Langlands duality and Poisson–Lie duality via cluster theory and tropicalization

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 Poisson–Lie $$G^*$$ respectively. The main result of this paper is the following relation between these objects: integral cone defined by cluster structure Berenstein–Kazhdan potential on double Bruhat cell $$G^{\vee ; w_0, e} \subset G^\vee isomorphic Bohr–Sommerfeld Poisson partial tropicalization $$K^* G^*$$ (the compact form $$K G$$ ). By Berenstein Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), first parametrizes canonical bases irreducible G-modules. corresponding points in second belong symplectic leaves labeled highest weight representation. As by-product our construction, we show volumes generic $$K^*$$ equal coadjoint orbits $${{\,\mathrm{Lie}\,}}(K)^*$$ . To achieve goals, make use (Langlands dual) varieties Fock Goncharov (Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009). These pairs whose seed matrices transpose each other. naturally isomorphism their tropicalizations. cones described above particular instance such an associated cells $$G^{w_0,