  Convolution in Borel-Moore homology plays an important role in Nakajima's
construction of representations of the Heisenberg algebra and of modified
enveloping algebras of Kac-Moody algebras. In its most basic form, convolution
between two quiver varieties is given by pullback and then pushforward via the
Hecke correspondence for quivers.
  In previous work we showed that the Hecke correspondence has a
Morse-theoretic interpretation in terms of spaces of flow lines. The goal of
this paper is to show that the topological information that defines generators
for Nakajima's representations can be encoded in the cup product for a Morse
complex defined on the smooth space of representations of a quiver without
relations, and then pulling back to the subvariety of representations that do
satisfy a given set of relations. The results are valid for the main motivating
example of Nakajima quivers, as well as other quivers with relations derived
from these (for example handsaw quivers).
  For the norm square of a moment map on the space of representations of a
quiver, the usual Morse-Bott-Smale transversality condition on the space of
flow lines fails, however a weaker version of transversality is still
satisfied. A major part of the paper is spent developing a general theory in
this setting of weak transversality from which one can recover the usual
construction of the differentials and cup product on the Morse complex by
adding an intermediate step of taking cup product with a certain Euler class,
which is explicitly computable for the space of representations of a quiver.
