Restriction of some unitary representations of O(1,N) to symmetric subgroups

Abstract

We find the complete branching law for the restriction of certain unitary representations of O(1, n + 1) to the subgroupsThe unitary representations we consider belong either to the unitary spherical principal series, the spherical complementary series or are unitarizable subquotients of the spherical principal series. In the crucial case 0 < m < n the decomposition consists of a continuous part and a discrete part. The continuous part is given by a direct integral of unitary principal series representations whereas the discrete part consists of finitely many representations which either belong to the complementary series or are unitarizable subquotients of the principal series. The explicit Plancherel formula is computed on the Fourier transformed side of the non-compact realization of the representations by using the spectral decomposition of a certain hypergeometric type ordinary differential operator. The main tool connecting this differential operator with the representations are second order Bessel operators which describe the Lie algebra action in this realization. To derive the spectral decomposition of the ordinary differential operator we use Kodaira's formula for the spectral decomposition of Schrödinger type operators.

On boundary value problems for some conformally invariant differential operators

“…For the real case we indicate in Appendix A how the full spectral decomposition of ∆ a in Ḣ 2−a 2 (R n ) is obtained. This was carried out in detail by Möllers-Oshima [13]. For the complex case the full decomposition is not yet know.…”

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

“…For the abelian case n = v = R n and n 1 = v 1 = R m these results were already obtained by Kobayashi-Speh [17] and by Möllers-Oshima [20]. Hence we assume z = {0} for the rest of this section which allows us to use the Plancherel formula for N an N 1 as formulated in Section 2.5 as well as Remark 2.3.…”

“…In the degenerate case where n = R n is abelian and n = R n−1 the operators D ν,k were constructed before by Juhl [12] and generalized to arbitrary signature in [15] (see Remark 3.11 for details). They were used to construct discrete components in the restriction of complementary series of SO(1, n + 1) to SO(1, n) by Kobayashi-Speh [17] and Möllers-Oshima [20]. For other rank one groups the abstract existence of the discrete components in the branching rule was previously established by Speh-Zhang [23] without giving an explicit embedding.…”

“…There is another somewhat opposite class of representations, the complementary series, which is of substancial interest in the spectral theory of locally symmetric spaces, in particular rank-one spaces. Recently various authors studied discrete components in the restriction of complementary series of rank one groups to symmetric subgroups, see the work of Kobayashi-Speh [17], Möllers-Oshima [20], Speh-Venkataramana [22], Speh-Zhang [23] and Zhang [25]. For rank one orthogonal groups the discrete spectrum is known by [20] and can explicitly be constructed in terms of Juhl's covariant differential operators [12], see [17,20,23].…”

“…Recently various authors studied discrete components in the restriction of complementary series of rank one groups to symmetric subgroups, see the work of Kobayashi-Speh [17], Möllers-Oshima [20], Speh-Venkataramana [22], Speh-Zhang [23] and Zhang [25]. For rank one orthogonal groups the discrete spectrum is known by [20] and can explicitly be constructed in terms of Juhl's covariant differential operators [12], see [17,20,23]. For the other rank one groups Speh-Zhang [23] found finitely many discrete components in the restriction, without providing an explicit construction of these components.…”