Several authors over the past decade or so have developed a "toric degeneration" for flag manifolds, which is a family of projective varieties, parametrized by $C$, whose fibre over 1 is the flag variety and whose fibre over 0 is a toric variety. Nohara, Nishinou and Ueda extend this to a "toric degeneration of integrable systems," which is a degeneration in the above sense that also preserves the structure of an integrable system. They construct a degeneration on flag manifolds that takes the Gelfand-Cetlin integrable system on the flag manifold to the integrable system coming from the torus action on the toric variety.

In a recent paper Baier, Florentino, Mourão, and Nunes study a different kind of deformation: they deform the complex structure on a symplectic toric manifold to obtain a link between the real and Kähler quantization of the manifold. They show that, under this deformation, elements of a canonical basis for the space of holomorphic sections tend to distributional sections supported on Bohr-Sommerfeld fibres. We apply BFMN’s deformation of the complex structure to the toric degeneration considered by NNU, to obtain a direct relationship between the real and Kähler quantizations of the flag variety. This is joint work in progress with Hiroshi Konno.