  Let (H_t) be the Ornstein-Uhlenbeck semigroup on R^d with covariance matrix I
and drift matrix \lambda(R-I), where \lambda>0 and R is a skew-adjoint matrix
and denote by \gamma_\infty the invariant measure for (H_t). Semigroups of this
form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are
normal on L^2(\gamma_\infty). We prove that if the matrix R generates a
one-parameter group of periodic rotations then the maximal operator associated
to the semigroup is of weak type 1 with respect to the invariant measure. We
also prove that the maximal operator associated to an arbitrary normal
Ornstein-Uhlenbeck semigroup is bounded on L^p(\gamma_\infty) if and only if
1<p\le \infty.
