The parabolic Verlinde formula: iterated residues and wall-crossings

We give a new proof for the parabolic Verlinde formula in all ranks based on a comparison of wallcrossings in geometric invariant theory and certain iterated residue functionals.On the way, we develop a tautological variant of Hecke correspondences, calculate the Hilbert polynomials of the moduli spaces, and present a new, transparent, local approach to the -shift problem of the theory.14D20, 14H60 1. Introduction 2259 2. Parabolic bundles 2265 3. Wall-crossing in the Verlinde formula 2269 4. The residue formula and the main result 2272 5. Wall-crossing in master space 2281 6. Wall-crossings in parabolic moduli spaces 2286 7. Tautological Hecke correspondences 2295 8.