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The main result of this note is that the shift of the parameter by 1 in theparameter space of decomposing measures in the problem of harmonic analysis onthe infinite-dimensional unitary group corresponds to the taking of the reducedPalm measure at infinity for our decomposing measures. The proof proceeds byfinite-dimensional approximation of our measures by orthogonal polynomialensembles. The key remark is that the taking the reduced Palm measure commuteswith the scaling limit transition from finite to infinite particle systems.

A Palm hierarchy for determinantal point processes with the confluent  hypergeometric kernel, the decomposing measures in the problem of harmonic  analysis on the infinite-dimensional unitary group