  The relaxed highest weight representations introduced by Feigin et al. are a
special 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 $\mathbb{C}^{\times}$. Secondly, 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 exact 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.
