GROUP-REPRESENTATIONS WEAKLY EQUIVALENT TO THE REGULAR REPRESENTATION

Let Gamma be a lattice in a connected real Lie group G which is simple, non compact and with centre reduced to {1}. Let pi be a unitary representation of G and let pi/(Gamma) be its restricition to Gamma. If pi is not trivial, we show that pi/(Gamma) weakly contains the regular representation lambda(Gamma) of Gamma. When pi is in the principal series or the discrete series of G, this implies that pi/(Gamma) and lambda(Gamma) are weakly equivalent. In the case G = PSL(2)(R) and when pi is in the complementary series, we obtain also results on pi/(Gamma).