  Let $\mathfrak{g}$ be a simple finite dimensional complex Lie algebra and let
$\widehat{\mathfrak{g}}$ be the corresponding affine Lie algebra. Kac and
Wakimoto observed that in some cases the coefficients in the character formula
for a simple highest weight $\widehat{\mathfrak{g}}$-module are either bounded
or are given by a linear function of the weight. We explain and generalize this
observation using Kazhdan-Lusztig theory, by computing values at $q=1$ of
certain (parabolic) affine inverse Kazhdan-Lusztig polynomials. In particular,
we obtain explicit character formulas for some $\widehat{\mathfrak{g}}$-modules
of negative integer level $k$ when $\mathfrak g$ is of type $D_n$, $E_6$,
$E_7$, $E_8$ and $k \geqslant -2, -3, -4, -6$ respectively, as conjectured by
Kac and Wakimoto.
  The calculation relies on the explicit description of the canonical basis in
the cell quotient of the anti-spherical module over the affine Hecke algebra
corresponding to the subregular cell. We also present an explicit description
of the corresponding objects in the derived category of equivariant coherent
sheaves on the Springer resolution, they correspond to irreducible objects in
the heart of a certain $t$-structure related to the so called non-commutative
Springer resolution.
