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Let N be the normalizer of the diagonal torus $T_{1} ≅ G_{m}$ in $S L_{2}$ . We prove localization theorems for $S L_{2}^{n}$ and $N^{n}$ for equivariant cohomology with coefficients in the (twisted) Witt sheaf, along the lines of the classical localization theorems for equivariant cohomology for a torus action. We also have an analog of the Bott residue formula for $S L_{2}^{n}$ and N. In the case of an $S L_{2}^{n}$ -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, which is always available if the subscheme of $T_{1}$ fixed points has dimension zero.

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9 - Atiyah–Bott Localization in Equivariant Witt Cohomology

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9 - Atiyah–Bott Localization in Equivariant Witt Cohomology

Summary