The nearby cycles of GKS kernels quantizing smoothed support-function Hamiltonians act on constructible sheaves by convolution with twisted polytope sheaves, matching the mirror Picard group action.
Achar, Perverse sheaves and applications to representation theory, Mathematical Surveys and Monographs, vol
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Contact isotopies in the coherent-constructible correspondence
The nearby cycles of GKS kernels quantizing smoothed support-function Hamiltonians act on constructible sheaves by convolution with twisted polytope sheaves, matching the mirror Picard group action.