Tensor operators and projection techniques in infinite dimensional representations of semi-simple Lie algebras

Polynomial identities satisfied by the infinitesimal generators of a semi-simple Lie group are employed to construct projection operators which project a tensor product representation V( lambda )(X)V (V( lambda ) finite dimensional and irreducible, V an infinite dimensional representation admitting an infinitesimal character) onto a primary subrepresentation. Such projection operators project out the generalised shift components of a tensor operator acting on an infinite dimensional representation V in direct analogy with the finite dimensional analysis of Bracken and Green (1971). Applications of these methods to evaluating the matrix elements of the generators of the locally compact groups U(p,q) and O(p,q) in unitary (discrete) irreducible representations are discussed.