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We find characterization for the distinguished varieties in the symmetrizedpolydisc $\mathbb G_n \; (n\geq 2)$ and thus generalize the work [\textit{J.Funct. Anal.}, 266 (2014), 5779 -- 5800] on $\mathbb G_2$ by the author andShalit. We show that a distinguished variety $\Lambda$ in $\mathbb G_n$ is apart of an algebraic curve, which is a set-theoretic complete intersection, andthat $\Lambda$ can be represented by the Taylor joint spectrum of $n-1$commuting scalar matrices satisfying certain conditions. An $n$-tuple ofcommuting Hilbert space operators $(S_1, \dots ,S_{n-1},P)$ for which$\Gamma_n=\overline{\mathbb G_n}$ is a spectral set is called a$\Gamma_n$-contraction. To every $\Gamma_n$-contraction $(S_1, \dots,S_{n-1},P)$ there is a unique operator tuple $(F_1, \dots , F_{n-1})$, calledthe $\mathcal F_O$-tuple of $(S_1, \dots ,S_{n-1},P)$, satisfying \[S_i-S_{n-i}^*P=D_PF_iD_P \,,\quad i=1, \dots ,n-1. \] We produce concretefunctional model for the pure isometric-operator tuples associated with$\Gamma_n$ and by an application of that model we establish that the$\Gamma_n$-contractions $(S_1, \dots ,S_{n-1},P)$ and $(S_1^*, \dots ,S_{n-1}^*,P^*)$ admit normal $\partial \overline{ \Lambda}_{\Sigma}-$dilationsfor a unique distinguished variety $\Lambda_{\Sigma}$ in $\mathbb G_n$, when$\Lambda_{\Sigma}$ is determined by the $\mathcal F_O$-tuple of $(S_1, \dots,S_{n-1}, P)$. Further, we show that the dilation of $(S_1^*, \dots,S_{n-1}^*,P^*)$ is minimal and acts on the minimal unitary dilation space of$P^*$. Also, we show interplay between the distinguished varieties in $\mathbbG_2$ and $\mathbb G_{3}$.

Distinguished varieties in a family of domains associated with spectral interpolation and operator theory