  We construct an algebra embedding of the quantum group $U_q(\mathfrak{g})$
into the quantum coordinate ring $\mathcal{O}_q[G^{w_0,w_0}/H]$ of the reduced
big double Bruhat cell in $G$. This embedding factors through the Heisenberg
double $\mathcal{H}_q$ of the quantum Borel subalgebra $U_{\geq0}$, 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.
