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Action of the group of diffeomorphisms on determinantal measures

The principal result of this communication is the quasi-invariance of a wide class of determinantal point processes on the real line under the group of diffeomorphisms with compact support. We recall that the space of configurations Conf(R) is the space of purely atomic Radon measures endowed with the vague topology. In other words, a configuration X = {x} is a subset of R without accumulation points. By g we denote a bounded measurable function on R such that the function g − 1 has compact support. To a function g there corresponds the multiplicative functional Ψg(X) = ∏ x∈X g(x) on Conf(R). Let μ be a σ-finite atomless Borel measure on R. Let L ⊂ L2(R, μ) be a closed subspace such that the projection Π onto it has kernel (we denote the kernel by the same symbol Π) of a special form: there exist an open subset U ⊂ R of full measure (μ(R \ U) = 0) and smooth functions A and B on U such that