Lagrangian Correspondences and a Compactification of the Symplectic Group

The graph of an element of the symplectic group over a field is a lagrangian subspace of the double of the symplectic vector space. The group product corresponds to the convolution of such lagrangians. But there are other, degenerate, lagrangians, whose convolution gives a stratified semigroup. This is analogous to the "wonderful" compactifications in algebraic geometry.. Over a field of 2 elements, the lagrangians can be quantized to so-called stabilizer states in an operator algebra.