§5 · Agent 5 · Functions without (A1)–(A5) · 2026-10-17T03:32:17Z

Proposition 5.8 · The support bound of H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) at function level over every \(F\), and the same-points question

proved audit: error

Error. Input (a) is false for the pair model of \(\GL_n/(\GL_p\times\GL_q)\) in residue characteristic 2, so step 4 fails there as written; Theorem 5.4 proves the orbit classification, not the compatibility with \(\tau\). The only involution with fixed group \(\Stab(x_0)\) is \(\theta=\operatorname{Ad}\varepsilon_0\) (\(\varepsilon_0=1\) on \(U_0\), \(-1\) on \(V_0\)), which does not normalize the diagonal torus; over \(\Q_2\) with \(n=2\) the point \(t^{(1,0)}x_0\) has label \(1\) while \(\tau(t^{(1,0)}x_0)\in K\) (for \(n=2\) the Cartan label of \(t^{(a,0)}x_0\) is \(\max(a-\val2,0)\)), and in characteristic 2 there is no such involution. For the spheres and for alternating forms, (a), (b) and the transfer are valid in every characteristic (the reflection defining \(\theta\) on the sphere is monomial; \(\tau(\omega)=\omega J^{-1}\) on forms). The stated conclusion is nevertheless true for the pair model too: Correction 10.4 proves the support bound for every dominant \(\lambda\) and every \(F\) by a direct argument. The same-points paragraph is correctly labelled heuristic. The conclusion still holds by another argument.
Used later by: Objection 7.24 · Objection: the support-bound check passes; the \(E_6/F_4\) expectation and two table cells are stale.
Corrected by: Correction 10.4 · Correction: the function-level support bound for the pair model of \(\GL_n/(\GL_p\times\GL_q)\) over every local field, Proposition X3.2 · Support bound and top orbit for the pair model, every dominant \(\lambda\), every \(F\), Proposition X3.3 · Cartan label of \(\tau(x)\) is \(\mu((\ell(x)-e)^+)\), \(e=\operatorname{val}2\), Theorem X3.4 · Corrected A5-8 for the three families, and (SG) for the pair model without \(\tau\).

The function-level content of H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) is the inclusion \(Kt^\lambda X(\cO)\subseteq\bigcup_{\nu\le\varphi(\lambda)}O_\nu\). The sheaf proof transfers to any local field once orbits are labelled through the Cartan embedding and the Cartan convolution bound is available; the node asserts this labelling for the spheres, the pair model of \(\GL_n/(\GL_p\times\GL_q)\) and alternating forms. Whether the two levels have the same support points is left heuristic.

Proposition 5.8.

For \(X\) one of the spheres, \(\GL_n/(\GL_p\times\GL_q)\) in the pair model of Theorem 5.4, or \(\GL_{2m}/\Sp_{2m}\), and for every non-archimedean local field \(F\),

$$K\,t^\lambda X(\cO)\ \subseteq\ \bigcup_{\nu\le\varphi(\lambda)}O_\nu\qquad(\lambda\in\Lambda_T^+),$$

that is, \(\supp(T_\lambda\ast\varphi_0)\) lies in the dominance lower set of \(\varphi(\lambda)\). This follows from two inputs over \(F\): (a) \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) with representatives satisfying \(\tau(x_\mu)=t^\mu\) for the Cartan embedding \(\tau\), hence \(\tau(O_\mu)\subseteq Kt^\mu K\); (b) the Cartan decomposition and its convolution bound. Input (a) holds for the three families by Lemma 5.2, Theorem 5.4 and Computation 5.5.

Heuristic, not proved here: over \(\F_q(\!(t)\!)\) the function-level support consists of all rational points of the sheaf-level support.

Dictionary. After Corollary C on the classical page, the function-level reading of H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) is the inclusion in the statement: the support of \(T_\lambda\ast\varphi_0\) lies in the dominance lower set of \(\varphi(\lambda)\).

  1. Input (a). If \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) with representatives satisfying \(\tau(x_\mu)=t^\mu\), then \(\tau(kx)=k\tau(x)\theta(k)^{-1}\) and \(\theta(K)=K\) give \(\tau(O_\mu)\subseteq Kt^\mu K\).
  1. Input (b). \(Kt^aK\cdot Kt^bK\subseteq\bigcup_{c\le a+b}Kt^cK\), and distinct dominant \(c\) give distinct double cosets (cited: the Cartan decomposition and its convolution bound for split \(G\) over any local field — Bruhat–Tits, Iwahori–Matsumoto; elementary for \(\GL_n\) through elementary divisors).
  1. Transfer of the sheaf argument. For \(y\in X(\cO)\),
$$\tau(kt^\lambda y)=k\,t^\lambda\,\tau(y)\,t^{-\theta\lambda}\,\theta(k)^{-1}\in Kt^\lambda K\,t^{-\theta\lambda}K ,$$

so the orbit \(\nu\) of \(kt^\lambda y\) satisfies \(\nu\le\lambda+(-\theta\lambda)^+\). The projection argument of Step 2 of Proposition III — \(p=\frac12(1-\theta)\) fixes \(\nu\) and \(\varphi(\lambda)\), kills the coroots of \(L\), and sends the other positive coroots into the positive cone — is purely combinatorial and gives \(\nu\le\varphi(\lambda)\).

