Arakawa, Suzuki and Tsuchiya constructed a correspondence between certain modules of the trigonometric Cherednik algebra $\mathfrak{ C}_N$ depending on a parameter $\kappa\in\mathbb{ C}$, and certain modules of the affine Lie algebra $\widehat{ \mathfrak{ sl}}_m$ of level $\kappa-m$. We give a detailed proof of this correspondence by working with the affine Lie algebra $\widehat{ \mathfrak{ gl}}_m$ alongside of $\widehat{ \mathfrak{ sl}}_m$. We also relate this construction to a correspondence between certain modules of the degenerate affine Hecke algebra $\mathfrak{ H}_N$ and all modules of $\mathfrak{ sl}_m$ or $\mathfrak{ gl}_m$. The latter correspondence was constructed earlier by Cherednik.
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