Strategy 2.1 · Strategy: change the \(\mathbb F_q\)-structure of a pair whose sheaf statement is known

Three Frobenius actions. For \(X=G/H\) over \(\mathbb F_q\), Frobenius acts on three pieces of data that the two levels treat differently.

1. Coweights. \(\mathrm{Fr}\) acts on \(\Lambda_T=X_*(T)\) by the \(*\)-action \(\sigma\) of the \(\mathbb F_q\)-structure of \(G\). For unramified \(G\) the Cartan decomposition gives \(K\backslash G(F)/K\cong(\Lambda_T^+)^\sigma\), so \(\mathcal H_K\) has the basis \(T_\lambda\), \(\sigma\lambda=\lambda\). The category \(\mathcal C_k^{\mathrm{Sat}}\) uses every \(\lambda\in\Lambda_T^+\).
2. Orbit labels. \(\mathrm{Fr}\) permutes the \(L^+G\)-orbits \(\mathcal O_\mu\), \(\mu\in\Lambda_S^+\). Only Frobenius-stable orbits have \(\mathbb F_q\)-points, so only they index \(K\)-orbits on \(X(F)\). The category \(\mathcal C_k\) contains every \(\mathrm{IC}_\mu\).
3. Multiplicity spaces. \(\mathrm{Fr}\) acts on \(E_{\nu,i}(\lambda)\) by the matrices \(A_{\nu,i}(\lambda)\) of the pages (IIb, step 4).

What (R) does. Under (R) both \(T\) and \(S\) are split, so actions 1 and 2 are trivial and only action 3 is left. A counterexample with a non-split \(\mathbb F_q\)-structure therefore has to break a splitness clause of (R).

Candidates. Take \(\mathbb F_q\)-forms of pairs whose sheaf statement is already proved over \(k\): tori, and the group case \((H\times H)/H\).

Plan.

• (a) A function-level obstruction lemma: Lemma 2.2.
• (b) The torus, at both levels: Counterexample 2.3.
• (c) The unitary and Galois twists of the group case: Setup 2.4 to Proposition 2.10.
• (d) Split \(G\) with non-split \(H\): Dead end 2.11.
• (e) The mechanism, and what it says about (R): Remark 2.12.

Expectation (heuristic). Suppose \(\sigma\) exchanges two factors of \(\Lambda_T\) but fixes the orbit labels. Then the rational leading terms \(\varphi((\Lambda_T^+)^\sigma)\) form a proper part of the labels, so (SG) should fail. Meanwhile the non-rational \(V_\lambda\) still supply every summand over \(k\).