I am trying to prove a version of quantum Schur-Weyl duality. I hope to be able to generalize the proof of the Schur-Weyl duality between $U\_q(\mathfrak{gl}\_n)$ and the Hecke algebra $H\_r$. So I am looking for a good reference for this with a careful proof. It would also be nice to see a proof that uses the quantum coordinate ring of $GL\_n$ instead of the enveloping algebra (and therefore is phrased in terms of decomposing $V^{\otimes r}$ as a left-comodule for this coordinate ring and a right $H\_r$-module).



Reference for quantum Schur-Weyl duality
This goes back to Jimbo I think. A reference is:
"A q-difference analogue of $U(\mathfrak g)$, Hecke algebra and the Yang-Baxter equation'', Lett. Math. Phys. 11 (1986).


It has been much studied though, so there are lots of subsequent papers, some of which might be closer to what you are looking for? For example [this](https://arxiv.org/abs/0806.0264) paper studies an analogue of Schur-Weyl duality for "walled Brauer algebras", and [this](https://arxiv.org/abs/math/0108038) paper studies a two-parameter version.


