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

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.

22 pages; added Section 5 in v2 to separate H-type results and rank one results

References in corpus (4)

• Symmetry breaking for representations of rank one orthogonal groups II
• Knapp-Stein Type Intertwining Operators for Symmetric Pairs II. -- The Translation Principle and Intertwining Operators for Spinors
• Tensor products of complementary series of rank one Lie groups
• Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups

Cited by in corpus (3)

• Symmetry breaking operators for real reductive groups of rank one
• The compact picture of symmetry breaking operators for rank one orthogonal and unitary groups
• On boundary value problems for some conformally invariant differential operators