completed 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 to construct a plectic Lie algebra action (whose meaning will be explained in the talk). This is joint work in progress with Yuanyang Jiang.