The residual spectrum of the group of type \(G_ 2\)
Let \(G\) be the split group of type \(G_2\) defined over a number field \(k\) and \(G({\mathcal A})\) its adele group. Let \(L_d^2\) be the discrete part of the space of square-integrable complex valued functions on \(G(k)\setminus G({\mathcal A})\), i.e. the direct sum of irreducible \(G({\mathcal A})\)-submodules. The author describes this space in terms of cuspidal representations of the standard Levi factors following Langlands' method. The most surprising result concerns the part of the discrete spectrum coming from a Borel subgroup; in addition to the trivial representation (the constants) and the representations attached to the grössencharacters of order 2 and 3, it contains a series of representations arising from the trivial character of the maximal torus. This series consists of irreducible quotients of an induced representation satisfying a strange condition on the cardinality of the ramified components \((|S|\neq 1)\). A part of this result is proved by C. Mœglin and J.-L. Waldspurger.
scientific article; zbMATH DE number 1060802
scientific article