# semisimplicity of braid reps?

Here's something I really feel I should know, but do not:

Let $q$ be some sufficiently nice complex number (just pretend we're working over $\mathbb Q(q)$, for example), and $V$ some simple representation of $U_q(\mathfrak g)$. Then you have the usual representation of the braid group $B_n$ on $V^{\otimes n}$. The question is: is this representation semisimple?

I haven't been able to find a direct reference to the problem, but it really sounds like it should be known so I thought I'd check here.

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A complete analysis of this is given in the paper by Orellana-Ram. 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.
I'm only on page 10 right now, but so far, it looks like the result that comes closest to what I wanted isn't quite there: they prove in Prop. 3.6 thet their modules $M\otimes V^{\otimes n}$ are what they call calibrated, i.e. semisimple under precisely the abelian subgroup of the affine braid group that comes from the "affine-ness" (the subgroup of $\tilde B_n$ generated by the $X^{\varepsilon_i}$, in the notation of the paper).