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

TL;DR

This paper establishes a deep connection between Langlands duality and Poisson-Lie duality through cluster theory and tropicalization, revealing isomorphisms between integral cones related to canonical bases and symplectic leaves.

Contribution

It introduces a novel isomorphism linking the cluster structure on the Langlands dual group to the Poisson-Lie structure on the dual group via tropicalization, unifying two duality concepts.

Findings

The integral cone from the cluster structure is isomorphic to the Bohr-Sommerfeld cone from the Poisson structure.

Symplectic volumes of leaves in the tropicalization match those of coadjoint orbits.

The isomorphism is realized through double cluster varieties and their tropicalizations.

Abstract

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 $G^{\vee; w_0, e} \subset G^\vee$.