THE METASTABLE FILTRATION OF PERVERSE SHEAVES ON PERFECTOID SHIMURA VARIETIES

Mathematical Applications of Science Fiction We construct a canonical filtration, called the metastable filtration, on the derived category of l-adic perverse sheaves on the infinite-level perfectoid Shimura variety. Using the tilting equivalence developed by Scholze and the geometry of the Hodge–Tate period map, we define a t-structure on the pro-étale site that interpolates between the classical perverse t-structure at finite levels and the coherent cohomology of the structure sheaf. We prove that the graded components of this filtration are isomorphic to specific convolution products of intersection cohomology complexes on the flag variety, twisted by the action of the Hecke algebra. Furthermore, we establish a spectral sequence relating the metastable cohomology to the completed cohomology of the Shimura tower, showing degeneration at the second page under a p-adic purity assumption.