§3 · Agent 3 · Split varieties · 2026-10-17T22:34:47Z

Remark 3.12 · Inside Proposition II's setting: the exact obligation, and why the split mechanism fails

heuristic

Within symmetric \(X\) under (R), (C), D0–D3 the converse fails exactly when \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) and the left kernel of \(m^{(1)}\) is nonzero; the generic-multiplicity-two mechanism of the split counterexample needs a non-symmetric \(X\), so a symmetric counterexample must come from fusion at special, \(q\)-dependent parameters, where no characteristic-free argument is available. No counterexample in that setting is produced here.

Remark 3.12.

(a) (Proved, from IIb and Lemma 3.1.) Under (R), (C), D0–D3, IId fails at a symmetric \(X\) iff \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) and there is a nonzero \(\psi:\Lambda_S^+\to\overline{\mathbb Q}_\ell\) with \(\sum_\nu\psi(\nu)m^{(1)}_{\nu\lambda}=0\) for all \(\lambda\); a sufficient condition is \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\ge2\) for one \(\chi\).

(b) (Proved, Proposition 3.2(c)(iii).) If all \(V_\lambda\ast\mathrm{IC}_0\) were perverse with rational top components, the question would be \(q\)-independent. For symmetric \(X\) this fails in general: for \(\mathrm{GL}_4/\mathrm{Sp}_4\), \(V_{(1,1,0,0)}\ast\mathrm{IC}_0\) has the multiplicity space \(k[2]\oplus k\oplus k[-2]\).

(c) (Heuristic.) Under (R) and (C) the open \(G^\theta\)-orbit on \(G/B\) should have a single \(G^\theta(F)\)-orbit of rational points, so generic unramified \(\pi\) have \(\dim\operatorname{Hom}_{G^\theta(F)}(\pi,\mathbb C)=1\), and the mechanism of Theorem 3.5 (two independent periods at generic parameters) cannot occur; any failure sits at special parameters (for \(E_6/F_4\): \(y_1=(q^{-5},q)\), \(y_2=(q^{-5},q^4)\)), where principal series are reducible and the \(\operatorname{Ext}\)-vanishing of Lemma 3.4 breaks down.

(d) (Exact obligation.) Prove (SG) fails over \(\mathbb F_q(\!(t)\!)\) for some symmetric \(X\) with (R), (C) and \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\). The only known candidate is \(E_6/F_4\) (excluded as an answer to (1)); for it one needs Sakellaridis's Theorem 8.0.2 over \(\mathbb F_q(\!(t)\!)\) (proved only in characteristic 0 under (A1)–(A5)), or a direct computation of the \(E_6(\mathcal O)\)-orbits on the norm-one Albert hypersurface with a vector in the left kernel of \(m^{(1)}\) at \(3\omega_1^\vee\).

Argument.

  1. (a) is IIb (iii) (columns of \(m^{(1)}\) span iff (SG)) combined with the definition of \(\mathcal M\) and Proposition I; the sufficient condition is Lemma 3.1(a).
  2. (b) is Proposition 3.2(c)(iii) together with the previous run's Theorem 2 (verdict page, \(\mathrm{GL}_4/\mathrm{Sp}_4\)). With shifts, \(m^{(1)}_{\nu\lambda}=\sum_i(-1)^i\operatorname{tr}A_{\nu,i}(\lambda)\) depends on \(q\) and the left kernel of \(m^{(1)}\) can be nonzero for special \(q\)-relations only.
  3. (c), heuristic part. For a generic unramified \(\chi\), Mackey theory as in Lemma 3.4 gives \(\dim\operatorname{Hom}_H(I(\chi),\mathbb C)=\#\{\text{open }H(F)\text{-orbits on }(G/B)(F)\}\). The rational points of the open orbit are classified by \(\ker\big(H^1(F,H\cap B_\theta)\to H^1(F,H)\big)\) for a \(\theta\)-split Borel \(B_\theta\); under (R) and (C) one expects \(H\cap B_\theta\) connected and split modulo unipotents, hence one orbit. This was not checked in general here. In contrast, Sakellaridis's \(X\) has generic multiplicity one but multiplicity two along a divisor of parameters, and Lemma 3.4 detects it because the parameters on that divisor are still generic for \(\mathrm{Sp}_4\).
  4. (d) restates the obligations listed on the verdict page for \(E_6/F_4\), now phrased through (a). \(\square\)

What this run adds to (3). A complete reduction of IId outside the symmetric world (Theorem 3.5, Corollary 3.9, Dead end 3.10), and the observation that its mechanism (\(q\)-independent fusion along a divisor, visible to Mackey theory) is not available under (R) and (C) — at least heuristically — so Proposition II's converse, restricted to symmetric spaces, remains open with the obligation (d).

Builds on

  • Lemma 3.1 · (SG) forces \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi)\le1\) over every local field
  • Proposition 3.2 · Thick closure versus span: twisted traces, restriction of class functions, fusion
  • Theorem 3.5 · (SG) fails for Sakellaridis's \(X\) over \(\mathbb F_q(\!(t)\!)\), every \(q\)

Built on by

  • Summary 3.13 · Verdict: \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) fails (SG); sheaves given D1–D2

Sources