On a correspondence of automorphic forms on orthogonal groups of order five
Let \(G_ A\) be the adelic group associated with a quasi-split group G defined over a global field k. A representation \(\pi\) of \(G_ A\) is automorphic if the action of \(\pi\) is, via right translations, on a space of automorphic forms on \(G_ A\). An irreducible automorphic representation \(\pi\) of \(G_ A\) is called L-equivalent to an automorphic representation \(\pi'\) of \(G_ A\) if there exists a constituent \(\tau\) of \(\pi'\) such that \(\pi\) is locally equivalent to \(\tau\) almost everywhere. According to a well-known conjecture, there exists for a given irreducible automorphic representation \(\pi\) of \(G_ A\), an indecomposable automorphic representation \(\pi'\) of \(G_ A\) which has a Whittaker model and is L-equivalent to \(\pi\). (In general, \(\pi'\) may not be expected to be irreducible.) The main result of the paper concerns the case when G is \(G_ 1=SO(4,1)\) or its inner form \(G_ 2=SO(3,2)\) over a global field k of characteristic \(\neq 2\) and then establishes, for an irreducible automorphic cuspidal representation \(\pi\) of \(G_ A\) (not belonging to the image of the \(\theta\)-correspondence from \(\tilde SL(2,A)\) to \(G_ A\) but having a Bessel model of special type), the existence of a unique irreducible automorphic cuspidal representation \(\pi'\) of \(G_ 2(A)\) such that \(\pi'\) has a Whittaker model and \(\pi_ p\) is equivalent to \(\pi_ p'\) for all p with \(G(k_ p)\simeq G_ 2(k_ p)\) and \(\pi_ p\) unramified. For \(G=GSp(4)\), it is shown that an irreducible automorphic cuspidal representation \(\pi\) of \(G_ A\), having a Whittaker model, is not L-equivalent to an Eisenstein series representation of \(G_ A\).
correspondence of irreducible automorphic representations
automorphic representations of orthogonal groups for five variables
scientific article; zbMATH DE number 4049905