  1. The three families. Input (a) holds for the spheres (Lemma 5.2), for \(\GL_n/(\GL_p\times\GL_q)\) in the pair model (Theorem 5.4) and for alternating forms (Computation 5.5). Hence H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) holds at function level for these families over every \(F\), not only over \(\F_q(\!(t)\!)\). Sharper elementary bounds were proved directly: by pole order for the spheres (Theorem 5.3), componentwise for \(\GL_n/(\GL_p\times\GL_q)\) (Theorem 5.4), and the two-sided bound \(b\le_{\mathrm{dom}}b'\le_{\mathrm{dom}}\varphi(\lambda^*(b))\) for the symplectic rows (Lemma 5.6).

The same set of points? (heuristic). Over \(F=\F_q(\!(t)\!)\) with the \(\F_q\)-model, \(Kt^\lambda X(\cO)\subseteq(L^+G\,t^\lambda L^+X)(\F_q)\) trivially. Equality would ask that every rational point of the image be reached by a rational pair \((g,z)\). This is a Lang-type statement that needs connected stabilizers (H3H3 · Constant coefficients: on each orbit the heart consists of finite sums of the constant sheaf.), i.e. \(Z_{G^\theta}(S)\) connected — true for all three families (section 4 of each families page) — and it is labelled heuristic here, not re-proved. The sheaf statement \(\supp(V_\lambda\ast\IC_0)\subseteq\overline{\cO_{\varphi(\lambda)}}\) is only an upper bound, so the inclusion is all that H5H5 · Nonzero top splitting: for every orbit O_μ some Satake object V_λ ∗ IC_0 is supported on its closure with a nonzero split restriction to O_μ.(a) uses at either level. For \(\operatorname{char}F=0\) there is no Frobenius-trace relation at all; the inclusion is proved directly.

Builds on

  • Strategy 5.1 · Strategy: triangular generation from explicit orbit geometry, without Theorem E

Built on by

Original record, as the agent wrote it at 03:32 UTC

Support bound: function-level content of H5(a), valid for every F; the same-points question

Function-level H5(a) is the inclusion \(K\,t^\lambda X(\mathcal O)\subseteq\bigcup_{\nu\le\varphi(\lambda)}O_\nu\), i.e. \(\operatorname{supp}(T_\lambda\ast\varphi_0)\) lies in the dominance lower set of \(\varphi(\lambda)\) (classical.html, dictionary after Cor. C). The sheaf proof (pdf/proposition-III.tex, Lemma "Step 2") transfers verbatim to \(F\)-points of any local field once two inputs hold over \(F\): (a) orbit labelling through the Cartan embedding: \(K\backslash X(F)\leftrightarrow\Lambda_S^+\) with \(\tau(x_\mu)=t^\mu\), hence \(\tau(O_\mu)\subseteq Kt^\mu K\) (\(\tau(kx)=k\tau(x)\theta(k)^{-1}\), \(\theta(K)=K\)); (b) \(Kt^aK\cdot Kt^bK\subseteq\bigcup_{c\le a+b}Kt^cK\) and distinct dominant \(c\) give distinct double cosets (cited: Cartan decomposition and its convolution bound for split \(G\) over any local field, Bruhat–Tits/Iwahori–Matsumoto; elementary for \(\mathrm{GL}_n\) via elementary divisors). Then \(\tau(kt^\lambda y)=kt^\lambda\tau(y)t^{-\theta\lambda}\theta(k)^{-1}\in Kt^\lambda Kt^{-\theta\lambda}K\) for \(y\in X(\mathcal O)\), so the orbit \(\nu\) of \(kt^\lambda y\) satisfies \(\nu\le\lambda+(-\theta\lambda)^+\), and the purely combinatorial projection argument of Step 2 gives \(\nu\le\varphi(\lambda)\). (a) is proved above for the spheres (node 2), \(\mathrm{GL}_n/(\mathrm{GL}_p\times\mathrm{GL}_q)\) (node 4, pair model) and alternating forms (node 5); so H5(a) holds at function level for these families over every \(F\), not only \(\mathbb F_q((t))\). In nodes 3–4 I in fact proved sharper, elementary bounds (pole order; componentwise \(\le\)), and in node 6 the two-sided sandwich \(b\le_{\mathrm{dom}}b'\le_{\mathrm{dom}}\varphi(\lambda^*(b))\).

Same set of points? Over \(F=\mathbb F_q((t))\) with the \(\mathbb F_q\)-model: \(K\,t^\lambda X(\mathcal O)\subseteq(L^+G\,t^\lambda L^+X)(\mathbb F_q)\) trivially; equality asks that an \(\mathbb F_q\)-point of the image set be reached by \(\mathbb F_q\)-rational \((g,z)\): this is a Lang-type statement needing connected stabilizers (H3), i.e. \(Z_{G^\theta}(S)\) connected (true for all families here, families/*.html §4) — labelled heuristic here, not re-proved. The sheaf statement \(\operatorname{supp}(V_\lambda\ast\mathrm{IC}_0)\subseteq\overline{O_{\varphi(\lambda)}}\) is only an upper bound, so the inclusion is all that H5(a) uses at either level. For \(F\) of characteristic \(0\) there is no Frobenius-trace relation at all; the inclusion is proved directly.

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