  The main result of this paper is that determinantal point processes on the
real line corresponding to projection operators with integrable kernels are
quasi-invariant, in the continuous case, under the group of diffeomorphisms
with compact support (Theorem 1.4); in the discrete case, under the group of
all finite permutations of the phase space (Theorem 1.6). The Radon-Nikodym
derivative is computed explicitly and is given by a regularized multiplicative
functional. Theorem 1.4 applies, in particular, to the sine-process, as well as
to determinantal point processes with the Bessel and the Airy kernels; Theorem
1.6 to the discrete sine-process and the Gamma kernel process. The paper
answers a question of Grigori Olshanski.
