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 (Borel-Moore) homology Z := H_(E _V E), which we call Steinberg algebra.
(Graded) Projective and simple Z-modules are controlled by the BBD-decomposition
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 Borel-Moore homology we can use localization to torus-fixed points.
Examples of Steinberg algebras include group rings of Weyl groups, Khovanov-Lauda-Rouquier 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 quasi-hereditary algebra KQ KAn.
7. An-equioriented.
For the An-equioriented quiver we find a cell decompositions of the quiver flag varieties, which are parametrized by certain multi-tableaux.