Conditional Measures of Determinantal Point Processes

Given one-dimensional determinantal point processes induced by orthogonal projections with integrable kernels satisfying a certain growth condition, it is proved that their conditional measures with respect to the configuration in the complement of a compact interval are orthogonal polynomial ensembles with explicitly found weights. Examples include the sine-process and the process with Bessel kernel. The main role in the argument is played by the quasi-invariance, established in [2], of our point processes under the group of piecewise isometries of Double-struck capital R.