Notes on the geometric Satake equivalence
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...It can be thought of as an analogue of hyperbolic localization [Br] for μl-actions....
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...ltiplicativegroupoverF. HereamaximaltorusofareductivegroupHisaclosedsubgroup whichisatorusandwhosebasechangetoanalgebraicclosureF ofF isamaximaltorusofSpec(F) Spec(F)H inthe“traditional” sense,seee.g.[Hu]. 11If His a split reductive group and Kˆ is a maximal torus, then the root datum of with respect to K is the quadruple (X (K F );X (K F );( H F ;K F ); _(H F ;K F )) where F is an algebraic closure o...
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...e k by G=H] k . Since both of these group schemesaresmooth,by[Mi,PropositionVII.10.1]thisimpliesthatGe k isalsoasmoothgroup, i.e.that(14.3)isanextensionofk-algebraicgroupsinthe“traditional”senseofe.g.[Hu]. The unipotentradicalofGe k hastrivialimageinthetorus He k, henceisincludedinG=H] k ; since the latter group is semisimple it follows that this unipotent radical is trivial, i.e. that Ge k is reducti...
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