The subrepresentation theorem for automorphic representations (Q1951197)

The purpose of this short note is to provide the proof of the analogue of the Casselman subrepresentation theorem in the setting of automorphic representations. The subrepresentation theorem of \textit{W. Casselman} [in: Proc. Int. Congr. Math., Helsinki 1978, Vol. 2, 557--563 (1980; Zbl 0425.22019)] is one of the most basic theorems in representation theory of reductive groups over local fields, saying that any irreducible representation (in the appropriate category) is a subrepresentation of certain parabolically induced representation from a simpler representation of an appropriate subgroup. It is a refinement of the subquotient theorem of \textit{Harish-Chandra} [Trans. Am. Math. Soc. 76, 26--65, 234--253 (1954; Zbl 0055.34002)], which claims that any irreducible representation is a subquotient of such induced representation. In the setting of automorphic representations, the subquotient theorem is due to \textit{R. P. Langlands} [Proc. Symp. Pure Math. 33, 1, 203--207 (1979; Zbl 0414.22021)]. It says that, for a connected reductive group \(G\) defined over an algebraic number field \(k\), any irreducible automorphic representation of \(G(\mathbb{A})\) in the sense of \textit{A. Borel} and \textit{H. Jacquet} [Proc. Symp. Pure Math. 33, 1, 189--202 (1979; Zbl 0414.22020)], where \(\mathbb{A}\) is the ring of adèles of \(k\), can be realized as a subquotient of a parabolically induced representation from a cuspidal automorphic representation of an appropriate subgroup. The paper under review refines this result of Langlands. It shows that any irreducible subrepresentation of the space of automorphic forms on \(G(\mathbb{A})\) may be realized as a subrepresentation of a parabolically induced representation from a cuspidal automorphic representation of an appropriate subgroup. The proof is a slight refinement of Langlands' original proof in [loc. cit.], combined with the spectral decomposition results in [\textit{C. Mœglin} and \textit{J.-L. Waldspurger}, Spectral decomposition and Eisenstein series. Cambridge: Cambridge Univ. Press (1995; Zbl 0846.11032)].