Serre weights for locally reducible two-dimensional Galois representations

Let be a totally real field, and a place of dividing an odd prime . We study the weight part of Serre’s conjecture for continuous totally odd representations that are reducible locally at . Let be the set of predicted Serre weights for the semisimplification of . We prove that, when is generic, the Serre weights in for which is modular are exactly the ones that are predicted (assuming that is modular). We also determine precisely which subsets of arise as predicted weights when varies with fixed generic semisimplification.