For Hamiltonian actions (e.g. on smooth complex projective varieties), one can use equivariant localization, as in http://front.math.ucdavis.edu/0503.5609 . Let $R(S^1) = Z[t^\pm]$, so the restriction map $K_T(S^2) \to K_T($fixed points$) = Z[t_1^\pm] \oplus Z[t_2^\pm]$ hits those pairs $(p(t_1),q(t_2))$ such that $p(1) = q(1)$.

For Hamiltonian actions (e.g. on smooth complex projective varieties), one can use equivariant localization, as in Harada and Landweber's Surjectivity for Hamiltonian G-spaces in K-theory. Let $R(S^1) = Z[t^\pm]$, so the restriction map $K_T(S^2) \to K_T($fixed points$) = Z[t_1^\pm] \oplus Z[t_2^\pm]$ hits those pairs $(p(t_1),q(t_2))$ such that $p(1) = q(1)$.