| Popis: |
Suppose k is a field, G is a connected reductive algebraic k-group, T is a maximal k-torus in G, and \(\Gamma \) is a finite group that acts on (G, T). From the above, one obtains a root datum \(\Psi \) on which \({{\,\mathrm{Gal}\,}}(k)\times \Gamma \) acts. Provided that \(\Gamma \) preserves a positive system in \(\Psi \), not necessarily invariant under \({{\,\mathrm{Gal}\,}}(k)\), we construct an inverse to this process. That is, given a root datum on which \({{\,\mathrm{Gal}\,}}(k)\times \Gamma \) acts appropriately, we show how to construct a pair (G, T), on which \(\Gamma \) acts as above. Although the pair (G, T) and the action of \(\Gamma \) are canonical only up to an equivalence relation, we construct a particular pair for which G is k-quasisplit and \(\Gamma \) fixes a \({{\,\mathrm{Gal}\,}}(k)\)-stable pinning of G. Using these choices, we can define a notion of taking “\(\Gamma \)-fixed points” at the level of equivalence classes, and this process is compatible with a general “restriction” process for root data with \(\Gamma \)-action. |
