§3 · Agent 3 · Hypotheses in char p · 2026-10-18T02:59:01Z

Proposition 3.7 · Char \(\ne2\): two colours, the valuation cone, spherical roots \(\gamma_1,\gamma_6\)

proved audit: minor defect ★ key step

Minor defect (second audit). Step 3 correctly shows that k[X] is factorial with constant units, that there are exactly two colours, and that their weights form a Z-basis of Z omega1 + Z omega6 made of dominant weights. But 'i.e. {omega1, omega6}' does not follow: {omega1, omega1+omega6} and {2omega1+omega6, omega1+omega6} are also such bases. Statement (2) (v_Di = e_i, weight monoid N omega1 + N omega6, V(omega1)^H != 0, which colour P_a1 moves), the upper bound in step 5 (l = f1|X1 of weight omega1 generating the 10-dimensional E) and Proposition 3.10(iv) all depend on this identification. Two sentences repair it, using a fact the agent states in Audit 3.9: the pinned diagram automorphism delta commutes with C and fixes sigma_L, hence commutes with theta0, preserves T, B, H and x0, permutes the colours and swaps omega1 and omega6; a delta-stable Z-basis of Z^2 inside N^2 is the standard one. I verified this repair. The rest I re-derived: the lower bound via kappa(s) over k((s)), giving v_kappa = -mu-bar with no dominant lift needed; the upper bound (weight of F1 = omega6; F1(x0) != 0 because my computation gives omega1, omega6-omega1 (level 0) and -omega6 (level 2) as the weights of V(omega1) in X(X)); and (4) (normalization per Sakellaridis p. 42, A2 Cartan matrix).

Replaces Helgason's theorem and Knop's computation of the valuation cone (S4-3 steps 3 and 6) by characteristic-free arguments: \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\); the colours are the zero loci of regular \(B\)-eigenfunctions of weights \(\omega_1,\omega_6\), with valuations \(e_1,e_6\); valuations along the arcs \(t^\mu x_0\) give the lower bound and localization to the two \(\theta\)-stable \(D_5\)-Levis, with the invariant quadratic form of \(\mathrm{Spin}_{10}\), the upper bound, so \(\mathcal V=\{\langle v,\gamma_1\rangle\le0,\ \langle v,\gamma_6\rangle\le0\}\) and \(\Lambda_X^+=\Lambda_X\cap\mathcal V\).

Proposition 3.7.

Let \(k\) be algebraically closed, \(\mathrm{char}\,k\ne2\), \(X=G_k/H_k\). Let \(\Lambda_X^+\subseteq\Lambda_X=\mathrm{Hom}(\mathcal X(X),\mathbb Z)\) be the restrictions to \(B\)-eigenfunctions of the \(G\)-invariant \(\mathbb Z\)-valued valuations of \(k(X)\) trivial on \(k\), and \(\mathcal V\) the cone they span.

