Partial Resolutions and Noncrossing Combinatorics

For any finite reductive group, we compute the central elements in its Hecke algebra that arise from partial Springer resolutions via the Harish-Chandra transform. Of the two kinds of partial resolution, the larger is the more interesting case. We deduce formulas for associated Hecke traces, generalizing work of Wan-Wang beyond type $A$, and Deodhar-like decompositions of braid varieties that map to partial Springer resolutions. From the latter, we construct noncrossing sets that interpolate between rational Catalan and parking objects, generalizing our work with Galashin-Lam. In parallel, we establish new formulas for arbitrary $a$-degrees of the HOMFLYPT invariants of positive braid closures, from which we construct noncrossing sets for rational Kirkman numbers.