§3 · Agent 3 · Split varieties · 2026-10-17T22:33:25Z

Corollary 3.9 · Given D1–D2 for \(X\): \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\), \(m^{(n)}=D_ncD_n'\)

proved audit: minor defect ★ key step

Conditionally on the sheaf-theoretic inputs D1–D2 for this non-symmetric \(X\), semismallness makes every Satake object perverse with multiplicities given by dual branching, every orbit label is a summand label, so \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\); and the Frobenius-trace matrix is the branching matrix up to diagonal scalings, so \((\mathrm{SG})_n\) is the surjectivity of restriction \(R(\check G)\to R(\check G_X)\), which fusion destroys — matching Theorem 3.5.

Corollary 3.9.

Assume for \(X\) the inputs D1 and D2 of the pages: a Frobenius-compatible formalism \(\mathcal C_k\) on \(L^+G\backslash LX\) with perverse t-structure whose simple objects are the \(\mathrm{IC}_\nu\) (\(\operatorname{End}=\overline{\mathbb Q}_\ell\)), proper base change for \(m_\lambda:\mathrm{Gr}_{\le\lambda}\tilde\times L^+X\to LX\), and the decomposition theorem for \(m_\lambda\). Then, over \(k=\overline{\mathbb F}_q\), \(q\) odd:

(a) \(V_\lambda\ast\mathrm{IC}_0\simeq\bigoplus_\nu\mathrm{IC}_\nu\otimes\overline{\mathbb Q}_\ell^{\,c_{\nu\lambda}}\) is perverse, with \(c\) the branching matrix of Proposition 3.8(c).

(b) \(\mathcal M=\Lambda_X^+\), hence \(\mathcal C^{\mathrm{Sat}}_k=\mathcal C_k\).

(c) \(m^{(n)}_{\nu\lambda}=c_{\nu\lambda}\,q^{n(\langle2\rho,\lambda\rangle-\delta(\nu))/2}\) for every \(n\).

(d) \((\mathrm{SG})_n\) holds iff \(R(\mathrm{SO}_5\times\mathrm{SO}_5)\otimes\mathbb Q\to R(\iota\check G_X)\otimes\mathbb Q\) is onto; it is not: \(\operatorname{tr}\mathrm{IC}_{(0,0,2)}\notin\mathcal H_{K_n}\ast\varphi_0\otimes\overline{\mathbb Q}_\ell\) for every \(n\) and \(q\).

Proof (given D1–D2).

  1. Perversity. Proposition 3.8(a) applied to every \(\lambda'\le\lambda\) is stratified semismallness of \(m_\lambda\) for the strata \(\mathrm{Gr}^{\lambda'}\tilde\times L^+X\) and \(\mathcal O_\nu\) (with \(\delta\) from Proposition 3.6(c)); hence \(m_{\lambda*}(\mathrm{IC}_{\mathrm{Gr}_{\le\lambda}}\tilde\boxtimes\mathrm{IC}_0)\) is perverse (the argument of Mirković–Vilonen for convolution; de Cataldo–Migliorini for semismall maps).
  2. Semisimplicity and multiplicities. By D2 it is a sum of \(\mathrm{IC}_\nu\otimes E_\nu\); local systems are constant since stabilizers are connected (Proposition 3.6(b)). \(E_\nu\) is the stalk at \(x_\nu\) in degree \(-\delta(\nu)\), i.e. \(H^{2f}(F^\lambda_\nu)\), \(f=\tfrac12(\langle2\rho,\lambda\rangle-\delta(\nu))\): lower strata \(\mathrm{Gr}^{\lambda'}\) contribute only in degrees \(<-\delta(\nu)\), since \(\mathrm{IC}_{\mathrm{Gr}_{\le\lambda}}|_{\mathrm{Gr}^{\lambda'}}\) lives in degrees \(\le-\langle2\rho,\lambda'\rangle-2\). So \(\dim E_\nu=c_{\nu\lambda}\) (Proposition 3.8(b)): (a).
  3. (b). Every \(\nu\) is relevant for some \(\lambda\) (Proposition 3.8(d)), so \(\mathrm{IC}_\nu\) is a summand and \(\mathcal M=\Lambda_X^+\). Proposition I applies: H1 from Proposition 3.6(a),(d), H2 from D1, H3 from Proposition 3.6(b).
  4. (c). The top components are geometrically irreducible and defined over \(\mathbb F_q\) (Proposition 3.8(b)), so \(\mathrm{Fr}^n\) acts on \(H^{2f}\) by \(q^{nf}\); only \(i=0\) occurs.
  5. (d). IIb (iii) holds verbatim for \(X\) given D1–D3 (D3 by Proposition 3.6(b)): \((\mathrm{SG})_n\) iff the columns of \(m^{(n)}\) span. By (c), \(m^{(n)}=D_n\,c\,D_n'\) with invertible diagonal matrices, so iff the columns of \(c\) span, i.e. iff restriction of characters is onto. The class function \(\chi\mapsto\chi(a,b,a)-\chi(b,a,b)\) kills every restricted character (Lemma 3.3(d)) and takes \(b^2-a^2+b^{-2}-a^{-2}\ne0\) on \(V_{(0,2,0)}\), the label \(\nu=(0,0,2)\). \(\square\)

Status. All geometric and combinatorial inputs are proved in Proposition 3.6–Proposition 3.8; the corollary is conditional only on D1–D2 for \(X\) (Dead end 3.10). Its conclusion (d) is consistent with, and independent of, the unconditional Theorem 3.5. It also realizes condition (iii) of Proposition 3.2(c): sheaves generate because \(\iota\check G_X\) is a subgroup, functions fail because of fusion.

Builds on

  • Proposition 3.2 · Thick closure versus span: twisted traces, restriction of class functions, fusion
  • Proposition 3.6 · Orbits \((d_1,d_2,e)\), connected stabilizers, \(\delta=3d_1+3d_2+e\)
  • Proposition 3.8 · Each \(m_\lambda\) is semismall; multiplicities are dual branching numbers

Built on by

  • Dead end 3.10 · What blocks an unconditional \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) for \(X\)
  • Summary 3.13 · Verdict: \(\mathrm{SL}_2^3\backslash\mathrm{Sp}_4^2\) fails (SG); sheaves given D1–D2
  • Audit 5.9 · Audit of A3-9, A3-2(c): given D1–D2, \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\)
  • Remark 6.2 · Registry after the split agent and the first audit: audit the split candidate and the mechanism
  • Theorem 6.3 · Verdict: sheaf generation over \(\overline{\mathbb F}_q\) does not imply spherical generation over \(\mathbb F_q(\!(t)\!)\)

Sources