Proposition VII as a paper · statement and complete proof (PDF) →

Proof

Steps 1–6 prove (a): the σ-even part of the target lies in the image of \(r\), the gluing bounds the σ-odd part from above, two exact identities bound it from below, and constant units forbid a single generator. Steps 7–10 transport this to \(C_K\) through Theorem 8.0.2. Steps 11–12 prove (c), the input of step 7, and step 13 concludes.

1. Method The σ-invariants form a polynomial ring

   \(\langle W_X,\sigma\rangle\) is the Weyl group of \(G_2\) on \(\Lambda_S\), so its invariants are polynomial, and \(S\) is free of rank two over them:

   $$S^\sigma=\mathbb C[\Lambda_S]^{W_X\times\{\pm1\}}=\mathbb C[M_{(1,1)},P],\qquad S=S^\sigma\oplus S^\sigma D .$$
   Proof

   • σ on invariants. \(\sigma\) acts on \(\Lambda_S\) as \(-w_0\), and \(w_0\) is trivial on \(W_X\)-invariants, so \(\sigma\) acts on \(S\) as \(-1\in\mathrm{GL}(\Lambda_S)\).
   • \(G_2\). Short roots \(W_X(1,1)=\{\pm(1,1),\pm(2,-1),\pm(-1,2)\}\), long roots \(\pm(3,0),\pm(0,3),\pm(3,-3)\). The short roots span \(\Lambda_S\), which is therefore the root lattice of \(G_2\), equal to its weight lattice. The fundamental weights \((1,1)\), \((3,0)\) have orbit sums \(M_{(1,1)}\), \(P\); Bourbaki, Lie VI §3.4, Thm. 1 gives the polynomial ring.
   • Rank two. \(S\) is generated by \(M_{(1,1)},M_{(3,0)},M_{(0,3)}\) (Proposition VIII, step 2), with $$M_{(0,3)}=P-M_{(3,0)},\qquad M_{(3,0)}M_{(0,3)}=M_{(3,3)}+3\in S^\sigma,\qquad M_{(3,0)}^2=PM_{(3,0)}-M_{(3,0)}M_{(0,3)},$$ so \(S=S^\sigma+S^\sigma D\), and the \(\sigma\)-eigenspaces make the sum direct.
2. Result The even part lies in the image

   Two exact expansions put both generators of \(S^\sigma\) in \(R\):

   $$r(m_{\omega_2^\vee})=c_8M_{(1,1)}+c_0,\qquad r(m_{\omega_1^\vee+\omega_6^\vee})-M_{(1,1)}^2=(c_8-2)P+(d_1-2)M_{(1,1)}+(d_0-6),\qquad S^\sigma\subseteq R .$$
   Proof

   • Expansion. For dominant \(\mu\), \(r(m_\mu)=\sum_\kappa\big(\sum_{\nu\in W\mu,\ \varphi(\nu)=\kappa}q^{\langle\rho_L,\nu\rangle}\big)M_\kappa\), summed exactly over orbits of 72 and 270 elements.
   • Coefficients. \(c_0=q^{\pm5}+q^{\pm4}+3q^{\pm3}+3q^{\pm2}+4q^{\pm1}\); \(r(m_{\omega_1^\vee+\omega_6^\vee})=M_{(2,2)}+c_8P+d_1M_{(1,1)}+d_0\) with \(d_1=q^{\pm6}+2q^{\pm5}+3q^{\pm4}+2q^{\pm3}+3q^{\pm2}+4q^{\pm1}+2\), \(d_0=3(q^{\pm6}+q^{\pm4}+q^{\pm2}+2)\); and \(M_{(1,1)}^2=M_{(2,2)}+2P+2M_{(1,1)}+6\).
   • Conclusion. \(M_{(1,1)}=(r(m_{\omega_2^\vee})-c_0)/c_8\in R\); then \(P\in R\), since \(R\) is a ring and \(c_8-2=q^{\pm3}+q^{\pm2}+q^{\pm1}\gt0\).
3. Result Gluing bounds the odd part

