  Let V be a two-dimensional absolutely irreducible p-adic Galois
representation and let Pi be the p-adic Banach space representation associated
to V via Colmez's p-adic Langlands correspondence. We establish a link between
the infinitesimal action of GL_2(Q_p) on the locally analytic vectors of Pi,
the differential equation associated to V via the theory of Fontaine and
Berger, and the Sen polynomial of V. This answers a question of Harris and
gives a new proof of a theorem of Colmez: Pi has nonzero locally algebraic
vectors if and only if V is potentially semi-stable with distinct Hodge-Tate
weights.
