Title:

Springer theory and the geometry of quiver flag varieties

1. Springer theory. For any projective map E ! V , Chriss and Ginzburg defined an algebra structure on the (BorelMoore) homology Z := H_(E _V E), which we call Steinberg algebra. (Graded) Projective and simple Zmodules are controlled by the BBDdecomposition associated to E ! V . We restrict to collapsings of unions of homogeneous vector bundles over homogeneous spaces because we have the cellular fibration technique and for equivariant BorelMoore homology we can use localization to torusfixed points. Examples of Steinberg algebras include group rings of Weyl groups, KhovanovLaudaRouquier algebras, nil Hecke algebras. 2. Steinberg algebras. We choose a class of Steinberg algebras and give generators and relations for them. This fails if the homogeneous spaces are partial and not complete flag varieties, we call this the parabolic case. 3. The parabolic case. In the parabolic cases, we realize the Steinberg algebra ZP as corner algebra in a Steinberg algebra ZB associated to Borel groups (this means ZP = eZBe for an idempotent element e 2 ZB). 4. Monoidal categories. We explain how to construct monoidal categories from families of collapsings of homogeneous bundles. 5. Construct collapsings. We construct collapsing maps over given loci which are resolutions of singularities or generic Galois coverings. For closures of homogeneous decomposition classes of the Kronecker quiver these maps are new. 6. Quiver flag varieties. Quiver flag varieties are the fibres of certain collapsings of homogeneous bundles. We investigate when quiver flag varieties have only finitely many orbits and we describe the category of flags of quiver representations as a _filtered subcategory for the quasihereditary algebra KQ KAn. 7. Anequioriented. For the Anequioriented quiver we find a cell decompositions of the quiver flag varieties, which are parametrized by certain multitableaux.
