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

Abstract

Let K be an algebraic number field, and let GK 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 GK as follows. It is well known that GK embeds diagonally as a lattice in GA, where A is the ring of adeles of K. Let \(\) be an irreducible unitary representation of GA. We show that \(\), the restriction of \(\) to GK, is irreducible and that \(\) is determined by \(\) up to unitary equivalence. Many of these restrictions are not in the support of the regular representation of GK.