Irreducible unitary representations of non-compact semisimple Lie groups and Casimir elements

Question

Consider a non-compact semisimple Lie group G. Then the irreducible unitary representations (UID) of G are infinite-dimensional.

1. Do the UID's of G modulo unitary equivalence correspond bijectively to the infinitesimal characters of the center of the universal enveloping algebra Lie G modulo the action of the Weyl group?
2. Is the equivalence class of each UID of G determined by the finitely many normalized Casimir values of the representation?

I would prove the first statement via Harish-Chandra’s theorem that infinitesimal equivalence implies unitary equivalence. Then the second statement follows from Harish-Chandra’s theorem that the center of the universal enveloping algebra of Lie G is generated by the polynomials in Lie H, which are invariant under the action of the Weyl group.