  We explain that the Pl\"ucker relations provide the defining equations of the
thick flag manifold associated to a Kac-Moody algebra. This naturally
transplant the result of Kumar-Mathieu-Schwede about the Frobenius splitting of
thin flag manifolds to the thick case. As a consequence, we provide a
description of the global sections of line bundles of a thick Schubert variety
as conjectured in Kashiwara-Shimozono [Duke Math. J. 148 (2009)]. This also
yields the existence of a compatible basis of thick Demazure modules, and the
projective normality of the thick Schubert varieties.
