Remark 2.12 · Remark: why the trace misses thick generation, and what (R) removes

1. Two statements. The first equivalence is IIa together with Lemma 2.8(2)–(3). The second is Proposition I. (G1) is Lemma 2.8(3)–(4): the summand \(\mathrm{IC}_c=V_{(c,0)}\ast\mathrm{IC}_0\) carries Weil structures, but only induced Weil objects of \(\mathrm{Sat}\) contribute to \(\mathcal H_K\ast\varphi_0\), and those see \(V_{(c,0)}\oplus V_{(0,c)}\) with an off-diagonal Frobenius. For (G3), a non-stable orbit has no rational points, so it does not index a \(K\)-orbit.

2. Forcing criterion.

• Support. \(\operatorname{tr}V_\lambda=\pm T_\lambda+\sum_{\mu<\lambda}a_{\lambda\mu}T_\mu\) for rational \(\lambda\), and \(\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{\mathcal O_{\varphi(\lambda)}}\) (D0). Induction on \(\lambda\), using that \(\varphi\) is order-preserving, gives \(\operatorname{supp}(T_\lambda\ast\varphi_0)\subseteq\bigcup_{\nu\le\varphi(\lambda)}O_\nu\).
• Top value. \(t^\lambda x_0\in\mathcal O_{\varphi(\lambda)}(\mathbb F_q)\) gives \((T_\lambda\ast\varphi_0)(t^\lambda x_0)\ge1\) (Lemma A(2)).
• Conclusion. Lemma 2.2(b) with \(\psi=\varphi|_{(\Lambda_T^+)^\sigma}\) shows that the labels outside \(\varphi((\Lambda_T^+)^\sigma)\) are not generated.
• Unitary twist. \(\theta(a,b)=(-b,-a)\) and \(\sigma(a,b)=(b,a)\) give \(\varphi(a,a)=(2a,2a)\), injective, with image \(2\Lambda_H^+\ne\Lambda_H^+\).

4. Under (R) and (C).

• (R) makes \(T\) split, so \(\sigma=1\), and \(S\) split, so \(t^\mu\in S(F)\) and every orbit is Frobenius-stable with a rational point.
• Injectivity on \(\Lambda_T^+\). A nonzero \(\kappa\in\ker\varphi=X_*(T)^\theta\) gives \(\varphi(N\xi+\kappa)=\varphi(N\xi)\), with both arguments dominant for \(\xi\) regular dominant and \(N\gg0\). So injectivity on \(\Lambda_T^+\) forces \(X_*(T)^\theta=0\), hence \(T=S\), \(Z_G(S)=T\) and \(Z_{G^\theta}(S)=T[2]\cong\mu_2^{\operatorname{rk}G}\), which is disconnected for \(p\) odd and rank \(\ge1\). Under (C), then, \(\varphi\) is never injective on \(\Lambda_T^+\).
• Injectivity modulo trivially acting cocharacters. Since \(T_\lambda\) and \(T_{\lambda+\kappa}\) act identically on \(C_K\) when \(t^\kappa\) acts trivially on \(X\), the relevant condition is injectivity modulo such \(\kappa\). By the same \(N\xi+\kappa\) argument, it forces every \(\kappa\in X_*(T)^\theta\) to act trivially on \(X\) after passing to a multiple, so \(Z':=(T^\theta)^\circ\) acts trivially on \(X\). Divide \(G\) by any simple factor that acts trivially on \(X\); such a factor only rescales Hecke operators by their degrees. The kernel of the action is then a normal subgroup whose identity component is a central torus, so \(Z'\) is central in \(G\). Every root is nonzero on \(S\), since a root vanishing on \(S\) would vanish on \(SZ'=T\); hence \(Z_G(S)=T\) and \(T=S\cdot Z'\).
  • Consider \(f:T\to S\), \(t\mapsto t\theta(t)^{-1}\). It is surjective (\(f(s)=s^2\) on \(S\)) with kernel \(T^\theta=Z_{G^\theta}(S)\), and \(X_*(f)=\varphi\). The cokernel of \(X_*(f)\) has order \(|\pi_0(\ker f)|\), so under (C) \(\varphi:\Lambda_T\to\Lambda_S\) is onto.
  • A preimage \(\lambda=\tfrac12\mu+\zeta\), \(\zeta\in X_*(Z')_{\mathbb Q}\), of a dominant \(\mu\) is dominant, because roots vanish on the central \(Z'\).
  • So every label lifts dominantly, and (SG) holds by Lemma 4.8 of the run. The forcing criterion yields nothing under (R) and (C), and a failure there must come from cancellations among Hecke operators sharing a leading orbit.
• The positive statement is the run's Lemma 4.8, quoted in the verdict: split symmetric spaces meeting Proposition III's two conditions (connected \(Z_{G^\theta}(S)\) and dominant lifts) satisfy (SG) over \(\mathbb F_q(\!(t)\!)\) under D0, D3.

5. Categorical trace. This item is heuristic; no statement of AGKRRV or Zhu on categorical traces is used or checked here. On the level of trace functions, the precise content is Lemma 2.8(4). In \(\mathrm{HH}_0(\mathcal C,\mathrm{Fr}^*)\), a retract \(M\xrightarrow{i}N\xrightarrow{p}M\) satisfies \([M,\beta]=[N,\mathrm{Fr}^*(i)\circ\beta\circ p]\) by cyclicity. The map \(\mathrm{Fr}^*(i)\beta p\) is a morphism of \(\mathcal C_k\) that need not come from \(\mathrm{Sat}\). Thick closure therefore enlarges the trace by exactly such classes, while (SG) uses only the classes coming from \(\mathrm{Sat}\).