2016/07/21 by Claude Eicher, Eicher, C.
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Coding theory and cryptography #FOS: Mathematics #Mathematical Approximation and Integration #Representation Theory (math.RT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1607.06342
openalex publication_date 2016/07/21 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
The relaxed highest weight representations introduced by Feigin et al. are a class of representations of the affine Kac-Moody algebra \mathfraksl2, which do not have a highest (or lowest) weight. We formulate a generalization of this notion for an arbitrary affine Kac-Moody algebra \mathfrakg. We then realize induced \mathfrakg-modules of this type and their duals as global sections of twisted D-modules on the Kashiwara flag scheme X associated to \mathfrakg. The D-modules that appear in our construction are direct images from subschemes of X that are intersections of finite dimensional Schubert cells with their translate by a simple reflection. Besides the twist λ, they depend on a complex number describing the monodromy of the local systems we construct on these intersections. We describe the global sections of the *-direct images as a module over the Cartan subalgebra of \mathfrakg and show that the higher cohomology vanishes. We obtain a complete description of the cohomology groups of the direct images as \mathfrakg-modules in the following two cases. First, we address the case when the intersection is isomorphic to \mathbbGm. Second, we address the case of the *-direct image from an arbitrary intersection when the twist is regular antidominant and the monodromy is trivial. For the proof of this case we introduce an auto-equivalence of the category of D-modules Hol(λ) induced by the automorphism of X defined by a lift of a simple reflection. These results describe for the first time explicit non-highest weight \mathfrakg-modules as global sections on the Kashiwara flag scheme and extend several results of Kashiwara-Tanisaki to the case of relaxed highest weight representations.