  We classify and explicitly construct the irreducible graded representations
of anti-spherical Hecke categories which are concentrated in one degree. Each
of these homogeneous representations is one-dimensional and can be
cohomologically constructed via a BGG resolution involving every (infinite
dimensional) standard representation of the category. We hence determine the
complete first row of the inverse parabolic $p$-Kazhdan--Lusztig matrix for an
arbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises
the Weyl--Kac character formula to all Coxeter systems (and their parabolics)
and proves that this generalised formula is rigid with respect to base change
to an arbitrary field.

Representation Theory;Combinatorics