Quantum groups, quantum tori, and the Grothendieck-Springer resolution

We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$ into the quantum coordinate ring ${\mathcal{O}}_q[G^{w_0,w_0}\mathrm{/}H]$ of the reduced big double Bruhat cell in $G$. This embedding factors through the Heisenberg double ${\mathcal{H}}_q$ of the quantum Borel subalgebra $U_{\ge 0}$, which we relate to ${\mathcal{O}}_q[G]$ via twisting by the longest element of the quantum Weyl group. Our construction is inspired by the Poisson geometry of the Grothendieck-Springer resolution studied by Evens and Lu, and the quantum Beilinson-Bernstein theorem investigated by Backelin, Kremnitzer, and Tanisaki.