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Invariant triple products. (English) Zbl 1140.22018

For irreducible admissible smooth representations \(\pi_1\), \(\pi_2\), and \(\pi_3\) of a semi-simple Lie group \(G\), one would like to know how large the space of \(G\)-invariant trilinear forms on \(\pi_1 \times \pi_2 \times \pi_3\) is. For \(G=\text{ PGL}_2({\mathbb R})\) this space was known to be at most one-dimensional (uniqueness of trilinear forms). In the paper under review the author proves that when \(G\) is a product of hyperbolic groups and \(\pi_i\) are spherical such uniqueness holds. He also gives some evidence that one expects uniqueness only in such a case. He also indicates how such uniqueness results are related to special values of \(L\)-functions and bounds for automorphic \(L^2\)-coefficients.

MSC:

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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References:

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