Achar recently introduced a “nearby cycles formalism” in the framework of chain complexes of parity sheaves. In this talk, I’ll explain joint work with Achar in which we compute the output of this functor in two related settings. The first is affine space, stratified by the action of a torus, and the second is the global Schubert variety associated to the first fundamental coweight of the group PGLn. The latter is a parity-sheaf analogue of Gaitsgory’s central sheaf construction. Much of the talk will focus on the representation theoretic context for the computation.