Invariant Differential Operators on H-Type Groups and Discrete Components in Restrictions of Complementary Series of Rank One Semisimple Groups

摘要

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups G to rank one subgroups G(1). For this we use the realizations of complementary series representations of G and G(1) on Sobolev-type spaces on the nilpotent radicals N and N-1 of the minimal parabolics in G and G(1), respectively. The groups N and N-1 are of H-type and we construct explicitly invariant differential operators between N and N-1. These operators induce the projections onto the discrete components. Our construction of the invariant differential operators is carried out uniformly in the framework of H-type groups and also works for those H-type groups which do not occur as a nilpotent radical of a parabolic subgroup in a semisimple group.