§4 · Agent 4 · Sheaf–function dictionary · 2026-10-17T02:40:19Z

Proposition 4.7 · The converse fails in general: \(\Csat=\cC_K\) gives detection by Hecke translates, not spherical generation

proved audit: minor defect

Minor defect (second audit). The proposition is correct (re-derived, including n ≤ dim ⊕i E{μ,i}(λ) by Vandermonde, and n = 1 under Chen–Yi's even-shift assertion, since all terms then have the same sign). Two wording defects: (1) 'not contained in the hyperplane complement of tr_n IC_μ' should read 'not contained in the hyperplane spanned by the tr_n IC_ν, ν ≠ μ'; (2) 'false as a formal implication of the dictionary' is not a theorem proved here — no model of the dictionary with Csat = C and (SG) false is exhibited (Sakellaridis's Example 8.0.1 is a function-level, p-adic statement under his hypotheses, for a variety whose sheaf level is unknown). What the argument shows is that the dictionary yields row-nonvanishing of the multiplicity matrix, which does not imply column-spanning; the status 'not proved' is right.

If \(\Csat=\cC_K\), every \(\IC_\mu\) is detected by some Hecke translate of \(\varphi_0\) over some \(\F_{q^n}(\!(t)\!)\): every row of the multiplicity matrix is nonzero, whereas (SG) asks that its columns span. The dictionary therefore does not yield the converse, whose truth for symmetric spaces stays open here.

Proposition 4.7.

Under the hypotheses of Theorem 4.6: if \(\Csat=\cC_K\), then for every \(\mu\in\Lambda_S^+\) there are \(\lambda\in\Lambda_T^+\) and \(n\le\dim\bigoplus_iE_{\mu,i}(\lambda)\) with $$\tr_n(V_\lambda)\ast_{K_n}\varphi_0\ \notin\ \operatorname{span}\{\tr_n\IC_\nu:\nu\ne\mu\}\subset C_{K_n}\otimes\overline{\Q}_\ell ,$$ i.e. the \(\cH_{K_n}\)-module \(\cH_{K_n}\varphi_0\) is not contained in the hyperplane complement of \(\tr_n\IC_\mu\). If Chen–Yi's even-shift (Tate-type) assertion in the proof of Theorem 93 is granted, \(n=1\) works.

Status. "\(\Csat=\cC_K\Rightarrow\)(SG)" is not proved, and is false as a formal implication of the dictionary; whether it fails for some symmetric \(X\) with \(Z_{G^\theta}(S)\) connected is open (no example on these pages, see Summary 4.13).

Proof of the proposition. Lemma 4.5 (part 3) gives a summand: \(E_{\mu,i}(\lambda)\ne0\) for some \(\lambda,i\). Lemma 4.4 (part 4(ii)) gives an \(n\) with \(m^{(n)}_{\mu\lambda}\ne0\); since \(m^{(n)}_{\mu\lambda}\) is a power sum over at most \(\dim\bigoplus_iE_{\mu,i}(\lambda)\) distinct eigenvalues, the Vandermonde argument finds such an \(n\) in that range. As \(m^{(n)}_{\mu\lambda}\) is the \(\tr_n\IC_\mu\)-coordinate of \(\tr_n(V_\lambda)\ast\varphi_0\) (Lemma 4.3), the translate is not in the span of the other \(\tr_n\IC_\nu\). \(\square\)

Why this is not (SG). (SG) says that the matrix \(M^{(1)}=(m^{(1)}_{\nu\lambda})_{\nu,\lambda}\) has column span equal to all finitely supported vectors on \(\Lambda_S^+\) (Lemma 4.3, Lemma 4.4); the proposition says that every row of \(M^{(n)}\) is nonzero for a suitable \(n\). Nonzero rows do not give spanning columns. The triangular criterion (Proposition B) needs, for each \(\mu\), a translate supported in the lower set \(\{\nu\le\mu\}\) and nonzero at \(\mu\). The summand \(\IC_\mu\subseteq V_\lambda\ast\IC_0\) produced by Lemma 4.5 lives inside \(\supp\subseteq\overline{\cO_{\varphi(\lambda)}}\), and when \(\mu\) has no dominant lift one has \(\mu<\varphi(\lambda)\) strictly (criterion page, step 6), so the translate is not supported in the lower set of \(\mu\). The categorical statement is closed under retracts (Lemma 4.5); the function statement is a linear-span statement in the Grothendieck group. This is exactly the gap between thick and stable spans ("Three spans compared", Thomason). Sakellaridis's Example 8.0.1 realises this failure shape at function level (\(W_X\)-nonconjugate points of \(A_X^*\) that are \(W\)-conjugate, so that the image of \(r\) is a proper subring), for a non-symmetric \(X\) not on these pages.

