The Embedded Eigenvalue Problem for Classical Groups

We report briefly on an endoscopic classification of representations by focusing on one aspect of the problem, the question of embedded Hecke eigenvalues. 1. The problem for G By “eigenvalue”, we mean the family of unramified Hecke eigenvalues of an automorphic representation. The question is whether there are any eigenvalues for the discrete spectrum that are also eigenvalues for the continuous spectrum. The answer for classical groups has to be part of any general classification of their automorphic representations. The continuous spectrum is to be understood narrowly in the sense of the spectral theorem. It corresponds to representations in which the continuous induction parameter is unitary. For example, the trivial one-dimensional automorphic representation of the group SL(2) does not represent an embedded eigenvalue. This is because it corresponds to a value of the one-dimensional induction parameter at a nonunitary point in the complex domain. For general linear groups, the absence of embedded eigenvalues has been known for some time. It is a consequence of the classification of Jacquet-Shalika [JS] and Moeglin-Waldspurger [MW]. For other classical groups, the problem leads to interesting combinatorial questions related to the endoscopic comparison of trace formulas. We shall consider the case that G is a (simple) quasisplit symplectic or special orthogonal group over a number field F . Suppose for example that G is split and of rank n. The continuous spectrum of maximal dimension is then parametrized by n-tuples of (unitary) idele class characters. Is there any n-tuple whose unramified Hecke eigenvalue family matches that of an automorphic representation π in the discrete spectrum of G? The answer is no if π is required to have a global Whittaker model. This follows from the work of Cogdell, Kim, Piatetskii-Shapiro and Shahidi 1991 Mathematics Subject Classification. Primary 22E55, 22E50; Secondary 20G35, 58C40.