§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:49:06Z

Conjecture 8.11 · Heuristic finer structure: \(F_k\cong Q^7\), seven shifted copies, rational smoothness, and the dual-group reading

heuristic audit: correct

Second audit: correct.

Not needed for the verdict: the special fibre is conjecturally the seven-dimensional quadric, which would give seven shifted copies of \(\IC_{\mu_k}\) and a rationally smooth orbit closure along \(\cO_{\mu_k}\). The numbers match the dual group \(\PGL_3\) and the coefficient \(c_8(q)\) of the function-level computation.

Conjecture 8.11.

Conjecture (heuristic; not used in Summary 8.10). \(F_k\cong Q^7\), the quadric of isotropic lines in the nine-dimensional representation of \(Z_k\simeq\Spin_9\). If so, the multiplicity space \(E_{\mu_k}\) is \(k^7\) in degrees \(-6,-4,\dots,6\), the relative Kostka–Foulkes polynomial \(P_{(3k,0);(3k-1,2)}\) equals \(1\), and \(\overline{\cO_{\mu_k'}}\) is rationally smooth along \(\cO_{\mu_k}\).

  1. The quadric. \(Z_k\) acts on \(F_k\) with eight torus-fixed points and a seven-dimensional orbit \(\Spin_9/P_1=Q^7\) (Computation 8.6); \(\chi(Q^7)=8\) matches the eight fixed points. If \(F_k=Q^7\), then \(H^*(F_k)=k\) in degrees \(0,2,\dots,14\), and Lemma 8.7 gives \(E_{\mu_k,n}=k\) for \(n=0,2,4,6\). Verdier self-duality of \(M_\lambda\) (in the \(\delta\)-renormalized duality, \(E_{\mu,-n}\simeq E_{\mu,n}^*\)) adds \(n=-2,-4,-6\). The remaining class of \(H^*(F_k)[D_k]\), in degree \(-D_k\), is the stalk of \(\IC_{\mu_k'}\): a single class, so \(P_{(3k,0);(3k-1,2)}=1\) and the closure is rationally smooth along \(\cO_{\mu_k}\). This is the shape found for \(\mathrm{GL}_4/\mathrm{Sp}_4\) in Theorem 4.9 (fibre \(Q^3\), three copies, one IC class). It matches Chen–Yi, Cor. 14 and Remark 16, together with [CMNO22, Thm. 1.9]: for a symmetric variety of splitting rank the relative Kostka–Foulkes polynomials are those of \(\check G_X=\PGL_3\), and \(K^{A_2}_{(2,2),(3,0)}(t)=t\) becomes \(1\) after the dimension normalization.
  2. Dual-group reading. Seven copies in degrees \(-6,\dots,6\), in steps of \(2\), is exactly \(c_8(q)-1=q^{-3}+q^{-2}+q^{-1}+1+q+q^2+q^3\) from Computation 6.5, where \(r(m_{\omega_1^\vee+\omega_6^\vee})=M_{(2,2)}+c_8(M_{(3,0)}+M_{(0,3)})+\cdots\). That is the graded multiplicity of the \(\check G_X\)-representation of highest weight \((3,0)\) in the restriction of the \(650\)-dimensional representation, graded by \(2\langle\rho_L,\nu\rangle\): the eight weights \(\nu\in W\lambda_1\) over \((3,0)\) have \(\langle\rho_L,\nu\rangle\in\{-3,\dots,3\}\) (Computation 8.6), and one of them belongs to the copy of \(V_{(2,2)}\). Nadler's Table 1 lists \(\check{\mathfrak h}=\mathfrak{sl}_3\) for E IV, and his Theorems 1.2.2 and 1.2.3 identify the (non-perverse, \(\Z\)-graded) specialization with restriction to \(\check H\subset\check G\). Identifying \(V_\lambda\ast\IC_0\) with that specialization needs the real–symmetric equivalence of [CN24], which is not among the sources. This is a gap, and the reason the proof avoids this route.
  3. Albert algebra. Model \(X=\{N=1\}\subset J=H_3(\mathbb O)\), with \(S=\{\mathrm{diag}(s_1,s_2,s_3):s_1s_2s_3=1\}\) acting by \(x_{ij}\mapsto s_is_jx_{ij}\) (weights \(2\varepsilon_i\) once and \(\varepsilon_i+\varepsilon_j\) eight times, \(3+24=27\); restricted roots \(\pm(\varepsilon_i-\varepsilon_j)\) of multiplicity \(8\)), \(\tau(x)=U_x\), and \(x_\mu=\mathrm{diag}(t^{\mu_1},t^{\mu_2},t^{\mu_3})\); \(\mu_1=(2,-1,-1)\) and \(\mu_1'=(2,0,-2)\). A lattice in position \(\lambda_1\) has elementary divisors \(\{2,1^8,0^9,(-1)^8,-2\}\), and \(F_1\) should be the quadric of isotropic lines in the traceless part of the spin factor \(H_2(\mathbb O)\) on the two equal entries (\(26=1+9+16\) under \(\Spin_9\subset F_4\)). Standard facts about \(J\) (Springer–Veldkamp, Ch. 5 and 7) were not needed and are not verified here.