Where the converse does hold. If \(Z_{G^\theta}(S)\) is connected and \(\varphi(\Lambda_T^+)=\Lambda_S^+\) (the two conditions of Proposition III), both levels hold independently (Proposition I; Lemma 4.8), so the biconditional holds trivially.

Builds on

  • Lemma 4.4 · Expansion of \(\tr(V_\lambda\ast\IC_0)\) in the IC basis: the coefficients are traces of Frobenius on multiplicity spaces
  • Lemma 4.5 · \(\Csat=\cC_K\) if and only if every \(\IC_\mu\) is a direct summand of a Satake object
  • Theorem 4.6 · Theorem: (SG) over \(\F_q(\!(t)\!)\) implies \(\Csat=\cC_K\) over \(\overline{\F}_q\)

Built on by

Original record, as the agent wrote it at 02:40 UTC

Converse fails in general: thick generation gives detection, not the row span; Example 8.0.1 shape

Proposition A4-7 (what \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) gives at function level). Hypotheses of A4-6. If \(\mathcal C^{\mathrm{Sat}}=\mathcal C\) then for every \(\mu\in\Lambda_S^+\) there are \(\lambda\in\Lambda_T^+\) and \(n\le\dim\bigoplus_iE_{\mu,i}(\lambda)\) such that $$\operatorname{tr}_n(V_\lambda)\ast_{K_n}\varphi_0\ \notin\ \operatorname{span}\{\operatorname{tr}_n\mathrm{IC}_\nu:\nu\ne\mu\}\subset C_{K_n}\otimes\overline{\mathbb Q}_\ell ,$$ i.e. the \(\mathcal H_{K_n}\)-module \(\mathcal H_{K_n}\varphi_0\) is not contained in the hyperplane complement of \(\operatorname{tr}_n\mathrm{IC}_\mu\). Proof. A4-5(iii) gives a summand; A4-4.4(ii) gives \(n\) with \(m^{(n)}_{\mu\lambda}\ne0\). \(\square\) (If Chen–Yi's even-shift/Tate-type assertion in the proof of Thm 93 is granted, \(n=1\) works.)

Why this is not (SG). (SG) is the statement that the matrix \(M^{(1)}=(m^{(1)}_{\nu\lambda})_{\nu,\lambda}\) has column span equal to all finitely supported vectors on \(\Lambda_S^+\) (A4-3, A4-4.1). A4-7 says every row of \(M^{(n)}\) is nonzero for suitable \(n\). "Every row nonzero" does not give "columns span": the triangular criterion (classical.html Prop. B) needs, for each \(\mu\), a translate supported in the lower set \(\{\nu\le\mu\}\) and nonzero at \(\mu\); the summand \(\mathrm{IC}_\mu\subseteq V_\lambda\ast\mathrm{IC}_0\) produced by A4-5 lives inside \(\operatorname{supp}\subseteq\overline{\mathcal O_{\varphi(\lambda)}}\), and when \(\mu\) has no dominant lift one has \(\mu<\varphi(\lambda)\) strictly (criterion.html Step 6), so the translate is not supported in the lower set of \(\mu\). The categorical statement is retract-closed (A4-5), the function statement is a \(K_0\)/linear-span statement; this is exactly the gap between \(\mathrm{Thick}\) and \(\mathrm{Stab}\) (thick-spans.html, Thomason). Sakellaridis's Example 8.0.1 realises the failure shape at function level (\(W_X\)-nonconjugate points of \(A_X^*\) conjugate under \(W\): the image of \(r\) is a proper subring), for a non-symmetric \(X\) not on these pages. Status: "\(\mathcal C^{\mathrm{Sat}}=\mathcal C\Rightarrow\) (SG)" is not proved and is false as a formal implication of the dictionary; whether it fails for some symmetric \(X\) with \(Z\) connected is open (no example on these pages: A4-13).

Where the converse does hold. If \(Z\) connected and \(\varphi(\Lambda_T^+)=\Lambda_S^+\) (Prop. III's two conditions): both levels hold independently (Prop. I; A4-8), so the biconditional holds trivially.

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