| Abstract: A realization of the coordinate ring
of the upper unipotent part of a complex semisimple group of simply laced
type in terms of Grothendieck groups of certain (geometrically defined)
finite dimensional algebras is provided. This isomorphism identifies classes
of irreducible representations with the dual canonical basis. The finite
dimensional algebras appearing in this realization are defined as certain
convolution algebras (in the sense of V. Ginzburg's theory), arising from
desingularizations of orbit closures of quiver representations. The proofs
use the methods of V. Ginzburg and the perverse sheaf realization of canonical
bases of G. Lusztig. |
