  Let $G$ be a connected semisimple Lie group. There are two natural duality
constructions that assign to it the Langlands dual group $G^\vee$ and the
Poisson-Lie dual group $G^*$. The main result of this paper is the following
relation between these two objects: the integral cone defined by the cluster
structure and the Berenstein-Kazhdan potential on the double Bruhat cell
$G^{\vee; w_0, e} \subset G^\vee$ is isomorphic to the integral Bohr-Sommerfeld
cone defined by the Poisson structure on the partial tropicalization of $K^*
\subset G^*$ (the Poisson-Lie dual of the compact form $K \subset G$). By [5],
the first cone parametrizes the canonical bases of irreducible $G$-modules. The
corresponding points in the second cone belong to integral symplectic leaves of
the partial tropicalization labeled by the highest weight of the
representation. As a by-product of our construction, we show that symplectic
volumes of generic symplectic leaves in the partial tropicalization of $K^*$
are equal to symplectic volumes of the corresponding coadjoint orbits in
$\operatorname{Lie}(K)^*$.
  To achieve these goals, we make use of (Langlands dual) double cluster
varieties defined by Fock and Goncharov [9]. These are pairs of cluster
varieties whose seed matrices are transpose to each other. There is a naturally
defined isomorphism between their tropicalizations. The isomorphism between the
cones described above is a particular instance of such an isomorphism
associated to the double Bruhat cells $G^{w_0, e} \subset G$ and $G^{\vee; w_0,
e} \subset G^\vee$.
