On local and global conjugacy

Local and global conjugacy issue is closely related to the multiplicity one property in representation theory and Langlands' program. We classify all even dimensional orthogonal irreducible representations which are locally conjugate but not globally conjugate in image in SO(2N)SO(2N), and with certain Langlands' functoriality these will lead to connected instances for the failure of multiplicity one for SO(2N)SO(2N) (named as LFMO). In this note, we gather most our local–global results in a purely representation theoretic way.