Langlands Duality and Poisson-Lie Duality via Cluster Theory and Tropicalization

Monday, 11 June, 2018

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Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to it the Langlands dual group G 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 [3], 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 \mathfrak{k}^*.
To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov [7]. 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.


Anton Alekseev
Arkady Berenstein
Benjamin Hoffman
Yanpeng Li