  We give a presentation of localized affine and degenerate affine Hecke
algebras of arbitrary type in terms of weights of the polynomial subalgebra and
varied Demazure-BGG type operators. We offer a definition of a graded algebra
$\mathcal{H}$ whose category of finite-dimensional ungraded nilpotent modules
is equivalent to the category of finite-dimensional modules over an associated
degenerate affine Hecke algebra. Moreover, unlike the traditional grading on
degenerate affine Hecke algebras, this grading factors through central
characters, and thus gives a grading to the irreducible representations of the
associated degenerate affine Hecke algebra. This paper extends the results
\cite[Theorem 3.11]{rouquier-qha}, and \cite[Main Theorem]{brundan-kleshchev}
where the affine and degenerate affine Hecke algebras for $GL_n$ are shown to
be related to quiver Hecke algebras in type $A$, and also secretly carry a
grading.
