Nearby cycles for parity sheaves on a divisor with simple normal crossings

TL;DR

This paper computes the nearby cycles functor for parity sheaves in specific geometric settings, including affine space with torus action and a Schubert variety related to PGL_n, extending the formalism introduced earlier.

Contribution

It provides explicit calculations of the nearby cycles functor for parity sheaves in new geometric contexts, connecting to Gaitsgory's central sheaf construction.

Findings

Explicit formulas for nearby cycles in affine space stratified by a torus.

Computation of nearby cycles on a Schubert variety related to PGL_n.

Extension of the parity sheaves formalism to new geometric settings.

Abstract

The first author recently introduced a "nearby cycles formalism" in the framework of chain complexes of parity sheaves. In this paper, we compute this functor in two related settings: (i) affine space, stratified by the action of a torus, and (ii) the global Schubert variety associated to the first fundamental coweight of the group $PGL_n$. The latter is a parity-sheaf analogue of Gaitsgory's central sheaf construction.