§8 · Agent 8 · E₆/F₄: sheaf level · 2026-10-17T03:30:00Z

Computation 8.6 · Torus-fixed points of \(F_k\) and a seven-dimensional orbit of \(Z_{G^\theta}(\mu_k)^\circ\)

proved audit: correct

Second audit: correct.

The special fibre \(F_k\) contains exactly eight torus-fixed points, and the connected centralizer \(Z_k\) of \(\mu_k\) in \(F_4\), of type \(B_4\), has an orbit of dimension at least seven through each of them. Hence \(\dim F_k\ge7\), more than the detection lemma needs.

Computation 8.6.

Let \(Z_k:=Z_{G^\theta}(\mu_k)^\circ\), viewed in \(L^+G\) as constant loops, and let \(T'=(T^\theta)^\circ\) be the torus of the black coroots.

  1. \(Z_k\) fixes \(x_{\mu_k}\), hence acts on \(F_k\).
  2. \(\dim Z_k=36\) (type \(B_4\)), with
$$\operatorname{Lie}Z_k=\mathfrak t^\theta\oplus\bigoplus_{\alpha\in\Phi_L}\mathfrak g_\alpha\oplus\bigoplus_{\{\alpha,\theta\alpha\}}k(X_\alpha+\theta X_\alpha),$$

where \(\Phi_L\) is the root system of the black \(D_4\) and the last sum runs over the eight \(\theta\)-orbits of roots of the \(D_5\)-Levi \(Z_G(\mu_k)\) outside \(D_4\). The summands are \(T'\)-weight spaces with distinct weights.

  1. The points \(t^\nu L^+G\) (\(\nu\in\Lambda_T\)) of \(F_k\) are exactly those with \(\nu\in W\lambda_k\) and \(\varphi(\nu)=\mu_k\). There are exactly eight, and \(\langle\rho_L,\nu\rangle\) takes the values \(-3,-2,-1,0,0,1,2,3\) on them.
  2. For each such \(\nu\), \(\dim\big(\operatorname{Lie}Z_k\cap\operatorname{Lie}P(-\nu)\big)=29\), where \(P(-\nu)\subseteq G\) is the stabilizer of \(t^\nu L^+G\). Hence \(\dim Z_k\cdot t^\nu L^+G\ge7\) and \(\dim F_k\ge7\).

The Levi. \(\mu_k=(3k,0)\) pairs to \(3k>0\) with \(\alpha_1\) and to \(0\) with \(\alpha_2,\dots,\alpha_6\), so \(Z_G(\mu_k)\) is the Levi subgroup of type \(D_5\) on nodes \(2,\dots,6\), for every \(k\). It is \(\theta\)-stable because \(\theta\mu_k=-\mu_k\).

