Is it true that all continuous finite dimensional $p$-adic representations of $Gal(\bar{K}/K)$ are semisimple, where $K$ is a number field, i.e. if $\rho:Gal(\bar{K}/K) \mapsto GL\_n(\bar{\mathbb{Q}}\_p)$ is a continuous representation, where $K$ is a number field (even $\mathbb{Q}$ if you want), is it semisimple?


Replacing $\mathbb{Q}\_p$ with $\mathbb{C}$, the result is indeed true, using the existence of an invariant Haar measure over the complex numbers. So the natural question (related) is if it exists a $\mathbb{Q}\_p$-valued Haar measure in a compact group? Of course some conditions must be removed from the classical formulation (for example it does not make sence to ask that if $f$ is a positive continuous function, then its integral is positive), but the existence of a non-trivial invariant measure such that the volume of the group is 1 should be enough. Any reference is welcome as well



semisimplicity of p-adic Galois representations
This can't possibly work. Many p-adic Galois representations are not semisimple. For instance, $\mathbb{Z}\_p$ occurs as a quotient of the Galois group of $\mathbb{Q}$ (as the Galois group of the cyclotomic $\mathbb{Z}\_p$-extension) and there are non-semisimple $\mathbb{Q}\_p$-representations of $\mathbb{Z}\_p$, e.g. the map sending $x \in \mathbb{Z}\_p$ to $\begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix}$. (This is pretty fortunate, since otherwise all Galois cohomology groups would be forced to be trivial.)


As for Haar measure: what would be a $\mathbb{Q}\_p$-valued Haar measure on $\mathbb{Z}\_p$? It would have to give $p^n \mathbb{Z}\_p$ measure $1/p^n$, but these are getting large rather than small p-adically; so you couldn't sensibly integrate continuous $\mathbb{Q}\_p$-valued functions against such a measure - the "Riemann sums" would not converge. The lack of anything resembling Haar measure is one of the reasons why p-adic analysis has a very different flavour from real analysis.


EDIT: As some other posters have pointed out, it's possible to define a nice $\mathbb{Q}\_p$-valued, and even $\mathbb{Q}$-valued, measure on a suitably nice class of sets of $\mathbb{Z}\_p$; e.g. you can clearly do this for open compact subsets. This then gives you a linear functional on locally constant functions (analogous to the step functions used in measure theory over $\mathbb{R}$). But this won't extend to a linear functional on continuous $\mathbb{Q}\_p$-valued functions, for the reasons I sketched above, and so it doesn't imply anything about semisimplicity of continuous representations.


