Let ξ(t), t ∈ [0, 1], be an α-stable Lévy process in ℝd. Denote by Pξ the measure generated by ξ in the Skorokhod space double-struck D([0, 1], ℝd. Under some conditions on the spectral measure of the process ξ, we construct a group of Pξ-preserving transformations of double-struck D([0, 1], ℝd). Bibliography: 12 titles. © 2008 Springer Science+Business Media, Inc.