Constructing Galois representations ramified at one prime

Let $n>1$, $e\ge 0$ and a prime number $p\ge 2^{n+2+2e}+3$, such that the index of regularity of $p$ is $\le e$. We show that there are infinitely many irreducible Galois representations $\rho: Gal(\bar{$\mathbb{Q}$}/$\mathbb{Q}$)\rightarrow {GL}_n($\mathbb{Q}$_p)$ unramified at all primes $l\mathrm{\ne }p$. Furthermore, these representations are shown to have image containing a fixed finite index subgroup of ${SL}_n({\mathbb{Z}}_p)$. Such representations are constructed by lifting suitable residual representations $\bar{\rho }$ with image in the diagonal torus in ${GL}_n({\mathbb{F}}_p)$, for which the global deformation problem is unobstructed.