Real McKay Correspondence: KR-Theory of Graded Kleinian Groups

This project considers the finite symmetry subgroups of the orthogonal group $\mathrm{O}(3)\subset \mathrm{G}\mathrm{L}(3,\mathbb{R})$ and the index $2$ containments $G⊲\widehat{G}$. The special orthogonal group $\mathrm{S}\mathrm{O}(3)\subset \mathrm{S}\mathrm{L}(3,\mathbb{R})$ admits a double cover from the spinor group $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(3)\cong \mathrm{S}\mathrm{U}(2)\subset \mathrm{S}\mathrm{L}(2,\mathbb{C})$, and lifting our subgroups up preserves the network of containments. Those subgroups not contained in $\mathrm{S}\mathrm{O}(3)\subset \mathrm{O}(3)$ are lifted to the pinor groups ${\mathrm{P}\mathrm{i}\mathrm{n}}_{\pm }(3)$ of which there are two choices. For the index $2$ containments $G⊲\widehat{G}$, we calculate the Real and complex Frobenius-Schur indicators, and apply Dyson’s classification of antilinear block structures to produce decorated McKay graphs for each case. We then explore $KR$-theory as introduced by Atiyah in 1966, which is a variant of topological $K$-theory for working with topological spaces equipped with an involution. The GIT quotient spaces ${\mathbb{C}}^2\mathrm{/}\mathrm{/}G$, can be equipped by an involution via the action of $\widehat{G}\mathrm{/}G$. In 1983, Gonzalez-Sprinberg and Verdier showed how one can view the McKay correspondence as an isomorphism between the $G$-equivariant $K$-theory $K_G({\mathbb{C}}^2)$ and the $K$-theory of the minimal resolution of the singularity $\widetilde{{\mathbb{C}}^2\mathrm{/}\mathrm{/}G}$. We use this to conjecture an analogous a form of the McKay correspondence for $C_2$-graded groups and $KR$-theory.