  The Yukawa-Higgs/Ginsparg-Wilson-fermion construction of chiral lattice gauge
theories described in hep-lat/0605003 uses exact lattice chirality to decouple
the massless chiral fermions from a mirror sector, whose strong dynamics is
conjectured to give cutoff-scale mass to the mirror fermions without breaking
the chiral gauge symmetry. In this paper, we study the mirror sector dynamics
of a two-dimensional chiral gauge theory in the limitof strong Yukawa and
vanishing gauge couplings, in which case it reduces to an XY model coupled to
Ginsparg-Wilson fermions. For the mirror fermions to acquire cutoff-scale mass
it is believed to be important that the XY model remain in its "high
temperature" phase, where there is no algebraic ordering--a conjecture
supported by the results of our work. We use analytic and Monte-Carlo methods
with dynamical fermions to study the scalar and fermion susceptibilities, and
the mirror fermion spectrum. Our results provide convincing evidence that the
strong dynamics does not "break" the chiral symmetry (more precisely, that the
mirror fermions do not induce algebraic ordering in two-dimensions), and that
the mirror fermions decouple from the infrared physics.
