Subregular nilpotent orbits and explicit character formulas for modules over affine Lie algebras

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.