  We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear
sigma models (GLSM) on the $\Omega$-deformed sphere, $S^2_\Omega$, which is a
one-parameter deformation of the $A$-twisted sphere. We provide an exact
formula for the $S^2_\Omega$ supersymmetric correlation functions using
supersymmetric localization. The contribution of each instanton sector is given
in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of
vanishing $\Omega$-deformation, the localization formula greatly simplifies the
computation of $A$-twisted correlation functions, and leads to new results for
non-abelian theories. We discuss a number of examples and comment on the
$\epsilon_\Omega$-deformation of the quantum cohomology relations. Finally, we
present a complementary Higgs branch localization scheme in the special case of
abelian gauge groups.
