Theorem Quasi-Split and Anisotropic Groups via the Indexed Root Datum
theorem admitted

Quasi-Split and Anisotropic Groups via the Indexed Root Datum

Let \(G\) be a connected reductive \(F\)-group with indexed root datum \[ {}_{F}\Psi(G)=(X,D,X^\vee,D^\vee,D_0,\Gamma_F). \] Then:

  1. \(G\) is quasi-split over \(F\) if and only if \(G\) contains a Borel subgroup defined over \(F\). Equivalently, \(D_0=\emptyset\).
  2. \(G\) is anisotropic over \(F\) if and only if \(G\) has no proper parabolic \(F\)-subgroup. Equivalently, \(D_0=D\).
Proof

Let \(S\) be a maximal \(F\)-split torus and \(T\supseteq S\) a maximal \(F\)-torus. Quasi-splitness is equivalent to the centralizer \(Z_G(S)\) being a torus. In that case a cocharacter of \(S\) with centralizer \(T\) defines a parabolic \(P(\lambda)\) that is solvable, hence a Borel subgroup defined over \(F\).

Conversely, if \(B\) is an \(F\)-defined Borel subgroup, then \(B=P(\lambda)\) for an \(F\)-defined cocharacter after conjugating the image of \(\lambda\) into \(S\). Since \(P(\lambda)\) is solvable, its Levi \(Z_G(\lambda)\) is a torus, forcing \(Z_G(S)=T\) and \(D_0=\emptyset\).

The anisotropic criterion follows because proper parabolic \(F\)-subgroups are exactly the rational parabolics detected by noncentral \(F\)-split cocharacters. Such cocharacters exist precisely when the distinguished subset does not equal all of \(D\).

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