Commutant lifting and interpolation on quotients of bounded symmetric domains

Abstract

Let $Ω\subseteq \mathbb C^d$ be a bounded symmetric domain, $G$ a finite complex reflection group acting on $\mathbb C^d$, and $\boldsymbol θ:Ω\to \boldsymbol θ(Ω)$ the associated proper holomorphic map factored by $G.$ In this paper, we investigate commutant lifting and interpolation by Schur functions on the quotient domain $\boldsymbol θ(Ω).$ For a given quotient module of the Hardy space $H^2(\boldsymbolθ(Ω))$, we obtain equivalent criteria for a contractive module map to admit a Schur-class lift: one in terms of the contractivity of an associated functional on a subspace of $L^1(\partial\boldsymbolθ(Ω))$, and another in terms of a geometric distance formula in the same $L^1$-space. Specializing to quotient domains of the polydisc factored by imprimitive finite complex reflection groups, we obtain a commutant lifting criterion formulated in terms of inner functions. Finally, we apply these operator-theoretic results to finite-point Nevanlinna-Pick type interpolation problems on $\boldsymbol θ(Ω)$. Since the symmetrized bidisc and the tetrablock arise as quotient domains of suitable bounded symmetric domains, these criteria apply in particular to those domains.