Proposition X4.3 · \(E_6/F_4\): type-\(G\) spherical roots, two colours; Statements 6.3.1, 7.1.5 hold

1. Roots. \(\theta=-w_L\) and \(w_L=-1\) on the span of \(\alpha_2,\dots,\alpha_5\). Projecting \(\alpha_1\) onto that span (it pairs only with \(\alpha_3\)) gives \(w_L\alpha_1=\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5\), the highest root of the \(D_5\) on nodes \(1,\dots,5\) (as on the family page). Hence \(\gamma_1=\alpha_1+w_L\alpha_1\) and \((\alpha_1,w_L\alpha_1)=2+(-1)\cdot2=0\). Pairing with the simple coroots gives \(\langle\alpha_1^\vee,\gamma_1\rangle=2\), \(\langle\alpha_6^\vee,\gamma_1\rangle=-1\) and \(0\) on the black coroots, i.e. \(\gamma_1=2\omega_1-\omega_6\). \(\gamma_6\) follows by \(1\leftrightarrow6\), \(3\leftrightarrow5\).
2. Lattices. \(G\) is simply connected, so \(\mathcal X(T)\) is the weight lattice. \(T^\theta\) is the torus of \(\alpha_2^\vee,\dots,\alpha_5^\vee\) (connected: \(S[2]\subseteq(T^\theta)^\circ\), family page §4) and \(B\cap H=B_L\cap L^\theta\) is a Borel subgroup of \(\mathrm{Spin}_8\) (Lemma 6.9). \(B\)-eigenfunctions on \(\mathring X=B/(B\cap H)\) have weights trivial on \(T\cap H=T^\theta\), and every such weight occurs; so \(\mathcal X(X)=\{\chi:\langle\alpha_j^\vee,\chi\rangle=0,\ j=2,\dots,5\}=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\). The cocharacter of \(A_X=T/T^\theta\) induced by \(\lambda\in\Lambda_T\) pairs with \(\chi\in\mathcal X(X)\) as \(\langle\lambda,\chi\rangle\); so \(\langle e_i,\omega_j\rangle=\delta_{ij}\).
3. Spherical roots. For a symmetric variety the valuation cone \(\mathcal V\) is the image of the antidominant chamber (Sakellaridis p. 8, citing [Kn91, §5]); the extremal rays of \(-\mathcal V^\vee\) are spanned by the restricted simple roots, whose primitive vectors in \(\mathcal X(X)\) are \(\gamma_1,\gamma_6\) (both primitive, since \(2\omega_1-\omega_6\) and \(-\omega_1+2\omega_6\) have coprime coordinates).
4. Normalization and type (p. 42, p. 44). \(\gamma_1\) is the sum of the orthogonal roots \(\alpha_1,w_L\alpha_1\). \(W(E_6)\) is transitive on ordered pairs of orthogonal roots: it is transitive on roots, and the roots orthogonal to a given root form an \(A_5\) on which its stabilizer, containing \(W(A_5)\), is transitive. So the pair is conjugate to \((\alpha_1,\alpha_6)\) and is simple for some basis; hence \(\gamma_1'=\gamma_1\). No root is proportional to \(\gamma_1\) (its \(\alpha_1\)-coefficient is \(2\) and \(\gamma_1/2\) is not in the root lattice), so \(\gamma_1\) has type \(G\). Type \(N\) roots are proportional to roots, so none occurs. Same for \(\gamma_6\).
