Relaxed Highest Weight Modules from $\mathcal{D}$-Modules on the Kashiwara Flag Scheme

The relaxed highest weight representations introduced by Feigin et al. are a class of representations of the affine Kac-Moody algebra $\hat{\mathfrak{sl}_2}$, which do not have a highest (or lowest) weight. We formulate a generalization of this notion for an arbitrary affine Kac-Moody algebra $\mathfrak{g}$. We then realize induced $\mathfrak{g}$-modules of this type and their duals as global sections of twisted $\mathcal{D}$-modules on the Kashiwara flag scheme $X$ associated to $\mathfrak{g}$. The $\mathcal{D}$-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 $\lambda$, 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 $\mathfrak{g}$ and show that the higher cohomology vanishes. We obtain a complete description of the cohomology groups of the direct images as $\mathfrak{g}$-modules in the following two cases. First, we address the case when the intersection is isomorphic to $\mathbb{G}_{\text{m}}$. 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 $\mathcal{D}$-modules $\text{Hol}(\lambda)$ 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 $\mathfrak{g}$-modules as global sections on the Kashiwara flag scheme and extend several results of Kashiwara-Tanisaki to the case of relaxed highest weight representations.