  We prove that irreducible representations of the elliptic affine Hecke
algebras of Ginzburg, Kapranov, and Vasserot are in one-to-one correspondence
with certain nilpotent Higgs bundles on the elliptic curve. The main tool we
use is the equivariant elliptic cohomology of the Steinberg variety of the
Springer resolution. As a by-product, we study representations at roots of
unity in type-$A$. As another by-product, we define a version of elliptic
Demazure-Lusztig operators with dynamical parameters that satisfy the braid
relations. We discuss speculative indications of this correspondence in 4d
$\mathcal N=2$ gauge theory.
