Claude Eicher (MIT), Relaxed highest weight representations over affine Kac-Moody algebras from D-modules on the Kashiwara flag scheme


The relaxed highest weight representations introduced by Feigin, Semikhatov, and Tipunin are a class of representations of the Lie algebra affine sl2 , which do not have a highest (or lowest) weight. We generalize this notion to an arbitrary affine Kac-Moody algebra g and realize induced g-modules of this type and their duals as global sections of twisted D-modules on the Kashiwara flag scheme associated to g. The D-modules that appear in our construction are direct images from subschemes given by the intersection of finite dimensional Schubert cells with their translate by a simple reflection. Besides the twist, they depend on a complex number describing the monodromy of the local systems we construct on these intersections. These results describe for the first time explicit non-highest weight g-modules as global sections on the Kashiwara flag scheme and extend results of Kashiwara-Tanisaki to the case of relaxed highest weight representations. This is based on arxiv:1607.06342 [math.RT]