The Verlinde formula, an expression for the Euler characteristic of line bundles on the moduli spaces of stable bundles on a curve, is a strikingly beautiful statement in enumerative geometry motivated by quantum physics. It has attracted a lot of attention over the years, and has a number of different proofs.

In Chapter 2 of this thesis, we give a new proof of the more difficult parabolic variant of this formula based on a comparison of wall-crossings in Geometric Invariant Theory and certain iterated residues calculus. On the way, we develop a tautological variant of Hecke correspondences, calculate the Hilbert polynomials of the moduli spaces, and present a new, transparent approach to the rho-shift problem of the theory.

In Chapter 3 we show that the residue/wall-crossing methods of Chapter 2 may be successfully employed to describe the pushforward maps in the K-theory of moduli spaces and present new, explicit formulas for the Euler characteristic of a wider class of vector bundles on the moduli space of stable parabolic bundles.

Our work was motivated by the results of Teleman and Woodward on the index of K- theory classes of moduli stacks.