One of the many contributions of Kostant is a rare gem which probably has not been sufficiently explored: a sheaf-theoretical model for geometric quantization associated to real polarizations. Kostant's model works very well for polarizations given by fibrations or fibration-like objects (like integrable systems away from singularities). For toric manifolds where the real polarization is determined by the fibers of the moment map, Kostant's model yields a representation space whose dimension is the number of integer points inside the corresponding Delzant polytope. We will discuss extensions of this model to consider almost toric manifolds and integrable systems with non-degenerate singularities where “unexpected” infinities can show up even if the manifold is compact.[-]

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.[-]