Torus actions, localization and induced representations on cohomology

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.