Crystal graphs and semicanonical functions for symmetrizable Cartan matrices

In joint work with B. Leclerc and J. Schröer we propose a 1-Gorenstein
algebra H, defined over an arbitrary field K, associated to the datum of a
symmetrizable Cartan Matrix C, a symmetrizer D of C and an orientation
$\Omega$. The H-modules of finite projective dimension behave
in many aspects like the modules over a hereditary algebra, and we can
associate to H a kind of preprojective algebra $\Pi$. If we look, for K
algebraically closed,  at the varieties of representations of $\Pi$ which
admit a filtration by generalized simples, we find that the components of
maximal dimension provide a realization of the crystal $B(-\infty)$
corresponding to C. For K beeing the complex numbers we can construct,
following ideas of Lusztig, an algebra of constructible functions which
contains a family of "semicanonical functions". Those are naturally
parametrized by the above mentioned components of maximal dimension.
Modulo a conjecture about the support of the functions in the "Serre ideal"
the semicanonical functions  yield a basis of the enveloping algebra U(n) of
the positive part of the Kac-Moody Lie algebra g(C).