  We study the antiferromagnetic {\it XY} model on a triangular lattice by
extensive Monte Carlo simulations, focusing on its ordering and critical
properties. Our result clearly shows that two separate transitions occur at two
distinct temperatures, the one at a higher temperature is associated with a
$Z_2$-symmetry breaking driven by the chirality, and the one at a lower
temperature is associated with the onset of the quasi-long-range order of the
{\it XY} spin. We carefully examine the critical properties of each transition
to find that the criticality of the chiral transition is consistent with the
standard two-dimensional Ising universality class, whereas that of the spin
transition might differ from the conventional Kosterlitz-Thouless (KT) one. The
observed non-KT nature of the spin criticality is consistent with the most
recent simulation result on the fully-frustrated {\it XY} model on a square
lattice.
