Proposition X4.7 · \(\operatorname{im}r=\{g:g(y_1)=g(\sigma y_1)\}\), of codimension one, every \(q\)

Write \(t=q^{1/2}\) and \(c_8:=q^{-3}+q^{-2}+q^{-1}+2+q+q^2+q^3\).

1. \(S^\sigma\). \(\sigma\) acts on \(\Lambda_S\) as \(-w_0\) (Computation 6.4(5)), so \(\langle W_X,\sigma\rangle=W_X\times\{\pm1\}\), the Weyl group of \(G_2\) acting on \(\Lambda_S\) with short roots \(W_X(1,1)\) and long roots \(\pm(3,0),\pm(0,3),\pm(3,-3)\). \(\Lambda_S\) is spanned by the short roots, so it is the root lattice of \(G_2\), which equals its weight lattice; the fundamental weights are \((1,1)\) and \((3,0)\), with \(W(G_2)\)-orbit sums \(M_{(1,1)}\) and \(P\). By Bourbaki, Lie VI §3.4, Thm. 1, \(S^\sigma=\mathbb C[M_{(1,1)},P]\) with \(M_{(1,1)},P\) algebraically independent.
2. \(S\) over \(S^\sigma\). \(S\) is generated by \(M_{(1,1)},M_{(3,0)},M_{(0,3)}\) (Lemma 6.2, Computation 6.4(4)). With \(M_{(0,3)}=P-M_{(3,0)}\), \(M_{(3,0)}M_{(0,3)}=M_{(3,3)}+3\in S^\sigma\) and \(M_{(3,0)}^2=P\,M_{(3,0)}-M_{(3,0)}M_{(0,3)}\), we get \(S=S^\sigma+S^\sigma M_{(3,0)}=S^\sigma+S^\sigma(M_{(3,0)}-M_{(0,3)})\); splitting into \(\sigma\)-eigenspaces gives the direct sum, with \(S^-=S^\sigma(M_{(3,0)}-M_{(0,3)})\).
3. \(S^\sigma\subseteq R\). Exact expansions: \(r(m_{\omega_2^\vee})=c_8M_{(1,1)}+c_0\) (Computation 6.4(6)); \(M_{(1,1)}^2=M_{(2,2)}+2P+2M_{(1,1)}+6\); and \(r(m_{\omega_1^\vee+\omega_6^\vee})=M_{(2,2)}+c_8P+d_1M_{(1,1)}+d_0\) with Laurent polynomials \(c_0,d_1,d_0\) in \(t\). So \(r(m_{\omega_1^\vee+\omega_6^\vee})-M_{(1,1)}^2=(c_8-2)P+(d_1-2)M_{(1,1)}+(d_0-6)\) with \(c_8-2=q^{-3}+q^{-2}+q^{-1}+q+q^2+q^3>0\). As \(R\) is a ring, \(M_{(1,1)},P\in R\), hence \(S^\sigma\subseteq R\).
4. Odd part. \(r\circ\sigma^*=\sigma^*\circ r\) and \(\sigma^*\) preserves \(\mathbb C[A^*]^W\) (Computation 6.4(5)), so \(R\) is \(\sigma\)-stable and \(R=S^\sigma\oplus(R\cap S^-)\), \(R\cap S^-=I\,(M_{(3,0)}-M_{(0,3)})\) for the ideal \(I:=\{a\in S^\sigma:a(M_{(3,0)}-M_{(0,3)})\in R\}\).
5. \(I\subseteq\mathfrak n\). By Theorem 6.6, \(R\subseteq\mathcal K\). For \(a\in S^\sigma\), \(g=a(M_{(3,0)}-M_{(0,3)})\) has \(g(y_1)-g(\sigma y_1)=2a(y_1)(M_{(3,0)}-M_{(0,3)})(y_1)\), and \((M_{(3,0)}-M_{(0,3)})(y_1)=q^6+q^3+q^{-9}-q^9-q^{-3}-q^{-6}\ne0\) for \(q>1\). So \(a(y_1)=0\).
6. \(\mathfrak n\subseteq I\). Exact identities in \(t\) (c6_image.py):

