Sen Operators and Lie Algebras arising from Galois Representations over $p$-adic Varieties

Any finite-dimensional $p$-adic representation of the absolute Galois group of a $p$-adic local field with imperfect residue field is characterized by its arithmetic and geometric Sen operators defined by Sen and Brinon. We generalize their construction to the fundamental group of a $p$-adic affine variety with a semi-stable chart, and prove that the module of Sen operators is canonically defined, independently of the choice of the chart. Our construction relies on a descent theorem in the $p$-adic Simpson correspondence developed by Tsuji. When the representation comes from a $\mathbb{Q}_p$-representation of a $p$-adic analytic group quotient of the fundamental group, we describe its Lie algebra action in terms of the Sen operators, which is a generalization of a result of Sen and Ohkubo. These Sen operators can be extended continuously to certain infinite-dimensional representations. As an application, we prove that the geometric Sen operators annihilate locally analytic vectors, generalizing a result of Pan.