Affine Hecke algebras and quiver Hecke algebras
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.