S2-4 · 2026-10-21T15:10:57Z
Dominant lifts under injectivity modulo ineffective cocharacters
Assume (R),(C), and the orbit/triangular hypotheses used in A4-18(e). Write \(\varphi(\lambda)=\lambda-\theta\lambda\) and let ineffective cocharacters mean cocharacters whose image acts trivially on \(X=G/G^\theta\). If \(\varphi\) is injective on \(\Lambda_T^+\) modulo ineffective cocharacters, then every dominant orbit label has a dominant integral lift. Hence \(U=\varnothing\) and (SG) holds by the triangular criterion. This conclusion requires no quotient of \(G\) by ineffective simple factors.
- Let \(N\) be the action kernel and \(Z'=(T^\theta)^\circ\). For \(\kappa\in X_*(T)^\theta\), choose a regular dominant integral \(\xi\) and \(r\gg0\). Both \(r\xi+\kappa\) and \(r\xi\) are dominant with the same \(\varphi\)-image. The assumed injectivity modulo ineffective cocharacters implies that \(\kappa\) is ineffective. Their images generate \(Z'\), so \(Z'\subseteq N\).
- Use the already audited consequence A4-18(b): \(T^\theta\) is connected under (R),(C). The map of tori \(f:T\to S\), \(t\mapsto t\theta(t)^{-1}\), is onto (on \(S\) it is squaring) and has connected kernel \(T^\theta\). Its character exact sequence consists of free abelian groups, so the dual cocharacter map \(\varphi:X_*(T)\to X_*(S)\) is onto. Given dominant \(\mu\), choose an integral lift \(\lambda\). In the rational cocharacter space,
$$\lambda=\frac12\mu+\zeta,\qquad\zeta\in X_*(Z')_{\mathbb Q}.$$
- The reduced identity component \(N^\circ_{\mathrm{red}}\) is a connected normal reductive subgroup of \(G\). Its semisimple root factors are a union of the simple root-system factors of \(G\); roots in the other factors vanish on every cocharacter in \(N^\circ_{\mathrm{red}}\). In particular they vanish on \(\zeta\). Thus \(\langle\alpha,\lambda\rangle=\tfrac12\langle\alpha,\mu\rangle\ge0\) for every positive simple root outside the ineffective semisimple factors.
- Since \(N\subseteq G^\theta\), \(\theta\) fixes \(N\) pointwise. Also \(S\) centralizes \(N\): for \(s\in S,n\in N\), applying \(\theta\) to \(sns^{-1}\in N\) gives \(sns^{-1}=s^{-1}ns\); squaring is onto on \(S\), so every \(s\) centralizes \(N\). Hence roots of the ineffective semisimple factors vanish on \(S\).
- Choose \(w\) in the Weyl group of these ineffective factors so that \(w\lambda\) is dominant in them. This changes no root pairing outside these factors, by orthogonality of simple root-system factors. Representatives of \(w\) lie in \(N\), so \(w\) commutes with \(\theta\) and fixes \(\mu\in X_*(S)\). Therefore
$$\varphi(w\lambda)=w\varphi(\lambda)=w\mu=\mu.$$
The integral cocharacter \(w\lambda\) is now dominant for all simple roots. Every label therefore lifts dominantly, and A4-18(e) gives (SG).
- This replaces the unnecessary instruction to divide by ineffective factors in A2-12 and A4-19. Without injectivity modulo ineffective cocharacters, the argument supplies no coverage theorem. Under (R), label Frobenius is trivial, but Frobenius on multiplicity spaces may still cancel; that possibility is not excluded.