Spectral decompositions of some representations of Lie groups
Let G be one of the groups \(SO_ 0(p,q)\), U(p,q) and Sp(p,q). They are the translation groups of hyperbolic spaces. Let \(P=MAN\) be a maximal parabolic subgroup of G. Here A is a one-dimensional vector subgroup of G corresponding to a Cartan subspace of the hyperbolic space. The group M is a product \(M_ 1M_ 2\) of its subgroups where \(M_ 2\) is isomorphic to \({\mathbb{Z}}_ 2\), U(1) and Sp(1) respectively. Consider representations of G induced by representations \(m_ 1m_ 2an\mapsto a^{\sigma}\delta (m_ 2)\) of P where \(a\mapsto a^{\sigma}\), \(\sigma\in {\mathbb{C}}\), is a quasicharacter of A and \(\delta\in \hat M_ 2\). For these representations, the following things are considered in this note: the K-spectrum, the reducibility, the restrictions \(SO_ 0(2p,2q)\) on U(p,q), \(SO_ 0(4p,4q)\) and U(2p,2q) on Sp(p,q), and U(p,q) on U(1)\(\times U(p-1,q)\). Then the authors study some quasiregular representations of G. Earlier the decomposition of the quasiregular representation on the hyperbolic spaces was got by the reviewer, J. Faraut, W. Rossmann and others. The authors note that for complex and quaternion cases this decomposition can be got easily from the restrictions above and the decomposition for real case. The decompositions in this note have a rather qualitative character.
scientific article; zbMATH DE number 4154763