Restriction to symmetric subgroups of unitary representations of rank one semisimple Lie groups

Abstract

ABSTRACT. We consider the spherical complementary series of rank one Lie groups H n = SO 0 (n, 1; F) for F = R, C, H. We prove that there exist finitely many discrete components in its restriction under the subgroup H n−1 = SO 0 (n − 1, 1; F). This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of G n = SU (n, 1), SU (n, 1) × SU (n, 1) and SU (2n, 2) and by the branching of holomorphic representations under the corresponding subgroup G n−1 .

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

“…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. This way of obtaining intertwining operators has also been investigated recently; see e.g.…”

“…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.…”

“…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.…”

“…In the present paper we shall find explicit embeddings for all discrete components given in [23]. For this we use the realization of complementary series representations on certain function spaces on a nilpotent group N, the nilpotent radical of a minimal parabolic.…”