The Poisson Lie Algebra, Rumin’s Complex and Base Change

Results from the forthcoming papers [4] and [8] are announced. We introduce a singular current construction, or base change, for pseudoalgebras which may be used to obtain a primitive Lie pseudoalgebra of type H from a suitable one of type K. When applied to representations, it derives the pseudo de Rham complex of type H from that of type K—which is related to Rumin’s construction from [15]—both with standard coefficients and with nontrivial Galois coefficients. In the latter case, the construction yields exact complexes of modules for the Poisson linearly compact Lie algebra \(P_{2N}\) exhibiting a nontrivial central action.