 
Summary: Clay Mathematics Proceedings
Volume 13, 2011
The Embedded Eigenvalue Problem for Classical Groups
James Arthur
This paper is dedicated to Freydoon Shahidi on the occasion of his sixtieth birthday.
Abstract. 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 spec
tral theorem. It corresponds to representations in which the continuous induction
parameter is unitary. For example, the trivial onedimensional automorphic repre
sentation of the group SL(2) does not represent an embedded eigenvalue. This is
because it corresponds to a value of the onedimensional induction parameter at a
nonunitary point in the complex domain. For general linear groups, the absence of
