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

Jan Möllers, Bent Ørsted, Genkai Zhang

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Abstract

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 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 nilpotent radical of a parabolic subgroup in a semisimple group.
Original languageEnglish
JournalJournal of Geometric Analysis
Volume26
Issue1
Pages (from-to)118-142
Number of pages25
ISSN1050-6926
DOIs
Publication statusPublished - 2016

Keywords

  • Complementary series
  • Invariant differential operators
  • Lie groups
  • Sobolev type spaces

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