Convolution algebras from quiver desingularizations

Markus Reineke (BUGH Wuppertal)

The aim of the talk is to provide 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. This isomorphism identifies classes of irreducible representations with the dual canonical basis.

The finite dimensional algebras appearing in this realization are defined as convolution algebras (in the sense of [CG]), arising from desingularizations of orbit closures of quiver representations introduced in [Re].

The proofs use the methods of [CG] and the perverse sheaf realization of canonical bases of [Lu]. Examples of these constructions will be discussed.

References

[CG] N. Chriss, V. Ginzburg: Representation Theory and Complex Geometry. Birkhäuser, Boston, 1997.

[Lu] G. Lusztig: Introduction to Quantum Groups. Progress in Mathematics 110, Birkhäuser, Boston, 1993.

[Re] M. Reineke: Quivers, desingularizations and canonical bases. To appear in: Studies in memory of Issai Schur, Birhäuser. Preprint math.AG/0104284.