  1. \(\mathcal X(X)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\).
  2. \(k[X]^\times=k^\times\), \(\mathrm{Pic}\,X=0\), \(k[X]\) is factorial. \(X\) has exactly two colours \(D_1=\mathrm{div}(f_1)\), \(D_6=\mathrm{div}(f_6)\), with \(f_i\in k[X]\) a \(B\)-eigenfunction of weight \(\omega_i\); \(v_{D_i}=e_i\); the weights of regular \(B\)-eigenfunctions form \(\mathbb N\omega_1\oplus\mathbb N\omega_6\); \(V(\omega_1)^H\ne0\ne V(\omega_6)^H\). \(D_1\) is moved by \(P_{\alpha_1}\) and stable under \(P_{\alpha_6}\); \(D_6\) the reverse.
  3. \(\mathcal V=C^-:=\{v:\langle v,\gamma_1\rangle\le0,\ \langle v,\gamma_6\rangle\le0\}\) and \(\Lambda_X^+=\Lambda_X\cap C^-\).
  4. The spherical roots are \(\gamma_1=2\omega_1-\omega_6\) and \(\gamma_6=-\omega_1+2\omega_6\); both are normalized (\(\gamma_i=\alpha_i+\beta_i\), a sum of two orthogonal roots simple for some basis) and of type \(G\); \(\mathrm{supp}\,\gamma_1=\{1,\dots,5\}\), \(\mathrm{supp}\,\gamma_6=\{2,\dots,6\}\), so \(L_{\gamma_1},L_{\gamma_6}\) are of type \(D_5\); \(\check\gamma_i=e_i\), \(\langle e_i,\gamma_j\rangle\) is the Cartan matrix of \(A_2\); \(W_X=\langle s_{\gamma_1},s_{\gamma_6}\rangle\simeq S_3\) acts on \(\Lambda_X\otimes\mathbb Q\) with fundamental domain \(\mathcal V\).
  1. (1). \(\mathring X=B/B_8\) (Proposition 3.2), so the weights of rational \(B\)-eigenfunctions are the characters of \(T\) trivial on \(T^\theta=T_8\), namely \((1-\theta^*)X^*(T)\); as the elementary divisors of \(1-\theta^*\) are \((1,1)\) (computed), this is the saturated lattice \(X^*(T)\cap(\mathbb Q\omega_1+\mathbb Q\omega_6)=\mathbb Z\omega_1\oplus\mathbb Z\omega_6\).
  2. Units, Pic. A unit of \(k[G/H]\) pulls back to a unit of \(k[G]\), which is constant (\(G\) semisimple, Rosenlicht). \(\mathrm{Pic}\,G=0\) for simply connected \(G\) (Popov 1974, Fossum–Iversen 1973; characteristic-free), so line bundles on \(G/H\) correspond to characters of \(H\) (as \(k[G]^\times=k^\times\)), and \(X^*(H)=X^*(F_4)=0\). \(X\) is smooth and affine (Lemma 3.3(4)), hence factorial.
  3. Colours. \(X\smallsetminus\mathring X\) has pure codimension 1 (\(\mathring X\) affine; EGA IV 21.12.7), and its components are exactly the \(B\)-stable prime divisors, i.e. the colours (\(X\) is homogeneous). Each colour is \(\mathrm{div}(f_D)\), \(f_D\in k[X]\) a \(B\)-eigenfunction (factoriality, constant units) of weight \(\lambda_D\), dominant (a \(U\)-invariant weight vector of a rational \(G\)-module has dominant weight; Jantzen, RAG II.2). Units of \(k[\mathring X]=k[U_P\times A_X]\) are \(k^\times\cdot X^*(A_X)\); each is \(c\prod f_D^{n_D}\) (its divisor lies on \(X\smallsetminus\mathring X\)). So \(\{\lambda_D\}\) is a basis of \(\mathbb Z\omega_1\oplus\mathbb Z\omega_6\) made of dominant weights, i.e. \(\{\omega_1,\omega_6\}\): two colours, \(v_{D_i}(f_j)=\delta_{ij}\), \(v_{D_i}=e_i\). Frobenius