Torus actions, localization and induced representations on cohomology

TL;DR

This paper investigates the action of Weyl groups on the cohomology of Springer varieties, providing new insights into the representation theory via torus actions, moment graphs, and fixed point sets, with applications to type A Springer varieties.

Contribution

It offers a criterion for lifting group actions to cohomology, simplifies proofs of Springer representations, and explores the structure of moment graphs and their induced group actions.

Findings

Weyl group actions on cohomology are characterized via fixed point sets.

A simple proof of the Alvis-Lusztig-Treumann Theorem in type A is provided.

The structure of moment graphs under group actions is described, revealing trivial and induced representations.

Abstract

This note is motivated by the problem of understanding Springer's remarkable action of the Weyl group $W=N_G(T)/T$ of a semi-simple complex linear algebraic group $G$, with maximal torus $T$, on the cohomology algebra of an arbitrary Springer variety in the flag variety of $G$ from the viewpoint of torus actions. Continuing the work [CK] which gave a sufficient condition for a group $\mathcal{W}$ acting on the fixed point set of an algebraic torus action $(S,X)$ on a complex projective variety $X$ to lift to a representation of $\mathcal{W}$ on the cohomology algebra $H^*(X)$ (over $\mathbb{C}$), we describe when the representation on $H^*(X)$ is equivalent to the representation of $\mathcal{W}$ on the cohomology $H^*(X^S)$ of the fixed point set. As a consequence of this theorem, we give a simple proof in type $A$ of the Alvis-Lusztig-Treumann Theorem, which describes Springer's representation of $W$ for Springer varieties corresponding to nilpotents in a Levi subalgebra of Lie$(G)$. In the final two sections, we describe the local structure of the moment graph $\mathfrak{M}(X)$ of a special torus action $(S,X)$, and we also show that if a finite group $\mathcal{W}$ acts on the moment graph of $X$, then $\mathcal{W}$ induces pair of actions on $H^*(X)$, namely the left and right or dot and star actions of Knutson [Knu] and Tymoczko [Tym] respectively. In particular, $W$ acts on the moment (or Bruhat) graph $\mathfrak{M}(G/P)$ of $(T,G/P)$ for any parabolic $P$ in $G$ containing $T$, and the right action of $W$ on $H^*(G/P)$ is an induced representation. Furthermore, we show the left action of $W$ on $H^*(G/P)$ is trivial.