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Root data with group actions

Abstract

Suppose kk is a field, GG is a connected reductive algebraic kk-group, TT is a maximal kk-torus in GG, and Γ\Gamma is a finite group that acts on (G,T)(G,T). From the above, one obtains a root datum Ψ\Psi on which Gal(k)×Γ\text{Gal}(k)\times\Gamma acts. Provided that Γ\Gamma preserves a positive system in Ψ\Psi, not necessarily invariant under Gal(k)\text{Gal}(k), we construct an inverse to this process. That is, given a root datum on which Gal(k)×Γ\text{Gal}(k)\times\Gamma acts appropriately, we show how to construct a pair (G,T)(G,T), on which Γ\Gamma acts as above. Although the pair (G,T)(G,T) and the action of Γ\Gamma are canonical only up to an equivalence relation, we construct a particular pair for which GG is kk-quasisplit and Γ\Gamma fixes a Gal(k)\text{Gal}(k)-stable pinning of GG. Using these choices, we can define a notion of taking "Γ\Gamma-fixed points" at the level of equivalence classes, and this process is compatible with a general "restriction" process for root data with Γ\Gamma-action.Comment: v2: one word inserted, one citation inserted, one reference updated, one misspelling correcte

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    Last time updated on 10/08/2021