  We construct level one dominant representations of the affine Kac-Moody
algebra $\widehat{\mathfrak{gl}}_k$ on the equivariant cohomology groups of
moduli spaces of rank one framed sheaves on the orbifold compactification of
the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity
$\mathbb{C}^2/\mathbb{Z}_k$. We show that the direct sum of the fundamental
classes of these moduli spaces is a Whittaker vector for
$\widehat{\mathfrak{gl}}_k$, which proves the AGT correspondence for pure
$\mathcal{N}=2$ $U(1)$ gauge theory on $X_k$. We consider Carlsson-Okounkov
type Ext-bundles over products of the moduli spaces and use their Euler classes
to define vertex operators. Under the decomposition
$\widehat{\mathfrak{gl}}_k\simeq \mathfrak{h}\oplus \widehat{\mathfrak{sl}}_k$,
these vertex operators decompose as products of bosonic exponentials associated
to the Heisenberg algebra $\mathfrak{h}$ and primary fields of
$\widehat{\mathfrak{sl}}_k$. We use these operators to prove the AGT
correspondence for $\mathcal{N}=2$ superconformal abelian quiver gauge theories
on $X_k$.
