Some results on multiplicities for SL(n) (Q1806719)

Let \(F\) be a number field; the author considers the multiplicity of \(L\)-packets of automorphic representations of \(\text{SL}(n)/F\). The main point of this paper is to present a conjecture that the multiplicity of an \(L\)-packet of a cuspidal automorphic representation is bounded -- perhaps even by \(\varphi(n)\), where \(\varphi\) denotes Euler's function. Here the author defines multiplicity as follows: the \(L\)-packet is defined by a cuspidal automorphic representation \(\widetilde{\pi}\) of \(\text{GL}(n)\); two representations \(\widetilde{\pi}_1\) and \(\widetilde{\pi}_2\) generate the \(L\)-packet if and only if \(\widetilde{\pi}_1\cong \widetilde{\pi}_2\otimes \omega\), where \(\omega\) is a Hecke character; the multiplicity is the number of equivalence classes of such cuspidal \(\widetilde{\pi}\) inside the equivalence class of such \(\pi\) under the analogous equivalence relation holding locally almost everywhere. The author gives a heuristic basis for this conjecture based on Langlands' Tannakian philosophy. He then analyzes, when \(n\) is prime, the case of cuspidal representations associated to a Hecke character of a cyclic extension \(E/F\). Here he gives a complete description.