A character of $U(\infty)$ is a continuous, positive definite class function $\chi: U(\infty) \longrightarrow \mathbb{C}$, which is normalized by $\chi(e) = 1$. Observe that the set of characters of $U(\infty)$ is a convex set, so we can talk about its extreme points, called the extreme characters of $U(\infty)$.
It is a classical theorem of Edrei and Voiculescu that the extreme characters of $U(\infty)$ are parametrized by points $\omega = (\alpha^+, \alpha^-, \beta^+, \beta^-, \gamma^+, \gamma^-)$ in an infinite-dimensional simplex $\Omega \subset \mathbb{R}_+^{\infty}\times\mathbb{R}_+^{\infty}\times\mathbb{R}_+^{\infty}\times\mathbb{R}_+^{\infty}\times\mathbb{R}_+\times\mathbb{R}_+$ and they are given explicitly as follows. Let $\chi_{\omega}$ be the character associated to the point $\omega\in\Omega$. Since $\chi_{\omega}$ is a class function, it is determined completely by its restriction to the diagonal matrices in $U(\infty)$, that is, the subset $\mathbb{T}^{\infty} \subset U(\infty)$, where $\mathbb{T} = \{z \in \mathbb{C} : |z| = 1\}$. Then one has that $\chi_{\omega}(u_1, u_2, \ldots)$ has the form $$\prod_{i=1}^{\infty}{\phi_{\omega}(u_i)},$$ where $\phi_{\omega}(u_i)$ is given explicitly by an infinite product. It is, at least initially, surprising that the extreme characters are multiplicative. Roughly, this is explained (in some papers of Grigori Olshanski) by the fact that extreme characters $\chi_{\omega}$ are limits of "extreme normalized characters" of $U(n)$, and the functional equation for characters of $U(n)$ becomes the multiplicativity relation for extreme characters of $U(\infty)$.
In http://arxiv.org/abs/q-alg/9709011, a work of Okounkov and Olshanski, Theorem 1.3. proves a generalization of Edrei-Voiculescu's theorem that characterizes the extreme points of a subset of functions $C(\mathbb{T}^{\infty}) \longrightarrow \mathbb{C}$ that depends on a parameter $\theta > 0$. When $\theta = 1$, the result is Edrei-Voiculescu's theorem. The extreme points have also a multiplicative structure, i.e., they look like $\Psi(u_1, u_2, \ldots) = \prod_{i=1}^{\infty}{\phi_{\omega}^{\theta}(u_i)}$. The multiplicativity can be "conceptually explained" (in the same way as above) in the cases $\theta = 1/2, 2$, where some representation theory is available.
My question: is there any explanation for the multiplicativity of the "$\theta$-characters" in Theorem 1.3. of the paper above, for general $\theta$ (or at least $\theta\in\mathbb{N}$), that is independent of the proof of the Theorem in that paper?
In that paper, the proof finds all extreme characters and it happens that they have that multiplicative structure. I'd like to know if there are more conceptual explanations, either coming from representation theory or elsewhere.