Affine Kazhdan-Lusztig polynomials on the subregular cell in non simply-laced Lie algebras: with an application to character formulae (with an appendix by Roman Bezrukavnikov, Vasily Krylov, and Kenta Suzuki)

Abstract

We extend the techniques in arXiv:2209.08865(1) to the non-simply-laced situation, and calculate explicit special values of parabolic affine inverse Kazhdan-Lusztig polynomials for subregular nilpotent orbits. We thus obtain explicit character formulas for certain irreducible representations of affine Lie algebras. As particular cases, we compute characters of simple vertex algebras $V_{k}(\mathfrak{g})$ for $k=-1,\ldots,-b$, where $b$ is the largest label of the highest short coroot $\theta^\vee$. Conjecturally, all ordinary modules over $V_{-b}(\mathfrak{g})$ are covered by our computations. As an application, we obtain the explicit formulas for flavoured Schur indices of rank one Argyres-Douglas 4d SCFTs with flavour symmetry $G_2$ and $B_3$. Our results are proved using the geometry of the Springer resolution. We identify the cell quotient of the anti-spherical module over $\widehat{W}$ corresponding to the subregular cell with a certain one-dimensional extension of a module defined by Lusztig. We describe the canonical basis in this module geometrically and 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 that we describe using an equivariant version of the derived McKay correspondence.