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Multiplicity one theorems: the Archimedean case. (English) Zbl 1239.22014
Multiplicity theorems on the irreducible representations of classical Lie groups and their subgroups are presented. One denotes by \(G\) the classical Lie groups \[ \mathrm {GL}_{n+1}(\mathbb R), \mathrm {GL}_{n+1}(\mathbb C), \mathrm {U}(p,q+1), \mathrm {O}(p,q+1), \mathrm {O}_{n+1}(\mathbb C), \mathrm {SO}(p,q+1), \mathrm {SO}_{n+1}(\mathbb C), \] and by \(G'\) respectively their subgroups \[ \mathrm {GL}_n(\mathbb R), \mathrm {GL}_n(\mathbb C), \mathrm {U}(p,q), \mathrm {O}(p,q), \mathrm {O}_n(\mathbb C), \mathrm {SO}(p,q), \mathrm {SO}_n(\mathbb C) \] embedded in \(G\) in the standard way. Then, it is proven that every irreducible Casselman-Wallach representation of \(G'\) occurs with multiplicity at most one in every irreducible Casselman-Wallach representation of \(G\). The main technical result of this paper is: There exists a real algebraic anti-automorphism \(\sigma\) on \(G\) preserving \(G'\) with the following property: every generalized function on \(G\) which is invariant under the adjoint action of \(G'\) is automatically \(\sigma\)-invariant. Combined with a version of the Gelfand-Kazhdan criterion, this result implies the above property on the multiplicity of the irreducible Casselman-Wallach representations of the respective classical groups. Similar results are proved for Jacobi groups with their respective subgroups.

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E30 Analysis on real and complex Lie groups
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