Ergodic decomposition of infinite unitarily-invariant measures on spaces of infinite complex matrices

The Pickrell family of meaures is a natural analogue, in infinite dimension, of the canonical invariant measure on the Grassmann manifold. By definition, the Pickrell measures are unitarily-invariant, and the question of their ergodic decomposition is closely related to the problem of harmonic analysis on the infinite unitary group. The main result of the talk is an explicit description of the ergodic decomposition for infinite Pickrell measures on spaces of infinite complex matrices. The main construction is that of sigma-finite analogues of determinantal measures on spaces of configurations. An example is the infinite Bessel point process, the scaling limit of sigma-finite analogues of Jacobi orthogonal polynomial ensembles. The main result states that the infinite Bessel point process (subject to an appropriate change of variables) is precisely the ergodic decomposition measure for infinite Pickrell measures.

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