  We give a new proof for the parabolic Verlinde formula in all ranks based on
a comparison of wall-crossings 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 rho-shift problem of the
theory.
