Cubic unipotent Arthur parameters and multiplicities of square integrable automorphic forms
Let \(G\) be the split group of type \(\text{G}_2\) over an algebraic number field \(F\). The discrete spectrum of the representation of \(G({\mathbb A})\) in \(L^2(G(F)\setminus G({\mathbb A}))\) decomposes into the cuspidal spectrum and the residual spectrum. The authors show that there are irreducible representations of \(G({\mathbb A})\) which occur in both the cuspidal and the residual spectrum and that the multiplicities in the cuspidal spectrum are not bounded. Let \(E\) be an etale cubic \(F\)-algebra. It determines a form \(G_E\) of \(\text{Spin}_8\) in which \(G\) is imbedded and on which a form \(S_E\) of the symmetric group \(\text{S}_3\) acts. The examples confirming the statements above are found in decomposing the restriction to \(S_E({\mathbb A})\times G({\mathbb A})\) of a certain representation of \(G_E({\mathbb A})\). Also a known description of the residual spectrum of \(G({\mathbb A})\) is used. The authors place their result into the context of Arthur's conjectures. They describe what should be the packet corresponding to the Arthur parameter associated to \(E\).
scientific article; zbMATH DE number 1965425