The embeddings of the discrete series in the principal series for semisimple Lie groups of real rank one

Abstract

We consider the problem of finding all the “embeddings” of a discrete series representation in the principal series in the case of a simple real Lie group G of real rank one. More precisely, we solve the problem when G is Spin ⁡ ( 2 n , 1 ) , SU ( n , 1 ) , SP ( n , 1 ) or F 4 ( n ⩾ 2 ) \operatorname {Spin} (2n,\,1),{\text {SU}}(n,\,1),\,{\text {SP}}(n,\,1)\,{\text {or}}\,{F_4}\,(n\, \geqslant \,2) . The problem is reduced to considering only discrete series representations with trivial infinitesimal character, by means of tensoring with finite dimensional representations. Various other techniques are employed.