Lifting representations of finite reductive groups II: Explicit conorms

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