  1. \(Z_k\) fixes \(x_{\mu_k}\). For \(h\in Z_k\), \(\tau(hx_{\mu_k})=h\,t^{\mu_k}\theta(h)^{-1}=ht^{\mu_k}h^{-1}=t^{\mu_k}=\tau(x_{\mu_k})\), and the Cartan embedding \(\tau\) is injective. The fibre \(F_k\) is stable under the stabilizer of \(x_{\mu_k}\) because \(m\) is \(L^+G\)-equivariant.
  2. Lie algebra. \(\operatorname{Lie}Z_k=(\operatorname{Lie}Z_G(\mu_k))^\theta\). The 24 black roots contribute their root spaces, on which \(\theta\) acts trivially, since \(\theta\) fixes \(Z_{G^\theta}(S)=\Spin_8\) pointwise (family page, section 4). The 16 roots of the \(D_5\)-Levi outside \(D_4\) form 8 pairs \(\{\alpha,\theta\alpha\}\) with \(\theta\alpha\ne\alpha\), because \(\theta=-w_L\) negates the \(\alpha_6\)-coefficient; each pair contributes the line \(k(X_\alpha+\theta X_\alpha)\). With \(\dim\mathfrak t^\theta=4\) this gives \(4+24+8=36\): type \(B_4\), the group \(\Spin_9\) fixing the sphere \(\Spin_{10}/\Spin_9\) inside the Levi. Under \(T'\) these lines have as weights the distinct roots of \(B_4\).
  3. Fixed points. By Lemma 8.4, \(F_k\subseteq\Gr^{\lambda_k}\), so \(t^\nu L^+G\in F_k\) iff \(\nu\in W\lambda_k\) and \(t^{-\nu}x_{\mu_k}\in L^+X\). Since \(\tau(t^{-\nu}x_{\mu_k})=t^{\mu_k-\varphi(\nu)}\) and \(\tau^{-1}(L^+G)=L^+X\) (Chen–Yi, proof of Lemma 40), the second condition is \(\varphi(\nu)=\mu_k\). Write \(\nu=\tfrac{3k}2\omega_1^\vee+m\) with \(m=\tfrac12(\nu+\theta\nu)\) in the rational span of \(\alpha_2^\vee,\dots,\alpha_5^\vee\). Then \(|\nu|^2=3k^2+|m|^2\), while \(|\lambda_k|^2=\tfrac43(c_k^2+c_k+1)=3k^2+1\); so \(|m|^2=1\). Moreover \(m\) lies in the single coset \(\omega_5^{D_4}+Q(D_4)\), because \(\Lambda_T\cap\operatorname{span}(D_4)=Q(D_4)\) (family page, section 4) and \(\tfrac32\omega_1^\vee+\omega_5^{D_4}=2\alpha_1^\vee+2\alpha_2^\vee+3\alpha_3^\vee+4\alpha_4^\vee+3\alpha_5^\vee+\alpha_6^\vee\in\Lambda_T\). In \(D_5\)-coordinates \(\varepsilon_1',\dots,\varepsilon_5'\), with the \(D_4\) on \(\varepsilon_2',\dots,\varepsilon_5'\), this coset has exactly eight vectors of norm \(1\), namely \(m=\pm\varepsilon_j'\) for \(j=2,\dots,5\). Conversely each such \(\nu\) lies in \(W\lambda_k\): the Weyl group \(W_L\) of \(D_4\) moves \(m\) to \(\omega_5^{D_4}\), and \(\nu'=\tfrac{3k}2\omega_1^\vee+\omega_5^{D_4}=c_k\omega_1^\vee+\omega_5^\vee-\omega_6^\vee\) satisfies \(s_6\nu'=\nu'+\alpha_6^\vee=\lambda_k\), using \(\alpha_6^\vee=-\omega_5^\vee+2\omega_6^\vee\). So there are exactly eight such points, and \(\langle\rho_L,\nu\rangle=\langle\rho_L,m\rangle\) takes the values \(-3,-2,-1,0,0,1,2,3\), matching the eight terms of \(c_8(q)\) in Computation 6.4 and Computation 6.5.
  4. Orbit dimension. For a root \(\gamma\), \(t^{-\nu}u_\gamma(c)t^\nu=u_\gamma(t^{-\langle\gamma,\nu\rangle}c)\), so the stabilizer of \(t^\nu L^+G\) in \(G\) is the parabolic \(P(-\nu)\) generated by \(T\) and the \(U_\gamma\) with \(\langle\gamma,\nu\rangle\le0\). Since \(\operatorname{Lie}P(-\nu)\) is \(T\)-stable and the summands of part 2 are \(T'\)-weight spaces, the intersection is computed summand by summand: a black root space lies in \(\operatorname{Lie}P(-\nu)\) iff \(\langle\alpha,\nu\rangle\le0\), and a pair line iff \(\langle\alpha,\nu\rangle\le0\) and \(\langle\theta\alpha,\nu\rangle\le0\). Roots of the \(D_5\)-Levi have no \(\alpha_1\)-component, so \(\langle\gamma,\nu\rangle=\langle\gamma,m\rangle\), independent of \(k\). Take \(m=\varepsilon_2'\). The six black roots \(\varepsilon_2'\pm\varepsilon_j'\) (\(j=3,4,5\)) are positive on \(m\); the pairs are \(\{\pm\varepsilon_j'+\varepsilon_1',\pm\varepsilon_j'-\varepsilon_1'\}\), and only the pair with \(+\varepsilon_2'\) has a positive member. Hence the intersection has dimension \(4+18+7=29\), and since the dimension of a stabilizer is at most that of its Lie algebra in every characteristic,
$$\dim\big(Z_k\cdot t^\nu L^+G\big)\ \ge\ 36-29\ =\ 7,\qquad\text{so}\qquad\dim F_k\ge7 .$$

A computer check gives the same counts \((6,1,7)\) at all eight points for \(k=1,3,5,7\).

