On the tensor product of a finite and an infinite dimensional representation

There are two main results in the paper. The first gives the infinitesimal character that can occur in the tensor product V V_λ of an irreducible finite dimensional representation V_λ and an irreducible infinite dimensional representation V of a semisimple Lie algebra . The statement is that the infinitesimal characters are x_{v + μi}, I = 1, 2,…, k, where μ_i are the weights of V_λand v is the “pseudo” highest weight of V.The second result proves that if V is a Harish-Chandra module (one which comes from a group representation), then V V_λ has a finite composition series. But then the irreducible components in the composition series have the infinitesimal characters given in the first results.