  Let $T$ be a compact torus and $(M,\omega)$ a Hamiltonian $T$-space. We give
a new proof of the $K$-theoretic analogue of the Kirwan surjectivity theorem in
symplectic geometry by using the equivariant version of the Kirwan map
introduced in one of R. Goldin's papers. We compute the kernel of this
equivariant Kirwan map, and hence give a computation of the kernel of the
Kirwan map. As an application, we find the presentation of the kernel of the
Kirwan map for the $T$-equivariant $K$-theory of flag varieties $G/T$ where $G$
is a compact, connected and simply-connected Lie group. In the last section, we
find explicit formulae for the $K$-theory of weight varieties.
