Explicit Shintani base change and the Macdonald correspondence for characters of GLn(k)

Let {Kd}d⩾1 denote the set of unramified extensions of a local field K and {kd}d⩾1 the respective residual field extensions. The authors recall Macdonald's parameterization [I.G. Macdonald, Zeta functions attached to finite general linear groups, Math. Ann. 249 (1980) 1–15] of the irreducible characters of GLn(kd) in terms of “I-equivalence classes” of tame n-dimensional representations of the Weil–Deligne group W′(Kd). Using Zelevinsky's PSH Hopf algebra theory [A. Zelevinsky, Representations of Finite Classical Groups, Lecture Notes in Math., vol. 869, Springer-Verlag, New York, 1981], they prove (see (1.1)) that Mn(kd)○bck↑kd=resK↓Kd○Mn(k), where Mn(k) denotes the Macdonald parameterization map for GLn(k), bck↑kd the Shintani base-change map for GLn, and resK↓Kd the restriction of n-dimensional representations from the Weil–Deligne group W′(K) to W′(Kd) for I-equivalence classes of tame representations. As Henniart [G. Henniart, Sur la conjecture de Langlands locale pour GLn, J. Théor. Nombres Bordeaux 13 (2001) 167–187] has shown, the same relation holds with Mn replaced by the local Langlands correspondence and finite-field base change replaced by local-field base change with no restriction to I-equivalence classes. In an Addendum the authors show (see (A.1)) that the map φ0 which sends a level-zero irreducible representation of GLn(K) to the reduction of its “tempered type” [P. Schneider, E.-W. Zink, K-types for the tempered components of a p-adic general linear group, J. Reine Angew. Math. 517 (1999) 161–208] connects the level-zero local-field Langlands parameterization to the finite-field parameterization of Macdonald. They also remark (see the concluding Remark) that φ0 is compatible with the Shintani and local-field base change maps.