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

TL;DR

This paper presents a topological and diagrammatic approach to Springer representations for two-block Springer fibers of types C and D, connecting geometric, algebraic, and combinatorial perspectives.

Contribution

It introduces a new elementary topological construction and diagrammatic description of Springer representations for types C and D with two Jordan blocks, linking to classical theory.

Findings

Decomposition of representations into irreducibles determined.

Explicit presentations of cohomology rings obtained.

Isomorphisms between Kazhdan-Lusztig modules and Specht modules established.

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.