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Root Data with Group Actions

  • Jeffrey D. AdlerEmail author
  • Joshua M. Lansky
Chapter
Part of the Progress in Mathematics book series (PM, volume 328)

Abstract

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 (GT). 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 (GT), on which \(\Gamma \) acts as above. Although the pair (GT) 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.

Keywords

Reductive algebraic group Root datum Quasi-semisimple automorphisms 

2010 Mathematics Subject Classification

Primary 20G15 20G40 Secondary 20C33. 

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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAmerican UniversityWashington, DCUSA

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