Hua-pickrell Measures on General Compact Groups

Take a generic subgroup G, endowed with its Haar measure, from U(n,K), the unitary group of dimension n over the field K of real, complex or quaternion numbers. We give some equalities in law for Z := det(Id − G), G ∈ G : under some general conditions, Z can be decomposed as a product of independent random variables, whose laws are explicitly known (Section 2). Consequently G, endowed with a generalization of its Haar measure (the Hua-Pickrell measure), can be generated as a product of independent reflections. This constitutes a generalization of the well known Ewens sampling formula, corresponding to G = Sn, the n-dimensional symmetric group (Section 3). Finally, explicit determinantal point processes can be associated to the spectrum induced by the Hua-Pickrell measures, implying asymptotics on correlation functions (Section 4).