  An $N$ ${L} \choose {L/2}$-dimensional representation of the periodic
Temperley-Lieb algebra $TL_L(x)$ is presented. It is also a representation of
the cyclic group $Z_N$. We choose $x = 1$ and define a Hamiltonian as a sum of
the generators of the algebra acting in this representation. This Hamiltonian
gives the time evolution operator of a stochastic process. In the finite-size
scaling limit, the spectrum of the Hamiltonian contains representations of the
Virasoro algebra with complex highest weights. The $N = 3$ case is discussed in
detail. One discusses shortly the consequences of the existence of complex
Virasoro representations on the physical properties of the systems.
