A Coefficient Calculus for Unitary (g,K)-Modules of SU(2,2)

RESUMO

We develop an explicit operator and coefficient calculus for admissible (g,K)-modules of SU(2,2), where g = sl(4,C) and K = S(U(2) x U(2)). The K-types are indexed by triples (n,k,m). We introduce operators A_delta and B_delta associated with the noncompact roots, together with auxiliary operators Q and R. We establish their commutator relations, the central coefficient relations, and, in the unitary setting, the corresponding adjoint and norm-factor identities. This reduces part of the analysis of (g,K)-modules to the study of scalar coefficients arising from compositions of these operators. The coefficient formula applies to weights of multiplicity one, including all boundary weights. We derive necessary sign restrictions and explicit relations constraining the coefficient space. These coefficients also determine irreducibility. Several families of unitary (g,K)-modules are analyzed according to the minimal value of n+m. For N=0, the construction yields two-parameter families of unitary modules and describes reducibility when certain coefficients vanish. For N>0, the method produces both larger families of K-types and multiplicity-free ladder-type families. The resulting patterns are compatible with the known description of the unitary dual of SU(2,2) due to Knapp and Speh. The paper provides an explicit framework for recovering unitary (g,K)-modules from their K-type structure and suggests a possible approach to other real reductive groups.