On the Macdonald correspondence

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