On the Conjugacy of Element-Conjugate Homomorphisms II

We therefore consider only connected groups, which we often choose to regard as connected reductive algebraic groups over R. This paper classifies connected, simply connected, acceptable, compact groups. As a byproduct of the method, we also treat the non-simply connected, exceptional groups. It turns out that the only acceptable, exceptional group is G2. The original motivation for this question is the construction of automorphic forms appearing with multiplicity greater than 1 in the space of cusp forms on the dual group of G [1]. Our methods also find application in differential geometry; we construct for each n ≥ 35, a non-isometric pair of isospectral manifolds with universal covering space Spin(n). This paper could not have existed in its present form without the generous help of R. Pink. It gives me great pleasure to acknowledge his assistance. I would also like to thank W. Ziller for a useful conversation about symmetric spaces and the Max Planck Institut für Mathematik, for its hospitality while this work was in progress.