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

We characterize all distinguished varieties in the symmetrized polydisc Gn (n ≥ 2) and thus generalize the work [J. Funct. Anal., 266 (2014), 5779- 5800] on G2 by the author and Shalit. We show that a distinguished variety in Gn is a part of an algebraic curve, which is a set-theoretic complete intersection, and that Λ can be represented by the Taylor joint spectrum of n - 1 commuting scalar matrices satisfying certain conditions. An n-tuple of commuting Hilbert space operators .S1; : : : ;Sn-1;P) for which An D Gn is a spectral set is called a An-contraction. For every An-contraction .S1; : : : ;Sn-1;P) there is a unique operator tuple .F1; : : : ;Fn-1), called the FO-tuple of .S1; : : : ;Sn-1;P), satisfying Si _ S*n-iP D DP FiDP; i D 1; : : : ;n - 1: We produce a concrete functional model for the pure isometric-operator tuples associated with An, and by an application of that model we establish that the Ancontractions .S1; : : : ;Sn-1;P) and .S*1 ; : : : ;S*n-1; P*) admit normal @ΛΣ- dilations for a unique distinguished variety ΛΣ in Gn, when ΛΣ is determined by the FO-tuple of (S1; : : : ;Sn-1;P). Further, we show that the dilation of (S*1 ; : : : ;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 G2 and G3.