   \(R\) is \(\sigma\)-stable, \(R=S^\sigma\oplus ID\) for an ideal \(I\) of \(S^\sigma\), and every element of \(R\) agrees at the glued points; since \(D(y_1)\ne0\), \(I\subseteq\mathfrak n\):

   $$R\subseteq\mathcal K,\qquad D(y_1)=-\big(q^3-q^{-3}\big)\big(q^{3/2}-q^{-3/2}\big)^2\big(q^3+1+q^{-3}\big)\ne0 .$$
   Proof

   • σ-stable. \(\sigma\) fixes \(\rho_L\) and normalizes \(W\) and \(W_X\), so \(r\circ\sigma^*=\sigma^*\circ r\); with step 2, \(R=S^\sigma\oplus ID\), \(I=\{a\in S^\sigma:aD\in R\}\).
   • Gluing. \(q^{\rho_L}y_1\) and \(q^{\rho_L}\sigma y_1\) are \(W(E_6)\)-conjugate to \(q^{\rho-\omega_4}\) (Proposition VIII, step 3), so \(g(y_1)=g(\sigma y_1)\) for \(g\in R\).
   • Values. \(M_{(3,0)}(y_1)=q^{-9}+q^3+q^6\), \(M_{(0,3)}(y_1)=q^{-6}+q^{-3}+q^9\). For \(a\in I\): \((aD)(y_1)-(aD)(\sigma y_1)=2a(y_1)D(y_1)\), so \(a(y_1)=0\).
4. Result Two exact identities

   Differences of \(\sigma\)-conjugate orbit sums produce both generators of \(\mathfrak n\) times \(D\), so \(I=\mathfrak n\):

   $$r(m_{\omega_1^\vee+\omega_3^\vee})-r(m_{\omega_5^\vee+\omega_6^\vee})=c_8\big(M_{(1,1)}-M_{(1,1)}(y_1)\big)D,\qquad r(m_{3\omega_1^\vee})-r(m_{3\omega_6^\vee})=\big(P-P(y_1)\big)D .$$
   Proof

   • Expansions. Orbits of 432 and 27 elements: \(r(m_{\omega_1^\vee+\omega_3^\vee})=c_8M_{(4,1)}+d\,M_{(0,3)}+(\text{terms in }M_{(2,2)},M_{(1,1)})\), \(r(m_{3\omega_1^\vee})=M_{(6,0)}+c\,M_{(0,3)}\), with $$d=c_8\,M_{(1,1)}(y_1)=c_8\,(q^{\pm5}+q^{\pm4}+q^{\pm1}),\qquad c=q^{\pm9}+q^{\pm6}+q^{\pm3}+2=P(y_1)+2,$$ the first an identity of Laurent polynomials in \(q\).
   • Products. \(\sigma^*m_\mu=m_{\sigma\mu}\), \(M_{(1,1)}D=M_{(4,1)}-M_{(1,4)}\), \(M_{(3,0)}^2=M_{(6,0)}+2M_{(0,3)}\), \(M_{(0,3)}^2=M_{(0,6)}+2M_{(3,0)}\); the \(\sigma\)-fixed terms cancel.
   • Generators. Both left sides lie in \(R\) and \(c_8\ne0\), so \(M_{(1,1)}-M_{(1,1)}(y_1)\), \(P-P(y_1)\in I\); they generate \(\mathfrak n\) because \(S^\sigma\) is the polynomial ring in \(M_{(1,1)},P\). With step 3, \(I=\mathfrak n\).
5. Result The image of r has codimension one

   The image is exactly the ring of functions agreeing at the glued points, and its cokernel is the character of \(\xi\):

   $$R=\mathcal K=S^\sigma\oplus\mathfrak nD,\qquad S=R\oplus\mathbb C\,M_{(3,0)},\qquad\ell(r(f)g)=f(\xi)\,\ell(g)\ \ \text{for}\ \ \ell(g)=g(y_1)-g(\sigma y_1).$$
   Proof

   • For \(g=a+bD\), \(a,b\in S^\sigma\): \(\ell(g)=2b(y_1)D(y_1)\), so \(g\in\mathcal K\iff b\in\mathfrak n\).
   • \(S/R\simeq S^\sigma/\mathfrak n=\mathbb C\), and \(M_{(3,0)}=\tfrac12(P+D)\equiv\tfrac12D\) modulo \(R\).
   • \(r(f)(y_1)=r(f)(\sigma y_1)=f(q^{\rho_L}y_1)=f(\xi)\). This proves the conjecture of run node A6-8.
6. Result Constant units, two generators

