Some irreducible unitary representations of G(K) for a simple algebraic group \(G\) over an algebraic number field \(K\) (Q1566425)

Let \(G\) be a simply connected simple linear algebraic group defined over a number field \(K\). The authors investigate the behaviour of irreducible unitary representations of the adelic group \(G({\mathbf A}_K)\) upon restriction to the global points \(G(K)\). They prove that the restriction to \(G(K)\) stays irreducible and that inequivalent representations have inequivalent restrictions. If all local groups \(G(K_v)\) have Kazhdan's property \((T)\), it is proved for inequivalent (irreducible unitary) representations \(\pi, \pi'\) which are not both weakly contained in the regular representation that \(\pi |G(K), \pi' |G(K)\) are not even weakly equivalent. A similar result is stated for the restrictions of representations of \(\prod_{v \in S} G(K_v)\) to \(G({\mathcal O}_S)\), where \(S\) is a finite set of places containing the archimedean places.