Dilation, functional model and a complete unitary invariant for $C{\mathrm{.}}_0{\mathrm{\Gamma }}_n$-contractions

A commuting tuple of operators $(S_1,\dots ,S_{n-1},P)$, defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc $${\mathrm{\Gamma }}_n=\{(\sum _{1\le i\le n}{z}_i,\sum _{1\le i<j\le n}{z}_i{z}_j,\dots ,\prod _{i=1}^n{z}_i):\mathrm{\mid }{z}_i\mathrm{\mid }\le 1,i=1,\dots ,n\}$$ is a spectral set, is called a ${\mathrm{\Gamma }}_n$-$\mathit{\text{contraction}}$. A ${\mathrm{\Gamma }}_n$-contraction is said to be $\mathit{\text{pure}}$ or $C{\mathrm{.}}_0$ if $P$ is $C{\mathrm{.}}_0$, that is, if ${P^{\ast }}^n\to 0$ strongly as $n\to \mathrm{\infty }$. We show that for any ${\mathrm{\Gamma }}_n$-contraction $(S_1,\dots ,S_{n-1},P)$, there is a unique operator tuple $(A_1,\dots , A_{n-1})$ that satisfies the operator identities $$S_i-S_{n-i}^{\ast }P=D_P{A}_i{D}_P,i=1,\dots ,n-1.$$ This unique tuple is called the $\mathit{\text{fundamental}}\mathit{\text{operator}}\mathit{\text{tuple}}$ or ${\mathcal{F}}_O$-tuple of $(S_1,\dots ,S_{n-1},P)$. With the help of the ${\mathcal{F}}_O$-tuple, we construct an operator model for a $C.0 ; \Gamma_n$-contractionandshowthatthereexist$n-1$operators$C_1,\dots, C{n-1}$suchthateach$S_i$canberepresentedas$S_i=C_i+PC_{n-i}^$. We find an explicit minimal dilation for a class of $C._0 ; \Gamma_n$-contractionswhose$\mathcal F_O$-tuples satisfy a certain condition. Also we establish that the ${\mathcal{F}}_O$-tuple of $(S_1^,\dots, S_{n-1}^,P^)$togetherwiththecharacteristicfunctionof$P$ constitute a complete unitary invariant for the $C{\mathrm{.}}_0$ ${\mathrm{\Gamma }}_n$-contractions. The entire program is an analogue of the Nagy-Foias theory for $C{\mathrm{.}}_0$ contractions.