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 ⊂ G H \subset 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.