The two-parameter free unitary Segal–Bargmann transform and its Biane–Gross–Malliavin identification

Motivated by a conditional expectation interpretation of the Segal–Bargmann transform, we derive the integral kernel for the large-N   limit of the two-parameter Segal–Bargmann–Hall transform over the unitary group U(N)U(N), and explore its limiting behavior. We also extend the notion of circular systems to more general elliptic systems, in order to give an alternate construction of our new two-parameter free unitary Segal–Bargmann–Hall transform via a Biane–Gross–Malliavin type theorem.