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Any finite-dimensional $p$-adic representation of the absolute Galois groupof a $p$-adic local field with imperfect residue field is characterized by itsarithmetic and geometric Sen operators defined by Sen and Brinon. We generalizetheir construction to the fundamental group of a $p$-adic affine variety with asemi-stable chart, and prove that the module of Sen operators is canonicallydefined, independently of the choice of the chart. Our construction relies on adescent theorem in the $p$-adic Simpson correspondence developed by Tsuji. Whenthe representation comes from a $\mathbb{Q}_p$-representation of a $p$-adicanalytic group quotient of the fundamental group, we describe its Lie algebraaction in terms of the Sen operators, which is a generalization of a result ofSen and Ohkubo. These Sen operators can be extended continuously to certaininfinite-dimensional representations. As an application, we prove that thegeometric Sen operators annihilate locally analytic vectors, generalizing aresult of Pan.

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