  We continue the study of the Drinfeld double of the Jordan plane, denoted by
$\mathcal D$ and introduced in arXiv:2002.02514.
  The simple finite-dimensional modules were computed in arXiv:2108.13849; it
turns out that they factorize through $U(\spl_2(\Bbbk))$. Here we introduce the
Verma modules and the category $\mathfrak O$ for $\mathcal D$, which have a
resemblance to the similar ones in Lie theory but induced from indecomposable
modules of the 0-part of the triangular decomposition. Accordingly, there is
the notion of highest weight rank (hw-rk).
  We classify the indecomposable modules of hw-rk one and find families of
hw-rk two. The Gabriel quiver of $\mathcal D$ is computed implying that it has
a wild representation type.
