§3 · Agent 3 · Split varieties · 2026-10-17T22:35:12Z

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

proved audit: relies on an error ★ key step

Sakellaridis's Example 8.0.1 is a split spherical variety for which (SG) is proved to fail over \(\mathbb F_q(\!(t)\!)\) for every \(q\) (characteristic-free), and for which \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) follows from proved orbit and fibre geometry once the sheaf-theoretic foundations D1–D2 are available for this non-symmetric \(X\); the formal mechanism is fusion of conjugacy classes under \(\check G_X\subset\check G\).

Summary 3.13.

(1) Split counterexample, one gap. For \(X=\mathrm{SL}_2^3\backslash(\mathrm{Sp}_4\times\mathrm{Sp}_4)\) (split, affine, connected stabilizers):

(2) Mechanism. \(\operatorname{Tr}(\mathrm{Fr},-)\) sends the Satake action to the Hecke action, but thick generation adds retracts whose Frobenius-twisted traces are not Hecke translates of \(\varphi_0\) (Proposition 3.2(a)). Model: \(\operatorname{Res}:\operatorname{Rep}\check G\to\operatorname{Rep}\check G_X\) is always Karoubi-generating, while its trace (restriction of class functions) is onto only without fusion (Proposition 3.2(b)). For \(X\) this model is exact: \(m^{(n)}=D_n\,c\,D_n'\) with \(c\) the branching matrix of \(\iota:\mathrm{SL}_2^3\to\mathrm{SO}_5^2\), and \((a,b,a)\sim(b,a,b)\) fuses (Lemma 3.3, Corollary 3.9). Forcing conditions: Proposition 3.2(c).

(3) Proposition II's setting. No counterexample produced; the converse there is equivalent to the vanishing of the left kernel of \(m^{(1)}\) whenever \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), and the remaining obligation is (SG) over \(\mathbb F_q(\!(t)\!)\) for \(E_6/F_4\) or another (R),(C)-symmetric space with fusion (Remark 3.12).

Chain of evidence.

  1. Lemma 3.1: (SG) \(\Rightarrow\) cyclic Hecke module \(\Rightarrow\) at most one period per irreducible unramified \(\pi\), over every local field.
  2. Lemma 3.3, Lemma 3.4, Theorem 3.5: for generic \(\pi\) of \(\mathrm{Sp}_4(F)\), the coinvariants under \(\mathrm{Sp}_2\) have two non-isomorphic \(\mathrm{SL}_2\)-quotients, so \(\pi\boxtimes\pi\) has two \(H\)-periods: (SG) fails, characteristic-free.
  3. Proposition 3.6, Computation 3.7, Proposition 3.8: orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\); fibres are tree configurations; defect \(=-2\sum(u_i-w_i)_+\le0\); top components \(=\) branching multiplicities; every label relevant (e.g. \((0,0,2)\) for \(((1,1),0)\)).
  4. Corollary 3.9: given D1–D2, every \(\mathrm{IC}_\nu\) is a summand (sheaves generate), and the trace matrix is the branching matrix up to scaling, so (SG) is restriction-surjectivity, which fusion kills — the same failure as step 2, seen from the sheaf side.
  5. Dead end 3.10: the one missing input. Remark 3.11: no \(n\)-dependent cancellation. Remark 3.12: the symmetric case.

Not claimed. No unconditional proof of \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\) for \(X\); no statement about \(E_6/F_4\) over \(\mathbb F_q(\!(t)\!)\); no use of a local geometric Langlands conjecture (BZSV appears only as a heuristic consistent with Corollary 3.9); computations (fibres.py, mass.py, branch.py, fusion.py) are checks of proved formulas, not evidence substituting for proofs.

Builds on

  • Theorem 3.5 · (SG) fails for Sakellaridis's \(X\) over \(\mathbb F_q(\!(t)\!)\), every \(q\)
  • Corollary 3.9 · Given D1–D2 for \(X\): \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), \(m^{(n)}=D_ncD_n'\)
  • Dead end 3.10 · What blocks an unconditional \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) for \(X\)
  • Remark 3.12 · Inside Proposition II's setting: the exact obligation, and why the split mechanism fails

Built on by

  • Remark 6.2 · Registry after the split agent and the first audit: audit the split candidate and the mechanism

Sources