Dual pairs of quantum moment maps and doubles of Hopf algebras

For any finite-dimensional Hopf algebra $A$ there exists a natural associative algebra homomorphism $D(A) \to H(A)$ between its Drinfeld double $D(A)$ and its Heisenberg double $H(A)$. We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck-Springer resolution arises in this context.