Representations of solvable Lie groups and geometric quantization (Q1819228)

The paper under review uses the theory of geometric quantization for solvable Lie groups developed by Kostant-Sourian and Auslander-Kostant to quantize the covering spaces of coadjoint orbits of solvable Lie groups. The main theorem is that for each integral coadjoint orbit \(O\) and each element \(\sigma\) in the fundamental group \(\pi_1(O)\) of \(O\), there is a corresponding irreducible unitary representation \(\pi(O,\sigma)\) of \(G\). Furthermore, \(\pi(O_1,\sigma_1)\) and \(\pi(O_1,\sigma_2)\) are equivalent if and only if \(O_1\) and \(O_2\) and \(\sigma_1=\sigma_2\). If \(G\) is of type I then every irreducible unitary representation of \(G\) is equivalent to a \(\pi(O,\sigma)\).