On the $\omega$-circular $\ast$-distribution for complex unimodular $\omega$

We consider a notion of $\omega$-circular $\ast$-distribution ${\mu }_{\omega }$, defined in terms of a parameter $\omega \in \mathbb{C}$ with $\mathrm{\mid }\omega \mathrm{\mid }=1$. This $\ast$-distribution appears as limit law in an algebraic central limit theorem (CLT) for $\omega$-commuting unitaries, and interpolates between the Fermionic setting (at $\omega =-1$) and the Bosonic one (at $\omega =1$). On a combinatorial level, ${\mu }_{\omega }$ is described in terms of pair-partitions of sets $\{1,\mathrm{.}\mathrm{.}\mathrm{.},2n\}$, where one has a total order among the $n$ pairs of the partition and where every pair is assigned a direction, to the effect that one can define the sign of a crossing between two pairs. The main goal of the paper is to study a significant subspace, with “Fock space” flavour, of the $L^2$-space associated to a natural multi-variate version of ${\mu }_{\omega }$. The inner product on this subspace is defined by a sesquilinear form $⟨\cdot ,\cdot {⟩}_{\omega }$ on non-commutative polynomials. We explicitly describe $⟨\cdot ,\cdot {⟩}_{\omega }$ in terms of some tractable non-negative definite functions on symmetric groups, and we prove that it is non-degenerate if and only if $\omega$ is not a root of unity.