Appendix: On parahoric subgroups

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Kloosterman sheaves for reductive groups (2010, methodology): “Under our assumption (X∗(Zη))π1(P1\{0,∞}) is torsion free and therefore as in [23] we have an exact sequence:”
The Satake isomorphism for special maximal parahoric Hecke algebras (2009, methodology): “set of affine roots �af(G(L), S) on AL are given by �af = {α + k | α ∈ �, k ∈ Z}, cf. [ HR ].”