5. Levis. \(\operatorname{supp}\gamma_1=\{1,2,3,4,5\}\), \(\operatorname{supp}\gamma_6=\{2,\dots,6\}\): both \(D_5\).
6. Colours. \(\mathrm{Pic}(X_{\bar F})\simeq\mathcal X(F_4)=0\) (\(G\) simply connected) and \(\bar F[X]^\times=\bar F^\times\) (\(G\) semisimple), so \(\bar F[X]\) is factorial and every colour is the divisor of an irreducible \(B\)-eigenfunction. The weights of \(\bar F[X]^{(B)}\) are the spherical weights \(a\omega_1+b\omega_6\), \(a,b\ge0\) (Helgason's condition: \(\langle\lambda,\bar\alpha_i\rangle/\langle\bar\alpha_i,\bar\alpha_i\rangle\) equals \(a\), resp. \(b\), for \(\bar\alpha_i=\gamma_i/2\)), each with a one-dimensional eigenspace; this monoid is free on two generators, so there are exactly two colours, with \(v_{D_i}(f_{\omega_j})=\delta_{ij}\), i.e. \(\check v_{D_i}=e_i\) (up to exchanging the names \(1\leftrightarrow6\), which depends on left/right conventions and is harmless by the diagram symmetry). Since \(\langle e_i,\gamma_j\rangle\) is the Cartan matrix of \(A_2\), \(e_i=\check\gamma_i\).
7. \(r\)-values. \(\langle\check\gamma_1,\rho_{P(X)}\rangle=\langle\alpha_1^\vee,\rho-\rho_L\rangle=1-(-3)=4\) (\(\langle\rho_L,\alpha_1^\vee\rangle=-3\), Computation 6.4); \(\rho-\rho_L\) vanishes on the black coroots, so it is a character of \(A_X\). Same for \(\gamma_6\).
8. \(\Theta\). Type-\(G\) colours are defined over \(F\), so \(\sigma=+\); the additions (2)–(3) of the definition on p. 59 concern type \(T\) only. The \(W_X=S_3\)-orbit of \(\check\gamma_1\) is the six coroots of \(\check\Phi_X\) and contains \(\check\gamma_6\); it is negation-closed. So \(\Theta=\check\Phi_X\times\{(+,4)\}\) with multiplicity one, and the cone of colour valuations \(\mathbb R_{\ge0}e_1+\mathbb R_{\ge0}e_6\) meets it in \(\check\Phi_X^+\).
9. Statement 7.1.5. Each coroot is positive or negative. \(w_{\gamma_1}\) is a simple reflection of \(\check\Phi_X\), so \(\{\check\theta\in\check\Phi_X^+:w_{\gamma_1}\check\theta<0\}=\{\check\gamma_1\}\); and the colours belonging to \(\gamma_1\) are those with \(\langle\check v_D,\gamma_1\rangle>0\) (Lemma 7.1.2, p. 58), i.e. \(D_1\) only (\(\langle\check\gamma_6,\gamma_1\rangle=-1\)). Same for \(\gamma_6\).
10. Statement 6.3.1. By steps 4–5 (no type \(N\), classical \(L_\gamma\)) and Theorem 6.3.2 (p. 45) it holds. Cross-check: \(\mathring XP_{\gamma_1}/R(P_{\gamma_1})\) is a spherical variety of rank one for the \(D_5\) group (the only spherical root in the span of \(\operatorname{supp}\gamma_1\) is \(\gamma_1\)), with spherical root of full support and no type \(N\); in the list of Theorem 6.11.1 (pp. 51–52) the only such entry is \(\mathrm{Spin}_9\backslash\mathrm{Spin}_{10}\), computed in §6.19 (p. 56) with \(n=5\): \(B_{w_\gamma}=-e^{\check\gamma}(1-q^{-n+1}e^{-\check\gamma})/(1-q^{-n+1}e^{\check\gamma})\), and \(n-1=4=\langle\check\gamma,\rho_{P(X)}\rangle\), matching formula (6.3).
11. \(\beta\). Substitute into (7.2). (Heuristic remark: this is Macdonald's \(c\)-function for the restricted root system \(A_2\) with all multiplicities \(8\), as expected for \(E_{6(-26)}\).)

Checked by exact computation in c1_a4.py: the roots and pairings of steps 1, 4, 7, the absence of any root \(\beta\) with \(\theta\beta=-\beta\), \(\Theta\) and Statement 7.1.5.