  The spin and the chirality orderings of the Heisenberg spin glass in two
dimensions with the nearest-neighbor Gaussian coupling are investigated by
equilibrium Monte Carlo simulations. Particular attention is paid to the
behavior of the spin and the chirality correlation lengths. In order to observe
the true asymptotic behavior, fairly large system size L\gsim 20 (L the linear
dimension of the system) appears to be necessary. It is found that both the
spin and the chirality order only at zero temperature. At high temperatures,
the chiral correlation length stays shorter than spin correlation length,
whereas at lower temperatures below the crossover temperature T_\times, the
chiral correlation length exceeds the spin correlation length. The spin and the
chirality correlation-length exponents are estimated above T_\times to be
\nu_SG=0.9+-0.2 and \nu_CG=2.1+-0.3, respectively. These values are close to
the previous estimates on the basis of the domain-wall-energy calculation.
Discussion is given about the asymptotic critical behavior realized below
T_\times.
