The representations of the oscillator group

Commun. Math. Phys., 4, pp 217-236, 1967, by R. F. Streater

ABSTRACT

Using the Mackey theory of induced representations, all the unitary continuous representations of the four-dimensional Lie group G generated by the canonical variables and a positive definite quadratic "Hamiltonian" are found. These are shown to be in a one to one correspondence with the orbits under G in the dual space dG* to the Lie algebra dG of G, and the representations are obtained from the orbits by inducing from one-dimensional representations, provided that complex subalgebras are admitted. Thus a construction analogous to that of Kirillov and Bernat gives all the representations of this group.

REMARKS

This problem, but not the method of solution, was suggested to me by B. Kostant; subsequently, using a different method, the result was extended to all solvable Lie groups of type I by Kostant, Moore and Auslander. The oscillator group is defined as the semidirect product of the line (time) with the Heisenberg group, with the action given by the dynamics. Thus, it is not realised as a matrix group. It is a solvable group, but not an exponential group. A Lie group is said to be exponential if any element of the group lies on a one-parameter subgroup. Dixmier had given a tractable criterion, used in my paper, to determine whether a group is exponential or not. P. Bernat had previously shown that Kirillov's method of orbits, using real polarisations, does give all the irreducible representations of an exponential solvable Lie group, by inducing from one-dimensional representations of maximal subgroups subordinate to the orbit in the dual to the real algebra, under the dual (the adjoint) action of the group. Here, I show that if we consider the complexification of the dual space, then we get all representations by holomorphic induction from one-dimensional representations of maximal complex subgroups subordinate to the orbit in the complex sense.

Further work on this topic has been done by Abdel Ghani Zeghie, in Sur les espaces-temps homogenes, pp 551-576, in The Epstein Birthday Schrift, Geometry and Topology Monographs, Vol I, ISSN 1464-8997, which can be read here. Another work (by Barbot and Zeghib) can be read in English here.