  In this article we define a generalization of Lusztig Lagrangian varieties in
the case of arbitrary quivers, possibly carrying loops. As opposed to the
Lagrangian varieties constructed by Lusztig, which consisted in nilpotent
representations, we have to consider here slightly more general
representations. That this is necessary is already clear from the Jordan quiver
case. Our proof of the Lagrangian character is based on induction, but with non
trivial first steps, consisting in the study of quivers with one vertex but
possible loops. From our proof emerges a new combinatorial structure on the set
of irreducible components, which is more general than the usual crystals, in
that there are now more operators associated to a vertex with loops. We finally
consider a convolution algebra of constructible functions on our varieties, and
construct a family of constructible functions naturally attached to the
irreducible components.