Verified by computation (elementary divisors): e6_fibre.py

Builds on

  • Computation 8.6 · Torus-fixed points of \(F_k\) and a seven-dimensional orbit of \(Z_{G^\theta}(\mu_k)^\circ\)
  • Theorem 8.8 · \(\IC_{3k\omega_1^\vee}\) is a direct summand of the Satake object \(V_{\lambda_k}\ast\IC_0\)
  • Computation 6.5 · \(E_6/F_4\): in every computed window the image of \(r\) has codimension one, cut out by a \(\sigma\)-odd functional

Built on by

  • Objection 7.31 · Objection to the mirror step and the verdict: generation holds, \(\sigma\) needs normalizing, and the divergence needs its hypotheses stated
Original record, as the agent wrote it at 03:49 UTC

Finer structure: the fibre is the 7-quadric, seven shifted copies, rational smoothness, and the dual-group reading

Not needed for A8-10; recorded as consistency checks and conjectures.

  1. Conjecture \(F_k\simeq Q^7\). \(Z_k\simeq\mathrm{Spin}_9\) acts on \(F_k\) with the 8 fixed points of \(T\) and a 7-dimensional orbit \(\mathrm{Spin}_9/P_1=Q^7\) (isotropic lines in the 9-dimensional representation; \(\chi(Q^7)=8\) matches the 8 fixed points). If \(F_k=Q^7\) exactly then \(H^*(F_k)=k\) in degrees \(0,2,\dots,14\), and A8-7 gives \(E_{\mu_k,n}=k\) for \(n=0,2,4,6\); Verdier self-duality of \(M_\lambda\) (in the δ-renormalized duality, \(E_{\mu,-n}\simeq E_{\mu,n}^*\)) adds \(n=-2,-4,-6\), and the remaining class of \(H^*(F_k)[D_k]\), in degree \(-D_k\), is the stalk of \(\mathrm{IC}_{\mu_k'}\): a single class, i.e. the relative Kostka–Foulkes polynomial \(P_{(3k,0);(3k-1,2)}=1\) and \(\overline{\mathcal O_{\mu_k'}}\) is rationally smooth along \(\mathcal O_{\mu_k}\). This is the same shape as (F2) for \(\mathrm{GL}_4/\mathrm{Sp}_4\) (fibre \(Q^3\), three copies, one IC class) and matches Chen–Yi Cor. 14/Remark 16 with [CMNO22, Thm 1.9]: the relative Kostka–Foulkes polynomials of a splitting-rank variety are those of \(\check G_X=\mathrm{PGL}_3\), and \(K^{A_2}_{(2,2),(3,0)}(t)=t\) becomes \(1\) after the dimension normalization.
  2. Dual-group reading. Seven copies in degrees \(-6,\dots,6\) step 2 is exactly \(c_8(q)-1=q^{-3}+q^{-2}+q^{-1}+1+q+q^2+q^3\) from A6-5 (\(r(m_{\omega_1^\vee+\omega_6^\vee})=M_{(2,2)}+c_8(M_{(3,0)}+M_{(0,3)})+\dots\)), i.e. the graded multiplicity of the \(\check G_X\)-representation of highest weight \((3,0)\) in \(650|_{\check G_X}\) with grading \(2\langle\rho_L,\nu\rangle\): the eight weights \(\nu\in W\lambda_1\) over \((3,0)\) have \(\langle\rho_L,\nu\rangle\in\{-3,\dots,3\}\) (A8-6(c)), one of them belonging to the copy of \(V_{(2,2)}\). Nadler's Table 1 (nadler-real.pdf) lists \(\check{\mathfrak h}=\mathfrak{sl}_3\) for E IV and Thm 1.2.2/1.2.3 identify the (non-perverse, Z-graded) specialization with restriction to \(\check H\subset\check G\); identifying \(V_\lambda\ast\mathrm{IC}_0\) with that specialization needs the real–symmetric equivalence [CN24], not in the sources — gap, and the reason the proof above avoids it.
  3. Albert algebra. With \(X=\{N=1\}\subset J=H_3(\mathbb O)\), \(S=\{\mathrm{diag}(s_1,s_2,s_3):s_1s_2s_3=1\}\) acting by \(x_{ij}\mapsto s_is_jx_{ij}\) (weights \(2\varepsilon_i\) once and \(\varepsilon_i+\varepsilon_j\) eight times: \(3+24=27\), restricted roots \(\pm(\varepsilon_i-\varepsilon_j)\) of multiplicity 8), \(\tau(x)=U_x\) and \(x_\mu=\mathrm{diag}(t^{\mu_1},t^{\mu_2},t^{\mu_3})\); \(\mu_1=(2,-1,-1)\), \(\mu_1'=(2,0,-2)\). The elementary divisors of a lattice in position \(\lambda_1\) are \(\{2,1^8,0^9,(-1)^8,-2\}\) (script), and \(F_1\) should be the quadric of isotropic lines in the traceless part of the spin factor \(H_2(\mathbb O)\) on the two equal entries (\(26=1+9+16\) under \(\mathrm{Spin}_9\subset F_4\)). Standard facts about \(J\) (Springer–Veldkamp, Octonions, Jordan algebras and exceptional groups, Ch. 5, 7) were not needed and are not verified here.

Sources