I will describe an approach to geometric quantization in terms of deformation quantization in the setting of algebraic symplectic structures. I will explain why it recovers the expected answer for cotangent bundles. One can also modify this approach to understand geometric quantization of (-1) and 1-shifted symplectic structures which I will describe for the case of a derived critical locus.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/HS2015