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

We construct an algebra embedding of the quantum group Uq(g) into a central extension of the quantum coordinate ring Oq[Gw0,w0/H] of the reduced big double Bruhat cell in G. This embedding factors through the Heisenberg double Hq of the quantum Borel subalgebra U≥0, which we relate to Oq[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 in [10], and the quantum Beilinson-Bernstein theorem investigated in [2] and [36].