I am trying to prove a version of quantum SchurWeyl duality. I hope to be able to generalize the proof of the SchurWeyl 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 leftcomodule for this coordinate ring and a right $H_r$module).
This goes back to Jimbo I think. A reference is: "A qdifference analogue of $U(\mathfrak g)$, Hecke algebra and the YangBaxter 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 paper studies an analogue of SchurWeyl duality for "walled Brauer algebras", and this paper studies a twoparameter version. 


Hi Jonah  Section 4 of this paper deduces a quantum SchurWeyl duality from quantum GL(n)GL(m) Howe duality. But this doesn't use the quantum coordinate ring. 


Georgia Benkart and Sarah Witherspoon worked on a 2parameter generalization, but also in quantized enveloping algebra language. See math/0108038. There is also wide available work on various qSchur algebras. It is hard to find what is not known as far as various basic analogues here are concerned. 


This is treated in section 8.6 of the book "Quantum Groups and Their Representations" by Klimyk and Schmudgen, although also not using the quantum coordinate algebra. 


Too late maybe but still  "Noncommutative symmetric functions V: a degenerate version of U_q(gl_N)" by Krob and Thibon has the quantum coordinate ring version very carefully (I think) written. It is aimed at the $q=0$ case but the generic $q$ is treated in detail too. 

