Parity-complex hypotheses for the stratifications used in this work

Ascertain whether the technical assumptions on the strata required for the theory of parity complexes in Juteau–Mautner–Williamson (JMW, §2.1) are satisfied for the stratifications considered here, specifically the natural stratifications of the connected components of Drinfeld’s compactification \overline{Bun}_B^\lambda and of the Zastava schemes ^\mu. Establishing these conditions would allow the full parity-complex formalism to apply to IC_{\overline{Bun}_B^\lambda} and IC_{^\mu}.

Background

The authors prove parity-vanishing statements for the intersection cohomology complexes IC_{\overline{Bun}B^\lambda} and IC{^\mu}, showing they behave as even (or odd) objects with respect to their natural stratifications. However, the full theory of parity complexes in JMW requires certain technical conditions on the stratification (e.g., assumptions listed in JMW §2.1).

Whether these technical conditions hold for the stratifications arising from Drinfeld’s compactifications and Zastava schemes is not established here, leaving open if one can invoke the complete parity-sheaf machinery in this context.