Discrete components in restriction of unitary representations of rank one semisimple Lie groups

Abstract

We consider spherical principal series representations of the semisimple Lie group of rank one G = SO(n, 1; K), K = R, C, H. There is a family of unitarizable representations π ν of G for ν in an interval on R + , the so-called complementary series, and subquotients or subrepresentations of G for ν being negative integers. We consider the restriction of (π ν , G) under the subgroup H = SO(n − 1, 1; K). We prove the appearing of discrete components. The corresponding results for the exceptional Lie group F 4(−20) and its subgroup Spin(8, 1) are also obtained.

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

“…But the only possible bounded differential operator constructed in this way is the restriction operator. This corresponds to the branching law for complementary series of F 4(−20) restricted to Spin(8, 1) which was already investigated by the third author, see [25,Theorem 4.4] for the corresponding statement. Therefore we will not treat the case v 1 = 0 in this paper and always assume that [v 1 , v 1 ] = z.…”

“…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].…”

“…Now we show the boundedness of the intertwining operators D ν,k . We shall need an elementary fact (see also[25, Lemma 3.5]): Suppose that α > −1, β ≥ 0 and β − α > 1. Then there exists a constant C > 0 such that ∞ n=0…”

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

“…This paper is a continuation of [22] by Speh-Venkataramana and [26] by Zhang. Consider the complementary series representations (π µ , H) of the Lie group H = SO 0 (n, 1; F) for F = R, C, H and its restriction to the subgroup H 1 = SO 0 (n − 1, 1; F).…”

Tensor products of complementary series of rank one Lie groups

“…Tensor products of representations of SL(2, C) (i.e., locally isomorphic to the Lorentz group SO 0 (3, 1)) have been studied by Naimark [19]; see also [20] where some complementary series representations were constructed using restriction of holomorphic representations, the same idea being used in the present paper in the construction of discrete components for the groups SU(n, 1) and Sp(n, 1). A related question is the branching of complementary series of rank one groups L = SO(n, 1), SU(n, 1), Sp(n, 1) under the subgroups G = SO(n − 1, 1), SU(n, 1), Sp(n, 1) and is studied in [18,27,29].…”