  Based on a pairing of two regular multiplier Hopf algebras $A$ and $B$,
Heisenberg double $\mathscr{H}$ is the smash product $A \# B$ with respect to
the left regular action of $B$ on $A$. Let $\mathscr{D}=A\bowtie B$ be the
Drinfel'd double, then Heisenberg double $\mathscr{H}$ is a Yetter-Drinfel'd
$\mathscr{D}$-module algebra, and it is also braided commutative by the
braiding of Yetter-Drinfel'd module, which generalizes the results in [10] to
some infinite dimensional cases.
