  We extend the problem of finding Hamiltonian-invariant volume forms on a
Poisson manifold to the problem of construction of Hamiltonian-invariant
generalized functions. For this we introduce the notion of generalized center
of a Poisson algebra, which is the space of generalized Casimir functions. We
study as the case when the set of test-objects for generalized functions is the
space of compactly supported smooth functions, so the case when the
test-objects are n-forms, where n is the dimension of the Poisson manifold. We
describe the relations of this problem with the homological properties of the
Poisson structure, with Bott connection for the corresponding symplectic
foliation and the modular class.
