Commutant lifting and interpolation on quotients of bounded symmetric domains

Abstract

Let $\Omega \subseteq {\mathbb{C}}^d$ be a bounded symmetric domain, $G$ a finite complex reflection group acting on ${\mathbb{C}}^d$, and $\mathbf{\theta }:\Omega \to \mathbf{\theta }(\Omega )$ the associated proper holomorphic map factored by $G.$ In this paper, we investigate commutant lifting and interpolation by Schur functions on the quotient domain $\mathbf{\theta }(\Omega ).$ For a given quotient module of the Hardy space $H^2(\mathbf{\theta }(\Omega ))$, 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 $L^1(\partial \mathbf{\theta }(\Omega ))$, 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 $\mathbf{\theta }(\Omega )$. Since the symmetrized bidisc and the tetrablock arise as quotient domains of suitable bounded symmetric domains, these criteria apply in particular to those domains.

This is a preliminary version; comments and feedback are welcome