The functorial source problem via dimension data

For an automorphic representation $π$ of Ramanujan type, there is a conjectural subgroup $\mathcal{H}_π$ of the Langlands L-group $^{L}G$ associated to $π$, called the {\it functional source} of $π$. The functorial source problem proposed by Langlands and refined by Arthur intends to determine $\mathcal{H}_π$ via analytic and arithmetic data of $π$. In this paper, we consider the functorial source problem of automorphic representations of a split group, a unitary group, or an orthogonal group which do not come from endoscopy and have minimal possible ramification. In this setting, $\mathcal{H}_π$ must be an S-subgroup of $^{L}G$. We approach the functorial source problem by proving distinction and linear independence among dimension data of S-subgroups. Nice results along this direction are shown in this paper. We define a notion of quasi root system and use it as the key tool for studying S-subgroups and their dimension data.

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