  The Grothendieck groups of the categories of finitely generated modules and
finitely generated projective modules over a tower of algebras can be endowed
with (co)algebra structures that, in many cases of interest, give rise to a
dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one
can construct an algebra called the Heisenberg double, which is a
generalization of the classical Heisenberg algebra. The aim of this paper is to
study Heisenberg doubles arising from towers of algebras in this manner. First,
we develop the basic representation theory of such Heisenberg doubles and show
that if induction and restriction satisfy Mackey-like isomorphisms then the
Fock space representation of the Heisenberg double has a natural
categorification. This unifies the existing categorifications of the polynomial
representation of the Weyl algebra and the Fock space representation of the
Heisenberg algebra. Second, we develop in detail the theory applied to the
tower of 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call
quasi-Heisenberg algebras. As an application of a generalized Stone--von
Neumann Theorem, we give a new proof of the fact that the ring of
quasisymmetric functions is free over the ring of symmetric functions.
