On the Macdonald correspondence

In 1980 Ian G. Macdonald established an explicit bijection between the isomorphism classes of the irreducible representations of ${\mathrm{GL}}_n(k)$, where $k$ is a finite field, and inertia equivalence classes of admissible tamely ramified $n$-dimensional Weil-Deligne representations of $W_F$, where $F$ is a non-archimedean local field with residue field $k$ and $W_F$ the absolute Weil group of $F$. We describe a construction of the Macdonald correspondence based on the specialization to ${\mathrm{GL}}_n(k)$ of Lusztig's classification of irreducible representations of finite groups of Lie type, and review some properties of the correspondence. We define $\epsilon$-factors for pairs of irreducible cuspidal representations of finite general linear groups, and show that they match with the expected Deligne $\epsilon$-factors under the Macdonald correspondence. We use these $\epsilon$-factors for pairs to obtain a characterization of the Macdonald correspondence for the irreducible cuspidal representations