On total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case

TL;DR

This paper investigates the restriction formulas of total Springer representations for symplectic Lie algebras in characteristic 2 and the exotic case, revealing their equivalence and implications for affine pavings of Springer fibers.

Contribution

It provides explicit restriction formulas for total Springer representations in characteristic 2 and the exotic case, and demonstrates their equivalence.

Findings

Restriction formulas for total Springer representations are established.

The formulas for the symplectic Lie algebra in characteristic 2 and the exotic case are shown to be equivalent.

Results have implications for the existence of affine pavings of Springer fibers.

Abstract

Let $W$ be the Weyl group of type $BC_n$. We first provide restriction formulas of the total Springer representations for the symplectic Lie algebra in characteristic 2 and the exotic case to the maximal parabolic subgroup of $W$ which is of type $BC_{n-1}$. Then we show that these two restriction formulas are equivalent, and discuss how the results can be used to examine the existence of affine pavings of Springer fibers corresponding to the symplectic Lie algebra in characteristic 2.