  We investigate the properties of the XY pyrochlore antiferromagnet with local
<111> planar anisotropy. We find the ground states and show that the
configurational ground state entropy is subextensive. By computing the free
energy due to harmonic fluctuations and by carrying out Monte Carlo
simulations, we confirm earlier work indicating that the model exhibits thermal
order-by-disorder leading to low temperature long-range order consisting of
discrete magnetic domains. We compute the spin wave spectrum and show that
thermal and quantum fluctuations select the same magnetic structure. Using
Monte Carlo simulations, we find that the state selected by thermal
fluctuations in this XY pyrochlore antiferromagnet can survive the addition of
sufficiently weak nearest-neighbor pseudo-dipolar interactions to the spin
Hamiltonian. We discuss our results in relation to the Er2Ti2O7 pyrochlore
antiferromagnet.
