§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:19:25Z

Corollary X4.8 · Under Theorem 8.0.2, \(C_K\) is not cyclic and needs exactly two generators

proved correction ★ key step

Correction. Corrects Theorem 6.6 · \(E_6/F_4\): two points of the shifted torus that are not \(W_X\)-conjugate are \(W(E_6)\)-conjugate, so \(r\) is not onto and (SG) fails under (A1)–(A5)Sakellaridis's standing assumptions for his Theorem 8.0.2 (the pages' Theorem E): (A1) a characteristic-0 field and a split group with simply connected derived group; (A2) an integral model of X with a smooth open Borel orbit and invariant measures; (A3a) rank-one models over O, (A3b) the Cartan decomposition, (A3c) Iwahori orbits (his Axioms 2.2.1, 2.4.1, 2.4.2); (A4) conditions on the spherical roots and colours; (A5) X homogeneous and affine with one open Borel orbit..

Transports Proposition X4.7 through Sakellaridis's isomorphism: for \(E_6/F_4\) no single function generates \(C_K\) over \(\mathcal H_K\), the basic function together with the indicator of the orbit \(3\omega_1^\vee\) does, and the cokernel of \(\mathcal H_K\varphi_0\subseteq C_K\) is one-dimensional with Satake parameter \(q^{\rho-\omega_4}\). It also records the little that holds without his hypotheses.

Corollary X4.8.

Let \(F\) be \(p\)-adic, \(p\ne2\), \(X=\mathcal M_\theta\) (Proposition X4.4), and assume the conclusion of Sakellaridis's Theorem 8.0.2 for \(X\) (it holds if (E) and (I) of Gap X4.6 hold, by Proposition X4.3–Proposition X4.5). Then:

  1. \(C_K\) is not a cyclic \(\mathcal H_K\)-module: no \(\varphi\in C_K\) has \(\mathcal H_K\varphi=C_K\).
  2. \(C_K=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{O_{3\omega_1^\vee}}=\mathcal H_K\varphi_0\oplus\mathbb C\,\mathbf 1_{O_{3\omega_6^\vee}}\) as vector spaces; in particular \(C_K=\mathcal H_K\varphi_0+\mathcal H_K\mathbf 1_{O_{3\omega_1^\vee}}\), and the minimal number of generators is \(2\).
  3. \(C_K/\mathcal H_K\varphi_0\simeq\mathbb C\), on which \(\mathcal H_K\) acts by the unramified character with Satake parameter \(q^{\rho-\omega_4}\in A^*/W\).
  4. \(\dim\operatorname{Hom}_{\mathcal H_K}(C_K,\chi_\xi)=2\) for \(\xi=q^{\rho-\omega_4}\): the \(K\)-invariant eigenfunctionals \(\varphi\mapsto P_\varphi(y_1)\) and \(\varphi\mapsto P_\varphi(\sigma y_1)\) are independent.

Without Sakellaridis's hypotheses the following hold and nothing more is proved here: (a) Proposition X4.7 is unconditional algebra about \(r\); (b) Proposition X4.5(1),(2) and Proposition X4.4 hold for \(\mathcal M_\theta\), \(p\ne2\); (c) the criterion "two linearly independent \(\mathcal H_K\)-eigenfunctionals on \(C_K\) with the same character \(\Rightarrow\) \(C_K\) not cyclic" is unconditional, since a cyclic module \(\mathcal H_K/\mathfrak a\) has \(\dim\operatorname{Hom}(\mathcal H_K/\mathfrak a,\chi)\le1\). Over \(\mathbb F_q(\!(t)\!)\) the paper gives nothing (it assumes characteristic \(0\)).

  1. Transport. Theorem 8.0.2 gives an \(\mathcal H_K\)-isomorphism \(C_K\simeq S\) with \(\varphi_0\mapsto1\) and \(\mathcal H_K\simeq\mathbb C[A^*]^W\) acting through \(r\) (after the translation \(\delta(X)^{1/2}A_X^*\to A_X^*\); the left/right and sign conventions change \(r\) by automorphisms of source and target and change none of the statements, Setup 6.1, Audit 6.7; the character \(q^{\rho-\omega_4}\) is fixed by \(-w_0\), so inversion does not move it). Then 1 and 3 are Proposition X4.7(4),(3), and \(\mathcal H_K\varphi_0\leftrightarrow R\).
  2. The indicator of \(O_{3\omega_1^\vee}\). In the proof of Theorem 8.0.2 (p. 64) the indicator of the \(K\)-orbit with label \(\check\lambda\) maps to \(P_{\check\lambda}\), whose lowest weight for the order defined by the cone \(\mathcal T\) of colour valuations is \(\check\lambda\), with non-zero coefficient (Remark after Theorem 7.2.1, p. 62). By Proposition X4.3, \(\mathcal T=\mathbb R_{\ge0}\check\gamma_1+\mathbb R_{\ge0}\check\gamma_6\), so in dominant coordinates \(P_{3\omega_1^\vee}=c\,M_\kappa+c_1M_{(1,1)}+c_0\) with \(c\ne0\), where \(\kappa\in\{(3,0),(0,3)\}\) depends on the left/right convention: the dominant \(\kappa'\) with \(\kappa-\kappa'\in\mathbb N\check\gamma_1+\mathbb N\check\gamma_6\) are \(\kappa,(1,1),(0,0)\). \(M_{(1,1)},1\in R\) and \(M_{(3,0)},M_{(0,3)}\notin R\) (Proposition X4.7: \(P\in R\), \(M_{(3,0)}\notin R\)), so \(P_{3\omega_1^\vee}\notin R\) and \(S=R\oplus\mathbb CP_{3\omega_1^\vee}\) by \(\dim S/R=1\). This gives 2; the same for \(3\omega_6^\vee\). It also settles the stronger reading flagged in the audit of Theorem 6.6: \(\mathbf 1_{O_{3\omega_1^\vee}}\notin\mathcal H_K\varphi_0\).
  3. Point 4. Evaluation at a point \(y\) of \(\operatorname{Spec}S\) is an algebra map \(S\to\mathbb C\); composed with \(C_K\simeq S\) it is an \(\mathcal H_K\)-eigenfunctional with character \(f\mapsto f(q^{\rho_L}y)\). At \(y_1,\sigma y_1\) the characters coincide (both \(\xi\)) and the functionals differ (on \(M_{(3,0)}\)). Every eigenfunctional with character \(\xi\) kills \(\mathfrak m_\xi S\), and \(\dim S/\mathfrak m_\xi S=2\) (Proposition X4.7(4)), so the dimension is exactly \(2\). By the criterion (c) this re-proves 1.
  4. What the hypotheses buy. Every statement 1–4 passes through Theorem 8.0.2; for \(\mathcal M_\theta\) the only unverified inputs of that theorem are (E) and (I) of Gap X4.6.

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