  The irreducible unitary representations of the Banach Lie group $U_0(\H)$
(which is the norm-closure of the inductive limit $\cup_k U(k)$) of unitary
operators on a separable Hilbert space $\H$, which were found by Kirillov and
Ol'shanskii, are reconstructed from quantization theory. Firstly, the coadjoint
orbits of this group are realized as Marsden-Weinstein symplectic quotients in
the setting of dual pairs. Secondly, these quotients are quantized on the basis
of the author's earlier proposal to quantize a more general symplectic
reduction procedure by means of Rieffel induction (a technique in the theory of
operator algebras). As a warmup, the simplest such orbit, the projective
Hilbert space, is first quantized using geometric quantization, and then again
with Rieffel induction. Reduction and induction have to be performed with
either $U(M)$ or $U(M,N)$. The former case is straightforward, unless the
half-form correction to the (geometric) quantization of the unconstrained
system is applied. The latter case, in which one induces from holomorphic
discrete series representations, is problematic. For finite-dimensional
$\H=\C^k$, the desired result is only obtained if one ignores half-forms, and
induces from a representation, `half' of whose highest weight is shifted by $k$
(relative to the naive orbit correspondence). This presumably poses a problem
for any theory of quantizing constrained systems.