$$r(m_{\omega_1^\vee+\omega_3^\vee})-\sigma^*r(m_{\omega_1^\vee+\omega_3^\vee})=(M_{(3,0)}-M_{(0,3)})\,(c_8M_{(1,1)}-d),$$ $$r(m_{3\omega_1^\vee})-\sigma^*r(m_{3\omega_1^\vee})=(M_{(3,0)}-M_{(0,3)})\,(P+2-c),$$

with \(c=q^{-9}+q^{-6}+q^{-3}+2+q^3+q^6+q^9\) (the coefficient in \(r(m_{3\omega_1^\vee})=M_{(6,0)}+c\,M_{(0,3)}\), Computation 6.5) and \(d\) the coefficient of \(M_{(0,3)}\) in \(r(m_{\omega_1^\vee+\omega_3^\vee})\); they use \(M_{(1,1)}M_{(3,0)}=M_{(4,1)}+M_{(2,2)}+M_{(1,1)}\) and \(M_{(3,0)}^2=M_{(6,0)}+2M_{(0,3)}\). Both left sides lie in \(R\), so \(c_8M_{(1,1)}-d\) and \(P+2-c\) lie in \(I\subseteq\mathfrak n\). Evaluating at \(y_1\) gives \(d=c_8M_{(1,1)}(y_1)\) and \(c-2=P(y_1)\), so \(M_{(1,1)}-M_{(1,1)}(y_1)\in I\) and \(P-P(y_1)\in I\). Since \(S^\sigma\) is a polynomial ring in \(M_{(1,1)},P\), these generate \(\mathfrak n\): \(I=\mathfrak n\).

7. \(R=\mathcal K\). For \(g=a+b(M_{(3,0)}-M_{(0,3)})\), \(g\in\mathcal K\iff b(y_1)=0\). So \(\mathcal K=S^\sigma\oplus\mathfrak n(M_{(3,0)}-M_{(0,3)})=R\). As \(P\in R\), \(S=R\oplus\mathbb C(M_{(3,0)}-M_{(0,3)})=R\oplus\mathbb CM_{(3,0)}\). The functional \(g\mapsto g(y_1)-g(\sigma y_1)\) identifies \(S/R\) with \(\mathbb C\), and \(r(f)g\mapsto f(q^{\rho_L}y_1)\,(g(y_1)-g(\sigma y_1))\) because \(r(f)(y_1)=r(f)(\sigma y_1)\); \(q^{\rho_L}y_1=q^{\lambda_1^*}\) is \(W\)-conjugate to \(q^{\rho-\omega_4}\) (Theorem 6.6).
8. Units and cyclicity. A unit of \(S\) is a unit of \(\mathbb C[\Lambda_S]\), i.e. \(cy^\kappa\); \(W_X\)-invariance forces \(\kappa\) to be \(S_3\)-fixed, so \(\kappa=0\). If \(S=R\,s\), then \(1=r(f)s\) makes \(s\) a unit, \(s\in\mathbb C^\times\), and \(S=R\): contradiction with 7. \(S=R+RM_{(3,0)}\) gives two generators. Fibre: \(y_1\) and \(\sigma y_1\) are not \(W_X\)-conjugate (Theorem 6.6), so they give distinct maximal ideals of \(S\) containing \(\mathfrak m_\xi S\) (\(\mathfrak m_\xi\subseteq\mathbb C[A^*]^W\) the ideal of \(\xi\)); by the Chinese remainder theorem \(\dim S/\mathfrak m_\xi S\ge2\), and \(\le2\) because \(S=R+RM_{(3,0)}\) and \(R/\mathfrak m_\xi R=\mathbb C\).

This proves the conjecture of Conjecture 6.8 (\(\operatorname{im}r=\mathcal K\)) without the unramifiedness check it left open. Only the algebra of \(r\) is used: no hypothesis of Sakellaridis enters.