A complete analysis of this is given in the paper by [Orellana-Ram][1]. Actually, they consider the action of the affine braid group on $M\otimes V^{\otimes n}$, but you can recover your case by taking $M$ to be the trivial module. I believe that this is is a semisimple representation (assuming you mean $V$ to be finite dimensional), but in any case it is all spelled out in the paper above. 


  [1]: http://front.math.ucdavis.edu/0401.5317