Generic representations in \(L\)-packets (Q2816457)

Let \(G\) be a quasi-split connected reductive group defined over a local field \(k\) of characteristic zero. The conjectural local Langlands program partitions the irreducible admissible representations of \(G\) into finite sets that are known as \(L\)-packets. The purpose of this work is to offer the construction of a map in order to restate the conjecture of \textit{W. T. Gan} et al. [in: Sur les conjectures de Gross et Prasad. I. Paris: Société Mathématique de France (SMF). 1--109 (2012; Zbl 1280.22019)] about the adjoint group action on generic representations in \(L\)-packets. The author also obtains, in a conceptual and uniform manner, the classification of the Knapp-Stein \(R\)-group and proves the conjecture for unramified \(L\)-packets.