On the Betti numbers of Springer fibers for classical types

TL;DR

This paper derives recursive formulas for Betti numbers of Springer fibers in classical types by analyzing graded Springer representations and their restrictions to maximal parabolic subgroups.

Contribution

It introduces a new formula for restricting Springer representations to parabolic subgroups, leading to recursive calculations of Betti numbers for classical types.

Findings

Recursive formulas for Betti numbers of Springer fibers.

Explicit restriction formulas for Springer representations.

Application to classical types of Weyl groups.

Abstract

For a Weyl group $W$ of classical type, we present a formula to calculate the restriction of (graded) Springer representations of $W$ to a maximal parabolic subgroup $W'$ where the types of $W$ and $W'$ are in the same series. As a result, we obtain recursive formulas for the Betti numbers of Springer fibers for classical types.