Affine Kazhdan-Lusztig polynomials on the subregular cell in non simply-laced Lie algebras: with an application to character formulae
Summary
Bezrukavnikov, Kac, and Krylov uses the geometry of the Springer fiber \(\mathcal B_e\) where \(e\) is a subregular nilpotent element to compute special values of affine Kazhdan-Lusztig polynomials when \(G\) is simply-laced. We extend the computation to non-simply-laced Lie groups \(G\). In particular, letting \(U\subset\mathcal N\) be the subset of the nilpotent cone of \(\mathfrak g^\vee\) consisting of regular and subregular nilpotent orbits. Let \(\widetilde{\mathcal N}\to\mathcal N\) be the Springer resolution and let \(\widetilde U\to U\) be the restriction to \(U\). We characterize the irreducible objects in the exotic \(t\)-structure of \(D^{\mathrm b}\mathrm{Coh}^{G^\vee}(\widetilde U)\), which is the categorification of the canonical basis of the anti-spherical representation of \(\mathcal H_{\ge\mathrm{subreg}}\). We then compute the class of the standard basis \(\mathcal O_{\widetilde U}(\lambda)\) in the equivariant \(K\)-theory \(K^{G^\vee}(\widetilde U)\) in terms of the irreducible objects, which computes Kazhdan-Lusztig polynomials.