Restriction of discrete series of \(\text{SU}(2,1)\) to \(\text{S}(\text{U}(1)\times\text{U}(1,1))\) (Q1328325)

The group \(G = \text{SU}(2,1)\) consists of complex unimodular \(3 \times 3\) matrices preserving the Hermitian form \(| z_ 1|^ 2 + | z_ 2 |^ 2 - | z_ 3 |^ 2\). This group has several discrete series of irreducible unitary representations (holomorphic, antiholomorphic, and nonholomorphic). The author studies decompositions of restrictions of these representations onto the subgroup \(G_ 1 =\text{S}(\text{U}(1)\times\text{U}(1,1))\). These decompositions are multiplicity free. Both the discrete and the continuous parts enter the decompositions. To carry out concrete computations on \(G\), a realization for the discrete series on this group is described. The realization is based on the embedding of the discrete series into nonunitary principal series. The representation space for each discrete series of \(G\) becomes the completion of harmonic functions on \(S^ 3\) under suitable modified norm. The harmonic functions on \(S^ 3\) are treated in some detail. A particular basis, for each homogeneous harmonic function space, is chosen and labeled in terms of invariants for the action of \(G_ 1\). The action of the Lie algebra \({\mathfrak g}_ 1\) of \(G_ 1\) in this realization of discrete series representations then comes out explicitly as shifts along those homogeneous harmonic functions. The author uses the Casimir operator of \(G_ 1\) to determine decomposition of representations of \(G\) onto \(G_ 1\).