Gauss--Berezin integral operators and spinors over supergroups $\OSp(2p|2q)$, and Lagrangian super-Grasmannians

We obtain explicit formulas for the spinor representation of the orthosymplectic supergroup $\OSp(2p|2q)$ and the corresponding super-Olshanski semigroup. Also we extend the representation to Lagrangian super-Grassmannian.