§6 · Agent 6 · E₆/F₄: restriction map · 2026-10-17T02:07:00Z

Lemma 6.2 · Two criteria for the surjectivity of \(r\): monoid generators suffice, and a glued pair of points forbids it

proved audit: correct

Second audit: correct. No defect. Generation criterion (leading term M_{kappa+kappa'} with coefficient 1, finite lower sets), gluing criterion (W-invariants agree on W-conjugate points; finite-group invariants separate orbits) and the finite-morphism remark all check.

Under Theorem E, spherical generation is the surjectivity of \(r\). Surjectivity can be tested on the orbit sums of finitely many monoid generators of \(\Lambda_X^+\), and it fails whenever two points of the shifted torus that are not \(W_X\)-conjugate become \(W\)-conjugate, the mechanism of Sakellaridis's Example 8.0.1.

Lemma 6.2.

Assume the identification of Theorem E: \(C_K\simeq\C[\delta(X)^{1/2}A_X^*]^{W_X}\) with \(\varphi_0\mapsto1\) and \(\cH_K\simeq\C[A^*]^W\) acting through \(r\), so that \(\cH_K\cdot\varphi_0\) corresponds to \(\operatorname{im}r\). For \(\kappa\in\Lambda_X^+\) put \(M_\kappa:=\sum_{\kappa'\in W_X\kappa}y^{\kappa'}\).

  1. Generation criterion. If \(g_1,\dots,g_s\) generate the monoid \(\Lambda_X^+\), then
$$r\ \text{is onto}\iff M_{g_1},\dots,M_{g_s}\in\operatorname{im}r.$$
  1. Gluing criterion. If \(y_1,y_2\in A_X^*\) are not \(W_X\)-conjugate but \(q^{\rho_L}y_1\) and \(q^{\rho_L}y_2\) are \(W\)-conjugate in \(A^*\), then \(r\) is not onto.
  2. Remark (not needed). \(r\) is a finite ring map, and \(r\) is onto if and only if \(\delta(X)^{1/2}A_X^*/W_X\to A^*/W\) is a closed immersion, if and only if this morphism is injective on points and unramified.

Both criteria are elementary consequences of the identification in Theorem E; \(r\) is the map of Setup 6.1.

Proof of the generation criterion.

  1. \(r\) is restriction of functions followed by a translation, so it is a ring homomorphism and \(\operatorname{im}r\) is a subring containing \(1\).
  2. The \(M_\kappa\), \(\kappa\in\Lambda_X^+\), form a basis of \(\C[\Lambda_S]^{W_X}\), since every \(W_X\)-orbit in \(\Lambda_S\) meets \(\Lambda_X^+\) exactly once.
  3. For \(\kappa,\kappa'\in\Lambda_X^+\) one has \(M_\kappa M_{\kappa'}=M_{\kappa+\kappa'}+\sum_{\kappa''<\kappa+\kappa'}n_{\kappa''}M_{\kappa''}\) in the \(W_X\)-dominance order: a product \(y^ay^b\) with \(a\in W_X\kappa\) and \(b\in W_X\kappa'\) has dominant representative \(\le\kappa+\kappa'\), with equality only for \(a=\kappa\), \(b=\kappa'\). Lower sets of the dominance order on \(\Lambda_X^+\) are finite.
  4. If \(\kappa=\sum_in_ig_i\), then \(\prod_iM_{g_i}^{n_i}=M_\kappa+(\text{lower terms})\). By induction along the dominance order every \(M_\kappa\) is a polynomial in \(M_{g_1},\dots,M_{g_s}\). So if these lie in the subring \(\operatorname{im}r\), then \(r\) is onto; the converse is trivial.

Proof of the gluing criterion.

