**A Nagy-Foias program for the C. _{0} operator tuples associated with the symmetrized polydisc**

A commuting tuple of operators (S_{1},…, S_{n-1},P), defined on a Hilbert space H, for which the closed symmetrized polydisc

Γ_{n} ={ (∑_{1≤ i ≤ n} z_{i},∑_{1≤i<j ≤ n}z_{i}z_{j},…, ∏_{i=1}^{n} z_{i} ): |z_{i}|≤1, i=1,…,n }

is a spectral set, is called a Γ_{n}-contraction. A Γ_{n}-contraction (S_{1},…, S_{n-1},P) is said to be C._{0} or *pure*,

if P*^{n} → 0 strongly as n\rightarrow \infty. We show an explicit construction of a Nagy-Foias type dilation and an operator model for the C._{0} Γ_{n}-contractions. Also we describe a complete unitary invariant for such operator tuples. This is an analogue of the Nagy-Foias complete unitary invariant for a contraction in terms of characteristic function.