Geometry of the fixed points loci and discretization of Springer fibers in classical types

Consider a simple algebraic group G of classical type and its Lie algebra g. Let (e, h, f) g be an sl₂-triple and Qₑ= CG (e, h, f). The torus Tₑ that comes from the sl₂-triple acts on the Springer fiber Bₑ. Let Bₑ^gr denote the fixed point loci of Bₑ under this torus action. Our main geometric result is that when the partition of e has up to 4 rows, the derived category Dᵇ (Bₑ^gr) admits a complete exceptional collection that is compatible with the Qₑ-action. The objects in this collection give us a finite set Yₑ that is naturally equipped with a Qₑ-centrally extended structure. We prove that the set Yₑ constructed in this way coincides with a finite set that has appeared in various contexts in representation theory. For example, a direct summand Jc of the asymptotic Hecke algebra is isomorphic to K₀ (Sh^Qₑ (Yₑ Yₑ). The left cells in the two-sided cell c corresponding to the adjoint orbit of e are in bijection with the Qₑ-orbits in Yₑ. Our main numerical result is an algorithm to compute the multiplicities of the Qₑ-centrally extended orbits that appear in Yₑ.