Lifting representations of finite reductive groups II: Explicit conorms

Abstract

Let k be a field, G˜ a connected reductive k-group, and Γ a finite group. In a previous work, the authors defined what it means for a connected reductive k-group G to be parascopic for (G˜,Γ). Roughly, this is a simultaneous generalization of several settings. For example, Γ could act on G˜, and G could be the connected part of the group of Γ-fixed points in G˜. Or G could be an endoscopic group, a pseudo-Levi subgroup, or an isogenous image of G˜. If G is such a group, and both G˜ and G are k-quasisplit, then we constructed a map Nˆst from the set of stable semisimple conjugacy classes in the dual G∧(k) to the set of such classes in G˜∧(k). When k is finite, this implies a lifting from packets of representations of G(k) to those of G˜(k).In order to understand such a lifting better, here we describe two ways in which Nˆst can be made more explicit. First, we can express our map in the general case in terms of simpler cases. We do so by showing that Nˆst is compatible with isogenies and with Weil restriction, and also by expressing it as a composition of simpler maps. Second, in many cases we can construct an explicit k-morphism Nˆ:G∧⟶G˜∧ that agrees with Nˆst. As a consequence, our lifting of representations is seen to coincide with Shintani lifting in some important cases.