Christof Geiß (UNAM, Mexico City):
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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 Ω. 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 Π. If we look, for K
algebraically closed, at the varieties of representations of Π which
admit a filtration by generalized simples, we find that the components of
maximal dimension provide a realization of the crystal B(-∞) corresponding to C.
For K being the complex numbers we can construct,
following ideas of Lusztig, an algebra of constructible functions which
contains a family of "semicanonical functions", which are naturally parametrized by
the above mentioned components of maximal dimensions.
Modulo a conjecture about the support of the functions in the "Serre ideal"
those functions yield a semicanonical basis of the enveloping
algebra U(n) of the positive part of the Kac-Moody Lie algebra g(C).