The parabolic Verlinde formula: iterated residues and wall-crossings

TL;DR

This paper introduces a novel proof of the parabolic Verlinde formula across all ranks by comparing wall-crossings in GIT and iterated residues, also developing related geometric and algebraic tools.

Contribution

It provides a new proof method for the parabolic Verlinde formula using wall-crossings and iterated residues, along with new geometric insights and calculations.

Findings

New proof of the parabolic Verlinde formula for all ranks

Development of a tautological variant of Hecke correspondences

Calculation of Hilbert polynomials of moduli spaces

Abstract

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.