   A single generator would be a unit, and \(S\) has only constant units; the fibre over \(\xi\) is two-dimensional:

   $$S^\times=\mathbb C^\times,\qquad S=R\,s\implies s\in S^\times\implies S=R,\qquad S=R\cdot1+R\cdot M_{(3,0)},\qquad\dim S/\mathfrak m_\xi S=2 .$$
   Proof

   • Units. The units of \(\mathbb C[\Lambda_S]=\mathbb C[u^{\pm1},v^{\pm1}]\) are \(c\,y^\kappa\); \(W_X\)-invariance gives \(s_1\kappa=s_6\kappa=\kappa\), so \(\kappa=0\).
   • Not cyclic. \(1=r(f)s\) makes \(s\) a unit, hence constant, and \(S=R\) would contradict step 5.
   • Fibre. \(y_1,\sigma y_1\) are not \(W_X\)-conjugate, so \(\mathfrak m_{y_1}\ne\mathfrak m_{\sigma y_1}\), both containing \(\mathfrak m_\xi S\); the Chinese remainder theorem gives \(S/\mathfrak m_\xi S\twoheadrightarrow\mathbb C^2\). Conversely \(1,M_{(3,0)}\) span \(S/\mathfrak m_\xi S\) over \(R/r(\mathfrak m_\xi)\), a quotient of \(\mathbb C\).
7. Result Theorem 8.0.2 holds; transport through it

   (Sak) holds for \(\mathcal M_\theta\) over \(\mathbb F_q(\!(t)\!)\) and over every \(p\)-adic \(F\), \(p\) odd. So \(\Phi\) identifies the Hecke module with \(S\), the basic function with \(1\), and the Hecke span of a function with a principal \(R\)-submodule:

   $$\Phi(\mathcal H_K\varphi_0)=R,\qquad\Phi(\mathcal H_K\phi)=R\,\Phi(\phi).$$
   Proof

   • \(p\)-adic \(F\) · proved · cited. Steps 11–12 verify every hypothesis of Theorem 8.0.2 for \(\mathcal M_\theta\), so the theorem applies. For ramified \(F\), see run sakellaridis-char-p, Audit 4.7, Audit 5.5, Audit 7.3.
   • \(\mathbb F_q(\!(t)\!)\) · proved · run sakellaridis-char-p, audited. Proposition VI transfers the theorem from the fields \(\mathbb Q_q(p^{1/e})\), \(e\gg0\). It uses the Cartan decomposition \(K\backslash X(F)\simeq\Lambda_S^+\), proved twice (Theorem 3.8, Proposition 4.5), and the equality of all structure constants over close fields (Proposition 4.6); then \(\mathbf 1_{O_\kappa}\mapsto P_\kappa\) is \(\mathcal H_K\)-linear and unique (Theorem 4.8, audited in Audit 5.6).
   • Normalizations. The translation \(\delta(X)^{1/2}A_X^*\to A_X^*\), the sign of the shift and left versus right actions change \(r\) by automorphisms of source and target (Proposition VIII, step 1; run nodes A6-1, A6-7). \(\xi\) is fixed by \(-w_0\), since \(\sigma\) fixes node 4, so inverting characters does not move it.
8. Result C_K is not cyclic; the cokernel

   Steps 5–7 give (b)(1) and (b)(2):

   $$\mathcal H_K\phi\ne C_K,\qquad C_K=\mathcal H_K\varphi_0+\mathcal H_K\Phi^{-1}(M_{(3,0)}),\qquad C_K/\mathcal H_K\varphi_0\simeq S/R\simeq\mathbb C,\ \ h\mapsto\hat h(\xi).$$
   Proof

   • \(\mathcal H_K\phi=C_K\) would give \(S=R\,\Phi(\phi)\), excluded by step 6.
   • \(S=R+R\,M_{(3,0)}\) transports to two generators; step 5 gives the cokernel and its character.
9. Result The second generator