  1. Every \(f\in\C[A^*]^W\) takes the same value at \(W\)-conjugate points, so \(f(q^{\rho_L}y_1)=f(q^{\rho_L}y_2)\); that is, \(g(y_1)=g(y_2)\) for every \(g\in\operatorname{im}r\).
  2. Invariants of a finite group acting on an affine variety separate its orbits: \(A_X^*/W_X=\operatorname{Spec}\C[A_X^*]^{W_X}\) is a geometric quotient. So some \(g\in\C[\Lambda_S]^{W_X}\) has \(g(y_1)\ne g(y_2)\), and such a \(g\) is not in \(\operatorname{im}r\). An explicit separating \(g\) for \(E_6/F_4\) is given in Theorem 6.6.

The remark. Since \(A_X^*\hookrightarrow A^*\) is a closed embedding, \(\C[A^*]\to\C[A_X^*]\) is onto. As \(\C[A^*]\) is finite over \(\C[A^*]^W\), so are \(\C[A_X^*]\) and its subring \(\C[A_X^*]^{W_X}\): \(r\) is finite. A finite ring map is onto exactly when the morphism of spectra is a closed immersion, and a finite morphism of complex varieties is a closed immersion if and only if it is injective on points and unramified. The gluing criterion is the statement about points; the tangent-space condition is not computed here (see Conjecture 6.8).

Builds on

  • Setup 6.1 · Conventions for Sakellaridis's restriction map \(r\), calibrated on Proposition F; the sign of the shift is irrelevant

Built on by

Original record, as the agent wrote it at 02:07 UTC

Two criteria: ring generation suffices for onto; a glued pair of points forbids onto

Both are elementary consequences of Theorem E's identification \(C_K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), \(\varphi_0\mapsto1\), \(\mathcal H_K\) acting through \(r\); under it \(\mathcal H_K\cdot\varphi_0\leftrightarrow\operatorname{im}r\).

Lemma A (generation). \(\operatorname{im}r\) is a subring containing \(1\). Let \(M_\kappa:=\sum_{\kappa'\in W_X\kappa}y^{\kappa'}\) (\(\kappa\in\Lambda_X^+\)); these form a basis of the target, and \(M_\kappa M_{\kappa'}=M_{\kappa+\kappa'}+\) (lower terms in the \(W_X\)-dominance order, whose lower sets are finite). If \(g_1,\dots,g_s\) generate the monoid \(\Lambda_X^+\), then \(M_{g_1},\dots,M_{g_s}\) generate the target as a ring (induction along the dominance order). Hence $$r\ \text{onto}\iff M_{g_1},\dots,M_{g_s}\in\operatorname{im}r.$$

Lemma B (gluing). If \(y_1,y_2\in A_X^*\) are not \(W_X\)-conjugate but \(q^{\rho_L}y_1\) and \(q^{\rho_L}y_2\) are \(W\)-conjugate in \(A^*\), then \(r\) is not onto. Proof: (1) every \(f\in\mathbb C[A^*]^W\) has \(f(q^{\rho_L}y_1)=f(q^{\rho_L}y_2)\), i.e. \(g(y_1)=g(y_2)\) for all \(g\in\operatorname{im}r\). (2) For a finite group \(W_X\) acting on the affine variety \(A_X^*\), the invariants separate orbits (\(A_X^*/W_X=\operatorname{Spec}\mathbb C[A_X^*]^{W_X}\) is the geometric quotient), so some \(g\in\mathbb C[\Lambda_S]^{W_X}\) has \(g(y_1)\neq g(y_2)\); such \(g\notin\operatorname{im}r\). (Explicit separating \(g\) given in A6-6.)

Remark (general criterion, not needed). \(\mathbb C[A^*]\to\mathbb C[A_X^*]\) is onto, so \(r\) is a finite ring map; \(r\) onto \(\iff\) \(\delta(X)^{1/2}A_X^*/W_X\to A^*/W\) is a closed immersion \(\iff\) injective on points and unramified. Lemma B is the point part; the tangent part is not computed here (see A6-8).

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