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Poisson-Lie duality and Langlands duality via Bohr-Sommerfeld

By Anton Alekseev

Appears in collection : International workshop on geometric quantization and applications / Colloque international "Quantification géométrique et applications"

Let $G$ be a connected semisimple Lie group with Lie algebra $\mathfrak{g}$. There are two natural duality constructions that assign to it the Langlands dual group $G^\lor$ (associated to the dual root system) and the Poisson-Lie dual group $G^∗$. Cartan subalgebras of $\mathfrak{g}^\lor$ and $\mathfrak{g}^∗$ are isomorphic to each other, but $G^\lor$ is semisimple while $G^∗$ is solvable. In this talk, we explain the following non-trivial relation between these two dualities: the integral cone defined by the Berenstein-Kazhdan potential on the Borel subgroup $B^\lor \subset G^\lor$ is isomorphic to the integral Bohr-Sommerfeld cone defined by the Poisson structure on $K^∗ \subset G^∗$ (the Poisson-Lie dual of the compact form $K \subset G$). The first cone parametrizes canonical bases of irreducible $G$-modules. The corresponding points in the second cone belong to integral symplectic leaves of $K^∗$. The talk is based on a joint work with A. Berenstein, B. Hoffman and Y. Li.

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Citation data

  • DOI 10.24350/CIRM.V.19464603
  • Cite this video ALEKSEEV Anton (10/10/18). Poisson-Lie duality and Langlands duality via Bohr-Sommerfeld. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19464603
  • URL https://dx.doi.org/10.24350/CIRM.V.19464603


  • Alekseev, A., Berenstein, A., Hoffman, B., & Li, Y. (2018). Langlands duality and Poisson-Lie duality via cluster theory and tropicalization. <arXiv:1806.04104> - https://arxiv.org/abs/1806.04104

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