   \(\Phi\) sends the orbit indicator to Sakellaridis’s polynomial, whose σ-odd part is a nonzero multiple of \(D\); so it is not in \(R\), and it spans the cokernel:

   $$\Phi(\mathbf 1_{O_{3\omega_1^\vee}})=P_{(3,0)}\propto(1+q^{-4})M_\kappa+(1-q^{-8})M_{(1,1)}+(1-q^{-4}-q^{-8}+q^{-12}),\qquad\kappa\in\{(3,0),(0,3)\}.$$
   Proof

   • Orbit images. \(\Phi(\mathbf 1_{O_\kappa})=P_\kappa\) is part of (Sak), step 7. Over a \(p\)-adic \(F\) it comes from the proof of Theorem 8.0.2 (p. 64); over \(\mathbb F_q(\!(t)\!)\) the transfer defines \(\Phi\) by the same polynomials, which depend only on \(q\) (run sakellaridis-char-p, Theorem 4.8, step 5).
   • Lowest weight. \(P_\lambda\), defined by (8.1), has lowest weight \(\lambda\) for \(\succ_2\), with nonzero coefficient (proof on p. 64, using the Remark after Theorem 7.2.1, p. 62). The colour valuations are \(e_1,e_6\), and the dominant \(\kappa'\) with \((3,0)-\kappa'\in\mathbb Ne_1+\mathbb Ne_6\) are \((3,0)\), \((1,1)\), \((0,0)\).
   • Explicitly. From (7.4) with \(\Theta=\check\Phi_X\times\{(+,4)\}\), \(P_{(3,0)}\) is \(1+t\) times the Hall–Littlewood polynomial of \(A_2\) at \(3\varpi_1\), \(t=q^{-4}\); the convention fixes \(\kappa\). Its \(\sigma\)-odd part is \(\pm\tfrac12(1+q^{-4})D\ne0\), so \(P_{(3,0)}\notin R\), and \(\dim S/R=1\) gives \(S=R\oplus\mathbb C\,P_{(3,0)}\). The label \(3\omega_6^\vee\) follows by \(\sigma\).
10. Result Two eigenfunctionals with one character

    Evaluation at the two glued points gives independent eigenfunctionals with the same character, which a cyclic module cannot carry:

    $$\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi_\xi)=2,\qquad\dim\operatorname{Hom}(\mathcal H_K/\mathfrak a,\chi)\le1\ \text{for every ideal }\mathfrak a .$$
    Proof

    • \(\phi\mapsto\Phi(\phi)(y)\), \(y\in\{y_1,\sigma y_1\}\), has character \(h\mapsto\hat h(q^{\rho_L}y)=\hat h(\xi)\); the two differ on \(\Phi^{-1}(M_{(3,0)})\).
    • Every eigenfunctional with character \(\xi\) kills \(\mathfrak m_\xi C_K\), and \(\dim S/\mathfrak m_\xi S=2\) (step 6). The criterion re-proves (b)(1); this is the multiplicity at glued points that Sakellaridis describes after Corollary 8.0.4.
11. Result Sakellaridis’s hypotheses hold for the model

    For \(\mathcal M_\theta\), \(p\) odd, every hypothesis of Theorem 8.0.2 other than (A3a), step 12, holds over every \(p\)-adic \(F\) (proved · cited, audit records S4, S4b) and over \(\mathbb F_q(\!(t)\!)\), except the analytic part of Statement 6.3.1 (proved · run sakellaridis-char-p, audited: Summary 3.12, Summary 5.13). The invariants are the same in both characteristics:

    $$\#\{\text{maximal-rank Borel orbits}\}=\frac{|W|}{|W_X|\,|W_{P(X)}|}=\frac{51840}{6\cdot192}=45,\qquad\Delta_X=\{2\omega_1-\omega_6,\ -\omega_1+2\omega_6\}\ \text{(type }G),\qquad W_X\simeq S_3 .$$
    Proof

