Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups

TY - JOUR

T1 - Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups

AU - Möllers, Jan

AU - Ørsted, Bent

AU - Zhang, Genkai

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

KW - Complementary series

KW - Invariant differential operators

KW - Lie groups

KW - Sobolev type spaces

UR - http://www.scopus.com/inward/record.url?scp=84953384424&partnerID=8YFLogxK

U2 - 10.1007/s12220-014-9540-z

DO - 10.1007/s12220-014-9540-z

M3 - Journal article

SN - 1050-6926

VL - 26

SP - 118

EP - 142

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 1

ER -