Deformations of certain reducible Galois representations III

Let p be an odd prime and q a power of p. We examine the deformation theory of reducible and indecomposable Galois representations (rho) over bar : G(Q) -> GSp(2n)(F-q) that are unramified outside a finite set of primes S and whose image lies in a Borel subgroup. We show that under some additional hypotheses, such representations have geometric lifts to the Witt vectors W(F-q). The main theorem of this paper is a higher- dimensional generalization of the result of [S. Hamblen and R. Ramakrishna, Deformations of certain reducible Galois II, Amer. J. Math. 130(4) (2008) 913-944] [5].