  Using sheaves of A^1-connected components, we prove that the Morel-Voevodsky
singular construction on a reductive algebraic group fails to be A^1-local if
the group does not satisfy suitable isotropy hypotheses. As a consequence, we
show the failure of A^1-invariance of torsors for such groups on smooth affine
schemes over infinite perfect fields. We also characterize A^1-connected
reductive algebraic groups over a field of characteristic 0.