Here \(7=\dim\Spin_9/P_1=\dim Q^7\), the quadric of isotropic lines in the nine-dimensional representation; see Conjecture 8.11.

Verified by computation: e6_fibre.py

Builds on

  • Lemma 8.3 · The Hecke label \(\lambda_k\): a dominant lift of \(\mu_k'\) whose cell has dimension \(\delta(\mu_k')\)
  • Lemma 8.4 · Both convolution fibres lie in the open cell \(\Gr^{\lambda_k}\)

Built on by

  • Theorem 8.8 · \(\IC_{3k\omega_1^\vee}\) is a direct summand of the Satake object \(V_{\lambda_k}\ast\IC_0\)
  • Conjecture 8.11 · Heuristic finer structure: \(F_k\cong Q^7\), seven shifted copies, rational smoothness, and the dual-group reading
  • Audit 7.29 · Audit of the fixed-point computation: 8 fixed points, \(\dim Z_k=36\), stabilizer algebra of dimension \(29\), so \(\dim F_k\ge7\)
Original record, as the agent wrote it at 03:30 UTC

Torus-fixed points of the special fibre and a seven-dimensional stabilizer orbit through them

Let \(Z_k:=Z_{G^\theta}(\mu_k)^\circ\subseteq G\subseteq L^+G\) (constant loops). \(\mu_k=(3k,0)\) pairs to \(3k>0\) with \(\alpha_1\) and to \(0\) with \(\alpha_2,\dots,\alpha_6\), so \(Z_G(\mu_k)\) is the Levi of type \(D_5\) on nodes \(2,\dots,6\) for every \(k\), θ-stable since \(\theta\mu_k=-\mu_k\).

(a) \(Z_k\) stabilizes \(x_{\mu_k}\) and acts on \(F_k\). For \(h\in Z_k\): \(\tau(hx_{\mu_k})=h\,t^{\mu_k}\theta(h)^{-1}=ht^{\mu_k}h^{-1}=t^{\mu_k}=\tau(x_{\mu_k})\), and \(\tau\) is injective. \(F_k\) is \(\operatorname{Stab}(x_{\mu_k})\)-stable by definition.

(b) Lie algebra of \(Z_k\). \(\operatorname{Lie}Z_k=\operatorname{Lie}Z_G(\mu_k)^{\theta}=\mathfrak t^\theta\oplus\bigoplus_{\alpha\in\Phi_L}\mathfrak g_\alpha\oplus\bigoplus_{\{\alpha,\theta\alpha\}}k(X_\alpha+\theta X_\alpha)\), where \(\Phi_L\) are the 24 roots of the black \(D_4\) (θ acts trivially on \(\mathfrak g_\alpha\), page §4: θ fixes \(\mathrm{Spin}_8\) pointwise) and the 16 roots of \(D_5\) outside \(D_4\) form 8 θ-pairs with \(\theta\alpha\ne\alpha\) (θ\(=-w_L\) negates the \(\alpha_6\)-coefficient; script asserts this). So \(\dim Z_k=4+24+8=36\) (type \(B_4\): \(\mathrm{Spin}_9\), the fixed group of the sphere \(\mathrm{Spin}_{10}/\mathrm{Spin}_9\) inside the Levi). Each displayed line is a full \(T^\theta\)-weight space, the weights being distinct \(B_4\)-roots.

