Abstract:Quantization of flag manifolds was studied by Guillemin and Sternberg, who showed that the quantizations using real and Kahler polarizations are "the same." However, their proof is in terms of the equality of two numbers (the dimension of a representation space, and the number of Bohr-Sommerfeld fibres), and does not give any direct relationship between the two quantizations.
We construct a family of complex structures on the flag manifold, depending on a real parameter s, so that elements of a canonical basis for the holomorphic sections (which are elements of the Kahler quantization) tend, in the limit as s goes to infinity, to distributional sections supported on the Bohr-Sommerfeld fibres (which can be seen as elements of the real quantization). We use two main techniques in the proof: the "toric degneration of integrable systems" of Nishinou, Nohara, and Ueda; and the deformation of complex structures constructed by Baier, Mourao, Florentino, and Nunes.
This is joint work in progress with Hiroshi Konno.