reciprocity: \(f_6\) is a highest weight vector of weight \(\omega_6\) in \(k[X]=\mathrm{Ind}_H^Gk\), so \(0\ne\mathrm{Hom}_G(V(\omega_6),\mathrm{Ind}_H^Gk)=(V(\omega_6)^*)^H=V(\omega_1)^H\) (\(V(\omega_6)\) minuscule, irreducible); symmetrically \(V(\omega_6)^H\ne0\). A \(B\)-eigenfunction of weight \(\chi\) is \(P_\alpha\)-semi-invariant iff \(\langle\chi,\alpha^\vee\rangle=0\) (the \(L_\alpha\)-module it generates is a quotient of a Weyl module of \(L_\alpha\)), so \(D_6\) is \(P_{\alpha_1}\)-stable and \(D_1\) is not.
  4. Lower bound \(C^-\subseteq\mathcal V\), and \(\Lambda_X\cap C^-\subseteq\Lambda_X^+\). Let \(\kappa\in\Lambda_S\) be dominant and \(\mu\in X_*(T)\) with \(\varphi(\mu)=\kappa\) (\(\varphi(\Lambda_T)=\Lambda_S\), family page §2). Put \(v_\kappa(f):=\mathrm{ord}_t\,f(g\,\mu(t)\,x_0)\), via \(k(X)\hookrightarrow k(G)(\!(t)\!)\) induced by the dominant map \(G\times\mathbb G_m\to X\), \((g,t)\mapsto g\mu(t)x_0\): a \(G\)-invariant \(\mathbb Z\)-valued valuation trivial on \(k\). Over \(k(\!(s)\!)\), \(s^2=t\): \(\tau(\kappa(s)x_0)=\kappa(s)^2=\kappa(t)=\tau(\mu(t)x_0)\), so \(\mu(t)x_0=\kappa(s)x_0\). For \(g=bu^-\) generic (\(b\in B\), \(u^-\in U^-\)), \(\kappa(s)^{-1}u^-\kappa(s)\to u_0\in U^-\cap Z_G(\kappa)\) (\(\kappa\) dominant) and \(u_0x_0\in\mathring X\) for generic \(u^-\) (\(\mathring X\) is open and contains \(x_0\)); hence \(\kappa(s)^{-1}u^-\kappa(s)x_0=b(s)x_0\) with \(b(s)\in B(k(U^-)[[s]])\) (Hensel for the smooth surjection \(U_P\times T\to\mathring X\)), and \(f_\chi(bu^-\kappa(s)x_0)=(\text{unit})\cdot s^{-\langle\kappa,\chi\rangle}f_\chi(x_0)\). So \(v_\kappa(f_\chi)=-\langle\kappa,\chi\rangle/2=-\langle\mu,\chi\rangle\) (for \(\chi\in\mathcal X(X)\), \(\theta\chi=-\chi\) and \(\langle\varphi(\mu),\chi\rangle=2\langle\mu,\chi\rangle\)). As \(\kappa\) runs over \(\Lambda_S^+\), \(-\bar\mu\) runs over \(\Lambda_X\cap C^-\) (\(\langle-\bar\mu,\gamma_i\rangle=-\tfrac12\langle\kappa,\gamma_i\rangle\)). This needs no dominant lift of \(\kappa\): the wall labels \(3k\omega_1^\vee,3k\omega_6^\vee\) (\(k\) odd) are covered.
  5. Upper bound \(\Lambda_X^+\subseteq C^-\). Let \(M_1\) be the standard Levi of \(\{1,\dots,5\}\), \(Q_1=M_1U_{Q_1}\). \(\theta\) negates \(\alpha_6\)-coefficients, so \(\theta(\Phi_{M_1})=\Phi_{M_1}\), \(\theta(U_{Q_1})=U_{Q_1}^-\), \(Q_1\cap H=M_1^\theta\), and as in Proposition 3.2(4) \(U_{Q_1}\times X_1\to X\), \(X_1:=M_1x_0\simeq M_1/M_1^\theta\), is an open immersion (\(16+10=26\)). With \(\mathrm{pr}:Q_1x_0\to X_1\), a \(G\)-invariant valuation \(v\) gives an \(M_1\)-invariant valuation \(v'=v\circ\mathrm{pr}^*\) of \(k(X_1)\) (\(\mathrm{pr}\) is \(Q_1\)-equivariant), and \(f_\chi|_{Q_1x_0}=\mathrm{pr}^*(f_\chi|_{X_1})\) (\(U_{Q_1}\subseteq U\)), so \(\langle v,\chi\rangle=v'(f_\chi|_{X_1})\). Let \(\ell=f_1|_{X_1}\) and \(E=\mathrm{span}(M_1\ell)\). \([M_1,M_1]=\mathrm{Spin}_{10}\) and \(\omega_1\) restricts to its vector weight (node 1 ends the long arm of the \(D_5\) diagram \(1-3-4\{2,5\}\)), so \(E\) is a nonzero quotient of the Weyl module of highest weight \(\omega_1\), which is the natural 10-dimensional module (irreducible for \(p\ne2\)): \(E\simeq V_{10}\) up to a central character. Let \(Q^*\in\mathrm{Sym}^2E\) be the \(\mathrm{Spin}_{10}\)-invariant nondegenerate quadratic form on \(E^*\) and \(F_1\in k[X_1]\) its image (a sum of products of two elements of \(E\)). \(F_1\) is \(M_1\)-semi-invariant; its weight vanishes on \(\alpha_1^\vee,\dots,\alpha_5^\vee\) and lies in \(2\omega_1+\mathbb Z\{\alpha_1,\dots,\alpha_5\}\), so it is \(\omega_6\) (since \(\gamma_1=2\omega_1-\omega_6\) lies in that lattice). \(F_1\ne0\): the evaluation \(\iota(x_0)\in E^*\) is \(T^\theta\)-fixed, so supported on the weights of \(E\) lying in \(\mathcal X(X)\), which are \(\omega_1\) and \(\omega_1-\gamma_1=\omega_6-\omega_1\) (the weights of \(V(\omega_1)\) in \(\mathcal X(X)\) are \(\omega_1,\omega_6-\omega_1,-\omega_6\), computed, and \(-\omega_6=\omega_1-(\gamma_1+\gamma_6)\) has nonzero \(\alpha_6\)-coefficient); \(\dot n_1\in M_1^\theta\) (\(\alpha_1,\beta_1\in\Phi_{M_1}\)) fixes \(\iota(x_0)\) and exchanges the two weight lines, so both coordinates are nonzero; these are the highest and lowest weights of \(E\), paired by \(Q^*\), so \(F_1(x_0)=c\,\ell_{\omega_1}(x_0)\ell_{\omega_6-\omega_1}(x_0)\ne0\). Hence \(F_1=c'f_6|_{X_1}\), \(c'\ne0\) (eigenfunction spaces of \(B\cap M_1\) on the spherical \(X_1\) are lines). As \(v'(e)\ge v'(\ell)\) for \(e\in E\), \(v'(F_1)\ge2v'(\ell)\): \(\langle v,\omega_6\rangle\ge2\langle v,\omega_1\rangle\), i.e. \(\langle v,\gamma_1\rangle\le0\). The Levi \(M_6\) of \(\{2,\dots,6\}\) (\(\theta\) negates \(\alpha_1\)-coefficients; \(\omega_6\) is its vector weight; \(\dot n_6\in M_6^\theta\)) gives \(\langle v,\gamma_6\rangle\le0\).
  6. (4). \(-\mathcal V^\vee=\mathrm{cone}(\gamma_1,\gamma_6)\) and \(\gamma_1,\gamma_6\) are primitive in \(\mathcal X(X)\). \(\gamma_1=\alpha_1+\beta_1\) with \((\alpha_1,\beta_1)=0\), and \(\alpha_1\pm\beta_1\) are not roots (computed); \(W(E_6)\) is transitive on ordered pairs of orthogonal roots (a statement about the finite root system, S4-3 step 4), so the pair is simple for some basis; a sum of two orthogonal roots has squared length 4, so no other positive multiple of \(\gamma_1\) is such a sum, and \(\gamma_1/2\) is not in the root lattice: normalized, type \(G\). Same for \(\gamma_6\). \(\langle e_1,\gamma_1\rangle=\langle\alpha_1^\vee,\gamma_1\rangle=2\), \(\langle e_1,\gamma_6\rangle=-1\), and symmetrically. \(W_X\) acts by \(s_{\gamma_1},s_{\gamma_6}\) (Proposition 3.6(2)), and \(C^-\) is the antidominant chamber of the restricted \(A_2\).

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