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

Let K be an algebraic number field, and let G(K) be the group of K-rational points of a simply connected simple linear algebraic group G defined over K. We construct a new family of irreducible unitary representations of G(K) as follows. It is well known that G(K) embeds diagonally as a lattice in G(A), where A is the ring of adeles of K. Let pi be an irreducible unitary representation of G(A). We show that pi\G(K), the restriction of pi to G(K), is irreducible and that pi is determined by pi\G(K) up to unitary equivalence. Many of these restrictions are not in the support of the regular representation of G(K).