Atiyah-Bott localization in equivariant Witt cohomology

Abstract

Let $N$ be a normalizer of the diagonal torus $T_1\cong \mathbb{G}_m$ in $\text{SL}_2$. We prove localization theorems for $\text{SL}_2^n$ and $N^n$ for equivariant cohomology with coefficients in the (twisted) Witt sheaf, along the lines of the classical Atiyah-Bott localization theorems for equivariant cohomology for a torus action. We also have an analog of the Bott residue formula for $\text{SL}_2^n$ and $N$. In the case of an $\text{SL}_2^n$-action, there is a rather serious restriction on the orbit type. For an $N$-action, there is no restriction for the localization result, but for the Bott residue theorem, one requires a certain type of decomposition of the fixed points for the $T_1$-action.