    • (A1), (A1ₚ), field and group. \(F\) has characteristic \(0\), or \(F=\mathbb F_q(\!(t)\!)\); \(G_{\mathcal O}\) is split, with connected fibres and simply connected derived group, for every \(p\).
    • (A2), the integral model. \(\theta_0\) is defined over \(\mathbb Z\) (audit node S4-11); \(H_{\mathcal O}\) is smooth with connected fibres of type \(F_4\); \(X\) is quasi-affine; \(B\cap H=B_L\cap L^\theta\), \(L^\theta=\mathrm{Spin}_8\); \(B/(B\cap H)\to X\) is an open immersion onto \(U\simeq U_P\times A_X\), and the open orbit equals \(U\) by EGA IV 21.12.7; invariant measures exist since \(\langle2\rho-2\rho_L,\alpha_j^\vee\rangle=0\), \(j=2,\dots,5\). Also \(X(\mathcal O)=Kx_0\) by Hensel and Lang. The first check cited Edixhoven, Prop. 3.4, Steinberg, Thm. 8.1, and Alper; over any complete discrete valuation ring with \(2\) invertible, Conrad’s Lemma 2.2.4 and Thm. 2.3.1 and a direct proof of connectedness replace them (run sakellaridis-char-p, Proposition 3.2).
    • (A3b), Cartan decomposition. Existence: Delorme–Sécherre, Thm. 1.1, give \(G(F)=\bigcup_{j\in J}H(F)\,y_j\,T(F)\,K\) over the \(H(F)\)-classes of maximal \((\theta,F)\)-split tori, and \(|J|=1\) because \(H^1(F,\mathrm{Spin}_8)=1\); so \(X(F)=K\cdot T(F)x_0\), and \(\dot n_i=\sigma_{\alpha_i}\theta(\sigma_{\alpha_i})\in H(\mathcal O)\cap N_G(T)\) (\(i=1,6\)) act on labels by the simple reflections of \(W_X\), so every \(K\)-orbit meets \(A_X^+x_0\) (audit nodes S4-12, S4-13). Distinctness from \(\tau(kx)=k\tau(x)\theta(k)^{-1}\), \(\theta(K)=K\) and \(G(F)=\bigsqcup_{\nu\in\Lambda_T^+}Kt^\nu K\). Over \(\mathbb F_q(\!(t)\!)\): Delorme–Sécherre’s theorem covers the field and its proof was checked (Audit 3.5); Bruhat–Tits III gives \(H^1(F,\mathrm{Spin}_8)=1\) in place of Kneser, and a direct argument gives one class of maximal \(\theta\)-split tori in place of Vust, Richardson and Helminck–Wang (Lemma 3.3, Proposition 3.4); so \(K\backslash X(F)\simeq\Lambda_S^+\) (Theorem 3.8; a second proof by counting, Proposition 4.5).
    • (A3c), Iwahori orbits. \(Jx_\kappa\subseteq B(\mathcal O)x_\kappa\): from \(J=U_P(\mathcal O)(J\cap L)U_P^-(\mathfrak p)\) at labels with a dominant lift, and at \(3k\omega_1^\vee,3k\omega_6^\vee\) (\(k\) odd) from the wall parabolic with Levi \(D_5\) and a central \(z\) with \(z^2=\zeta(\varpi)\), \(\zeta=3\omega_1^\vee\). Over \(\mathbb F_q(\!(t)\!)\) this needs \(F(\sqrt t)/F\) separable, which holds as \(p\) is odd (Audit 3.9).
    • (A4), spherical roots and colours. \(\gamma_1=2\omega_1-\omega_6=\alpha_1+w_L\alpha_1\) and \(\gamma_6=-\omega_1+2\omega_6\) have type \(G\), and no maximal-rank pair has type \(N\), as no root lies in \(\mathbb Q\omega_1\oplus\mathbb Q\omega_6\); \(L_\gamma\) is of type \(D_5\); colours have valuations \(e_i=\check\gamma_i\); \(\Theta=\check\Phi_X\times\{(+,4)\}\); Statement 7.1.5 holds, and Statement 6.3.1 by Theorem 6.3.2. Over \(\mathbb F_q(\!(t)\!)\) the types, colours and valuation cone are reproved without characteristic-0 inputs (in place of Knop and Helgason), and of Statement 6.3.1 the combinatorial part is checked (Proposition 3.6, Proposition 3.7, Proposition 3.10).
    • (A5), one open orbit. \(X_F\) is affine, and \(B\cap H\) is connected, solvable and split, so \(\mathring X(F)\) is one \(B(F)\)-orbit, for every \(p\) odd (Lemma 3.3).
    • Cited, not re-proved. Delorme–Sécherre, Thm. 1.1; Bruhat–Tits; Hensel and Lang; Conrad, Edixhoven and Steinberg; EGA IV; SGA 3, Exp. XXIII; and Sakellaridis’s Theorem 8.0.2 with the results of his 2008 paper it uses. The first check’s characteristic-0 citations (Kneser, Vust, Richardson, Helminck–Wang, Alper, Knop) are replaced by arguments valid for every odd \(p\), in both characteristics (Audit 3.1, Summary 3.12), and its simplification “\(F\) unramified over \(\mathbb Q_p\)” is removed (Audit 4.7, Audit 5.5). Audit nodes S4-3, S4-4, S4-5, S4-10 to S4-13. Names such as Proposition 3.2 or Audit 3.5 in this step are nodes of the run sakellaridis-char-p.
12. Result (A3a) holds for the model

