In this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of $\mathscr{D}$-modules of the basic affine space, and one in terms of intertwining operators for universal Verma modules. We also construct natural collections of isomorphisms parameterized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Braverman and Finkelberg.

We prove an analogue of Koszul duality for category $ \mathcal{O} $ of a reductive group $G$ in positive characteristic $\ell $ larger than $1$ plus the number of roots of $G$ . However, there are no Koszul rings, and we do not prove an analogue of the Kazhdan–Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of $G$ plus $2$ .

We present an analysis of the properties of H i holes detected in 20 galaxies that are part of “The H i Nearby Galaxy Survey”. We detected more than 1000 holes in total in the sampled galaxies. The holes are found throughout the disks of the galaxies, out to the edge of the H i disk. We find that shear limits the age of holes in spirals. Shear is less important in dwarf galaxies which explains why H i holes in dwarfs are rounder, on average than in spirals. Shear is particularly strong in the inner part of spiral galaxies, limiting the lifespan of holes there and explaining why we find that holes outside R25 are larger and older. We proceed to derive the surface and volume porosity and find that this correlates with the type of the host galaxy: later Hubble types tend to be more porous. The size distribution of the holes in our sample follows a power law with a slope of aν ~ −2.9. Assuming that the holes are the result of massive star formation, we derive values for the supernova rate (SNR) and star formation rate (SFR) which scales with the SFR derived based on other tracers. If we extrapolate the observed number of holes to include those that fall below our resolution limit, down to holes created by a single supernova, we find that our results are compatible with the hypothesis that H i holes result from star formation.

In this paper we construct, for F1 and F2 subbundles of a vector bundle E, a ‘Koszul duality’ equivalence between derived categories of 𝔾m-equivariant coherent(dg-)sheaves on the derived intersection , and the corresponding derived intersection . We also propose applications to Hecke algebras.