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 $\mathfrak{g}$. Let $(e,h,f)\subset \mathfrak{g}$ be an ${\mathfrak{s}\mathfrak{l}}_2$-triple and $Q_e=C_G(e,h,f)$. The torus $T_e$ that comes from the ${\mathfrak{s}\mathfrak{l}}_2$-triple acts on the Springer fiber ${\mathcal{B}}_e$. Let ${\mathcal{B}}_e^{gr}$ denote the fixed point loci of ${\mathcal{B}}_e$ under this torus action. Our main geometric result is that when the partition of $e$ has up to $4$ rows, the derived category $D^b({\mathcal{B}}_e^{gr})$ admits a complete exceptional collection that is compatible with the $Q_e$-action. The objects in this collection give us a finite set $Y_e$ that is naturally equipped with a $Q_e$-centrally extended structure. We prove that the set $Y_e$ constructed in this way coincides with a finite set that has appeared in various contexts in representation theory. For example, a direct summand $J_c$ of the asymptotic Hecke algebra is isomorphic to $K_0(S{h}^{Q_e}(Y_e\times Y_e)$. The left cells in the two-sided cell $c$ corresponding to the adjoint orbit of $e$ are in bijection with the $Q_e$-orbits in $Y_e$. Our main numerical result is an algorithm to compute the multiplicities of the $Q_e$-centrally extended orbits that appear in $Y_e$.