Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group \(G^\vee \), and its Poisson–Lie dual group \(G^*\), respectively. 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 Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), 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 \({{\,\mathrm{Lie}\,}}(K)^*\). To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009). 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 \(G^{\vee ; w_0, e} \subset G^\vee \).

设G为连通半单李群。有两个自然对偶构造分配给G：分别是它的朗兰兹对偶群\(G^\vee \)和泊松-李对偶群\(G^*\)。本文的主要结果是这两个对象之间的关系如下：簇结构定义的积分锥和双Bruhat单元上的Berenstein-Kazhdan势\(G^{\vee ; w_0, e} \subset G^ \vee \)同构于由\(K^* \subset G^*\)（紧致形式\(K \subset G\））。由 Berenstein 和 Kazhdan（在：当代数学，第 433 卷。美国数学学会，普罗维登斯，第 13-88 页，2007 年）中，第一个锥体参数化了不可约G模的规范基础。第二个锥体中的对应点属于由表示的最高权重标记的部分热带化的积分辛叶。作为我们构建的副产品，我们证明了\(K^*\)的偏热带化中一般辛叶的辛体积等于\({{\,\mathrm{谎言}\,}}(K)^*\). 为了实现这些目标，我们利用 Fock 和 Goncharov 定义的（Langlands 双）双簇变体（Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009）。这些是成对的簇品种，它们的种子矩阵相互转置。它们的热带化之间存在自然定义的同构。上述锥体之间的同构是与双 Bruhat 单元\(G^{w_0, e} \subset G\)和\(G^{\vee ; w_0, e} \subset G^\vee \)。