The Closed Stratum of a Parahoric Deligne-Lusztig Variety

Abstract

In 1976, Deligne and Lusztig constructed irreducible representations for the rational points of a reductive algebraic group over a finite field by identifying the representations within the cohomology of a particular affine variety. We study an analogy of this construction to the case of an inner form of an unramified group over a local field, which replaces the single affine variety with an inverse system of affine varieties of increasing dimension defined over the residue field. Each of these varieties is equipped with a natural stratification, and the minimal strata, also called the closed strata, has particularly accessible structure. We study the geometry of the closed strata. When the inner twist is given by a coxeter element we prove that the varieties in the inverse system are all maximal, in the sense that the variety has as many points as are permitted by the Weil bound for its Betti numbers. We also show that the torus weight spaces of the cohomology are all supported in one degree. Despite the relevance to representation theory, our methods almost entirely algebro-geometric, and rely on a detailed description of the variety in terms of Lie-theoretic data.