S3-2 · 2026-10-21T15:10:33Z

The unitary example satisfies connectedness and fails the split clauses of (R)

Let \(q\) be odd, \(H\ne1\) split connected reductive over \(\mathbb F_q\), \(T_H\) split maximal, and \(\vartheta\) an \(\mathbb F_q\)-involution acting by inversion on \(T_H\). For \(G=\operatorname{Res}_{q^2/q}H\), \(\theta=\vartheta\sigma\), and \(X=G/G^\theta\), the groups \(G^\theta\) and \(Z_{G^\theta}(S)\) are geometrically connected, where \(S_k=\{(s,s):s\in T_H\}\) is split and maximal theta-split. A theta-stable Borel pair over \(\mathbb F_q\) exists, but its maximal torus cannot be split. Hence precisely the requirements that \(T\supseteq S\) and \(T_0\) be split in (R) fail. D0, D1 and D3 hold as in the geometric group case; D2 holds geometrically for every label and has its finite-field purity interpretation only for Frobenius-stable labels (or after passage to \(\mathbb F_{q^2}\)).

  1. In \(G_k=H\times H\), Frobenius acts by \((h_1,h_2)\mapsto(Fh_2,Fh_1)\) and \(\theta(h_1,h_2)=(\vartheta h_2,\vartheta h_1)\). Thus \((G^\theta)_k=\{(h,\vartheta h)\}\cong H\), which is connected. The diagonal torus \(S\) is inverted by \(\theta\), and Frobenius acts on it as the split Frobenius of \(T_H\); it is split. Maximality follows from the geometric group case, whose symmetric rank is \(r=\operatorname{rk}H\).
  2. A graph element \((h,\vartheta h)\) centralizes \(S\) exactly when \(h\in Z_H(T_H)=T_H\). Hence the geometric centralizer is a torus and is connected. Similarly, the centralizer of \((c,c)\) in the fixed group is geometrically \(Z_H(c)\), a connected Levi subgroup. This verifies (C), rather than assuming the relevant fixed group is split.
  3. Put \(F'=\vartheta F\) on \(H_k\). Since \(\vartheta\) commutes with \(F\) and \(\vartheta^2=1\), \((F')^2=F^2\); \(F'\) is the Frobenius of the descended finite-field form, and is a Steinberg endomorphism in Milne's Definition 27.53. Lang's theorem applies to this smooth connected group (Milne, Proposition 27.54). Choose a Borel pair \((B_2,T_2)\) fixed by \(F'\): explicitly, if \(F'\) sends any pair to its conjugate by \(h\), solve \(g^{-1}F'(g)=h^{-1}\) and conjugate that pair by \(g\). Then \((FB_2\times B_2,FT_2\times T_2)\) is fixed both by the product Frobenius and by \(\theta\). This gives the rational theta-stable Borel pair.
  4. By S3-1, every maximal torus in \(G\) has split rank at most \(r<2r\). Thus neither a maximal \(T\supseteq S\) nor the maximal torus \(T_0\) in the preceding pair is split. Definition over \(\mathbb F_q\), the rational involution, and existence of the rational theta-stable pair do hold. These are exactly the two failed clauses in the displayed definition of (R).
  5. Geometrically the quotient is \(H\), with action \((g_1,g_2)x=g_1x\vartheta(g_2)^{-1}\). The Cartan orbit and convolution identifications of A4-2 and A4-3 give D0 and the finite-type models underlying D1. The stabilizers are connected \(J_c=L^+H\cap t^cL^+Ht^{-c}\), with connected Levi and pro-unipotent radical, so finite-quotient Lang gives D3. The geometric decomposition remains valid. Frobenius interchanges a Satake label \((a,b)\) with \((b,a)\); a single such object descends to \(\mathbb F_q\) precisely when its label is stable. Its purity assertion over \(\mathbb F_q\) must carry that qualification. After quadratic extension all labels are stable. This is the qualification already present in A4-1, item 5; the rank repair does not remove it.

Builds on: S3-1 · A4-2 · A4-3