Orbital Parameters for Induced and Restricted Representations

Abstract

General formulas for the spectral decomposition of both induced and restricted representations are laid out for the case of connected Lie groups H c G. The formulas-which detail the actual spectrum, the multiplicits, and the spectral measure-are in terms of the usual parameters in the so-called orbit method.A proof of these formulas is given in the nilpotent situation.The proof is much simpler than a previously obtained proof using nilpotent algebraic geometry.It is also capable of generalization to nonnilpotent groups.With that in mind, many new examples are presented for semisimple and symmetric homogeneous spaces.Also, a start is made in the case of exponential solvable homogeneous spaces with the treatment of both normal and conormal subgroups.