    (A3a), rank-one models over \(\mathcal O\) (Sakellaridis’s Axiom 2.2.1), holds for \(\mathcal M_\theta\) over every discrete valuation ring with \(2\) invertible, mixed or equal characteristic, \(p=3\) included; for the \(45\times6=270\) pairs of a Borel orbit and a simple root (proved · run sakellaridis-char-p, audited: Theorem 2.8, Summary 6.9):

    $$YP_\alpha/R(P_\alpha)\simeq\begin{cases}\mathrm{PGL}_2\backslash\mathrm{PGL}_2&90\ \text{pairs }(Y,\alpha),\ \text{type }G,\\ B_2\backslash\mathrm{PGL}_2=\mathbb P^1_{\mathcal O}&180\ \text{pairs, type }U.\end{cases}$$
    Proof

    Proof outline

    • Tori. Counting \(\theta\)-eigenspaces on \(\mathfrak g\) (\(\dim\mathfrak g^\theta=52\), \(\dim\mathfrak g^{-\theta}=26\)) shows that every \(\theta\)-stable maximal torus is \(F_4\)-conjugate to \(T\) (Proposition 2.3, repaired in Audit 6.3).
    • Orbits. Over an algebraically closed field of characteristic \(\ne2\), every Borel orbit meets \(N_G(T)x_0\); the orbits are the 45 orbits \(B\dot wx_0\), all of maximal rank, \(p=3\) included (Proposition 2.3).
    • Orbit schemes. The stabilizers \(B_x\) are smooth over \(\mathbb Z[1/2]\), and \(B/B_x\to X\) is an immersion with smooth image (Lemma 2.4).
    • Rank-one quotients. \(YP_\alpha/R(P_\alpha)\) is a point (type \(G\), 90 pairs) or \(\mathbb P^1_{\mathcal O}\) (type \(U\), 180 pairs); types \(T\) and \(N\) never occur (Proposition 2.5, Computation 2.6; the table recomputed independently in Audit 6.1).
    • Sakellaridis’s scheme structure. By induction on closures from the closed orbit, the flat closure of each generic orbit has the expected special fibre; so Sakellaridis’s \(\mathcal O\)-scheme \(Y\) is the orbit scheme, and the complement clause holds for all 270 pairs, read inside closures (Lemma 2.7, Theorem 2.8; audited in Audit 6.6, Audit 6.7).
13. Conclusion C_K needs exactly two generators

    For \(E_6/F_4\), model \(\mathcal M_\theta\), over \(\mathbb F_q(\!(t)\!)\) and over every \(p\)-adic field, \(p\) odd: (SG) fails, and \(C_K\) needs exactly two generators, \(\varphi_0\) and \(\mathbf 1_{O_{3\omega_1^\vee}}\).

    The cokernel of \(\mathcal H_K\varphi_0\) is the character \(q^{\rho-\omega_4}\). No hypothesis remains open: the proof rests on Proposition VI and on the cited inputs of step 11. The algebra of (a) holds for every \(q\) and strengthens Proposition VIII, step 3 (run node A6-6), which stands as stated. At the sheaf level, over \(\overline{\mathbb F}_q\), every \(\mathrm{IC}\) sheaf is a summand of a Satake object (Proposition VIII).