(c) Fixed points. \(t^\nu L^+G\in F_k\iff\nu\in W\lambda_k\) and \(\varphi(\nu)=\mu_k\): indeed \(\tau(t^{-\nu}x_{\mu_k})=t^{\mu_k-\varphi(\nu)}\) and \(\tau^{-1}(L^+G)=L^+X\) (Chen–Yi, proof of Lemma 40). Write \(\nu=\tfrac{3k}2\omega_1^\vee+m\) with \(m=\nu^\theta\in\operatorname{span}_{\mathbb Q}(\alpha_2^\vee,\dots,\alpha_5^\vee)\). Then \(|\nu|^2=3k^2+|m|^2\) and \(|\lambda_k|^2=\tfrac43(c_k^2+c_k+1)=3k^2+1\), so \(|m|^2=1\); moreover \(m\in\omega_5^{D_4}+Q(D_4)\) (one coset, since \(\Lambda_T\cap\operatorname{span}(D_4)=Q(D_4)\), page §4, and \(\tfrac32\omega_1^\vee+\omega_5^{D_4}=(2,2,3,4,3,1)\in\Lambda_T\)). That coset has exactly 8 vectors of norm 1 (in \(D_5\)-coordinates \(\varepsilon_1',\dots,\varepsilon_5'\) with \(D_4\) on \(\varepsilon_2',\dots,\varepsilon_5'\): \(m=\pm\varepsilon_j'\), \(j=2,\dots,5\)). Conversely each such \(\nu\) lies in \(W\lambda_k\): \(W_L\) moves \(m\) to \(\omega_5^{D_4}\), and \(\nu'=\tfrac{3k}2\omega_1^\vee+\omega_5^{D_4}=c_k\omega_1^\vee+\omega_5^\vee-\omega_6^\vee\) satisfies \(s_6\nu'=\nu'+\alpha_6^\vee=\lambda_k\) (\(\alpha_6^\vee=-\omega_5^\vee+2\omega_6^\vee\)); \(\nu'\in\Lambda_T\) because \(\nu'=(2,2,3,4,3,1)+\tfrac{k-1}2\cdot3\omega_1^\vee\). So \(F_k^T\) has exactly 8 points, and \(\langle\rho_L,\nu\rangle\) takes the values \(-3,-2,-1,0,0,1,2,3\) on them (script), matching the eight terms of \(c_8(q)\) in A6-4/A6-5.

(d) Orbit dimension. For \(\nu=\tfrac{3k}2\omega_1^\vee+m\) the stabilizer of \(t^\nu L^+G\) in \(G\) is \(P(-\nu)=\{\langle\gamma,\nu\rangle\le0\}\) (\(t^{-\nu}u_\gamma(c)t^\nu=u_\gamma(t^{-\langle\gamma,\nu\rangle}c)\)). Since \(\operatorname{Lie}P(-\nu)\) is \(T\)-stable and the lines of (b) are \(T^\theta\)-weight spaces, \(\operatorname{Lie}Z_k\cap\operatorname{Lie}P(-\nu)=\mathfrak t^\theta\oplus\bigoplus_{\alpha\in\Phi_L,\langle\alpha,\nu\rangle\le0}\mathfrak g_\alpha\oplus\bigoplus_{\text{pairs with both}\le0}k(X_\alpha+\theta X_\alpha)\). Roots of the \(D_5\)-Levi have no \(\alpha_1\)-component, so \(\langle\gamma,\nu\rangle=\langle\gamma,m\rangle\), independent of \(k\). With \(m=\varepsilon_2'\): long roots \(\varepsilon_2'\pm\varepsilon_j'\) (\(j=3,4,5\)) give 6 positive values; the pairs are \(\{\pm\varepsilon_j'+\varepsilon_1',\pm\varepsilon_j'-\varepsilon_1'\}\) and only the pair with \(+\varepsilon_2'\) has a positive member. Hence \(\dim(\operatorname{Lie}Z_k\cap\operatorname{Lie}P(-\nu))=4+18+7=29\), and $$\dim\big(Z_k\cdot t^\nu L^+G\big)\ \ge\ 36-29\ =\ 7,\qquad\text{so}\qquad\dim F_k\ge7 .$$ (\(\dim\operatorname{Stab}\le\dim(\operatorname{Lie}Z_k\cap\operatorname{Lie}P)\) holds in every characteristic.) Script: for all 8 fixed points and \(k=1,3,5,7\) the counts are \((6,1,7)\). \(\square\)

(\(7=\dim\mathrm{Spin}_9/P_1=\dim Q^7\), the quadric of isotropic lines in the 9-dimensional representation; see A8-11.)

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