Explicit Shintani base change and the Macdonald correspondence for characters of GL\(_n(k)\)
Let \(K\) be a local field with residual field \(k\), and let \(W'_K = \mathbb C \rtimes W_K\) be the Weil-Deligne group of \(K\). Given an inertial subgroup \(I\) and a wild ramification subgroup \(P\) of \(W_K\), let \(\text{Rep}_n (W'_K/P)_I\) be the set of all \(n\)-dimensional admissible representations of \(W'_K/P\) modulo the equivalence relation determined by \(I\). If \(\Omega_n (k)\) denotes the set of all irreducible characters of \(GL_n (K)\), it was shown by \textit{I.~G.~Macdonald} [Math. Ann. 249, 1--15 (1980; Zbl 0417.14016)] that there is a bijective correspondence \(\mathcal M_n (k): \Omega_n (k) \to \text{Rep}_n (W'_K/P)_I\). Let \(K_d\) be an unramified extension of \(K\) of degree \(d \geq 1\) with the associated residual field extension \(k_d\). In this paper the authors use Zelevinsky's PSH Hopf algebra theory to prove that \(\mathcal M_n (k_d) \circ \text{bc}_{k \uparrow k_d} = \text{res}_{K \downarrow K_d} \circ \mathcal M_n (k)\), where \(\text{bc}_{k \uparrow k_d}\) is the Shintani base change map for \(GL_n\) and \(\text{res}_{K \downarrow K_d}\) is the restriction of \(n\)-dimensional representations from the Weil-Deligne group \(W' (K)\) to \(W' (K_d)\) for \(I\)-equivalence classes of tame representations. The same relation holds with \(\mathcal M_n\) replaced by the local Langlands correspondence and the finite-field base change replaced by a local base change with no restriction to \(I\)-equivalence classes.
scientific article; zbMATH DE number 5304642
scientific article