Quantization and Representations of Solvable Lie Groups

Introduction. In this note, we will announce a characterization of a connected, simply connected Type I solvable Lie group, G, and present a complete description of the set of all unitary equivalence classes of irreducible unitary representations of G together with a construction of an irreducible representation in each equivalence class. This result subsumes the results previously obtained on nilpotent Lie groups and solvable Lie groups of exponential type of Kirillov [3] and Bernât [2], respectively. Our result is made possible by a merging of a new general geometric approach to representation theory, based on the use of symplectic manifolds and quantization, of the second author with a detailed analysis of the Mackey inductive procedure which augments the results in [ l ] .