Endomorphisms of a representation Let $G$ be a group acting continuously on a free $\mathbb{Z}_p$-module of finite rank. Assume that $End_{G}(T)$ and $End_G(T/p)$ are the homotheties.
Is it possible that $End_{G}(T/p^n)$ contains more than the homotheties ?
When $T/p$ is absolutely irreducible, my guess would be that $T$ is also absolutely irreducible, and so is $T/p^n$ (I am aware that this heuristic is quite bad because what does it mean for a representation with coefficients in a ring to be irreducible when the ring is not simple ?)
 A: First a general remark: there is an article of Carayol in the $p$-adic monodromy volume (edited by Mazur and Stevens) which deals with the issue of passing information from $T/p$ to $T$ (and in particular with the fact that it doesn't make rigorous sense to say that $T$ is abs. irred.).  You could also look at Mazur's original article on deformations of Galois representations.

Now for your question:  Let $\varphi: T/p^n \to T/p^n$.  Reducing mod $p$,
$\varphi$ becomes a scalar.  Let $a \in \mathbb Z_p$ be a lift of this scalar.
Then $\varphi -a$ acts trivially on $T/p$, and so maps $T/p^n$ to $pT/p^nT
\cong T/p^{n-1}$.   Thus $\varphi-a$ may be obtained as $p$ times an endomorphism
of $T/p^{n-1}T$.  Arguing by induction on $n$, we find that $\varphi - a$
is scalar, and hence so is $\varphi$ itself.
Thus your question has a positive answer (and in fact it is enough to assume
that $T/pT$ has only scalar endomorphisms).  Also, one can replace $\mathbb Z_p$
by any complete local Noetherian $\mathbb Z_p$-algebra (the same argument will work, and it is related to the fact that if $T/pT$ has only scalar endomorphisms
then it has a representable deformation functor).
