Two-block Springer fibers of types C and D: a diagrammatic approach to Springer theory

Abstract

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component group actions admit a diagrammatic description in terms of cup diagrams which appear in the definition of arc algebras of types B and D. We determine the decomposition of the representations into irreducibles and relate our construction to classical Springer theory. As an application we obtain presentations of the cohomology rings of all two-block Springer fibers of types C and D. Moreover, we deduce explicit isomorphisms between the Kazhdan-Lusztig cell modules attached to the induced trivial module and the irreducible Specht modules in types C and D.

Schur–Weyl duality, Verma modules, and row quotients of Ariki–Koike algebras

“…A homeomorphic topological model has been built for these varieties [Kho04,Weh09] and the action of the symmetric group on the top degree cohomology has a skein theoretic interpretation [RuTy11,Rus11]. In types C and D, the geometry and topology of two-row Springer fibers have been studied in [EhSt16a,Wil18,StWi19,ILW22]. As in type A, the irreducible components and their intersections are smooth and they admit explicit descriptions in terms of cup diagrams.…”