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

Abstract

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 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 nilpotent radical of a parabolic subgroup in a semisimple group.2010 MSC: Primary 22E46; Secondary 22E25.

On boundary value problems for some conformally invariant differential operators

“…T = ∂ 2 ∂t 2 . Following [12] we inductively define a sequence (D s,k ) k of differential operators on H 2n+1 depending on a parameter s ∈ C by Now for −2n < a ≤ 2 and 0 ≤ 2k < − a 2 we define a differential restriction operator D a,k : C ∞ (H 2n+1 ) → C ∞ (H 2n−1 ) by D a,k u(z ′ , t ′ ) := (D − a+2n 4 ,k u)(z ′ , 0, t ′ ). In [12,Theorem 4.1] we show that D a,k extends to a bounded operatorḢ…”

“…Using the group Fourier transform on H 2n+1 one can define natural Sobolev type spaceṡ H s (H 2n+1 ), 0 ≤ s < n + 1, analogous to the real case (see Section 2.3.2 for details). In [12] we prove that the restriction to H 2n−1 = {(z, t) ∈ H 2n+1 : z n = 0} ⊆ H 2n+1 defines a continuous linear operatorḢ s (H 2n+1 ) →Ḣ s−1 (H 2n−1 ) whenever s > 1.…”

“…Using the group Fourier transform on H 2n+1 one can define natural Sobolev type spaces Ḣs (H 2n+1 ), 0 ≤ s < n + 1, analogous to the real case (see Section 2.3.2 for details). In [12] we prove that the restriction to…”

“…were first constructed in [12] and exist also on more general two-step nilpotent groups. In Theorem B.1 we study the mixed boundary value problems…”

Symmetry breaking operators for real reductive groups of rank one

“…The existence of the meromorphic family of distributions u A λ,ν and a generic multiplicity one statement were previously obtained by Möllers-Ørsted-Oshima [MØO16] (see also [Möl17]), but the precise multiplicities also for singular parameters as well as the detailed study of the meromorphic nature of u A λ,ν were missing so far. Only the differential symmetry breaking operators corresponding to the distributions u C λ,ν were previously constructed by Möllers-Ørsted-Zhang [MØZ16a], but in particular the sporadic differential operators in Theorem E were not known before.…”