§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:13:58Z

Theorem (cited) X4.1 · Sakellaridis's Theorem 8.0.2 and Corollary 8.0.4 with every hypothesis in his own terms

proved audit: minor defect ★ key step

Minor defect (second audit). I checked every quotation and page number against the PDF (printed page = PDF page): §1.1 p.2, §1.7 pp.6–7, §2.1 p.8, Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. For E6/F4 it holds over every O with 2 invertible (VI run). p.9, Axioms 2.4.1/2.4.2 pp.10–11, Statement 6.3.1 pp.44–45, Theorem 6.3.2 p.45, §7 p.58, Statement 7.1.5 and Theorem 7.2.1 p.60, §8 p.63, Example 8.0.1 and Theorem 8.0.2 p.64, Corollary 8.0.4 p.65. All are correct. One standing assumption is missing: the opening of §6 (p.41) assumes no type-N reflections, i.e. no maximal-rank pair (Y,α) of type N, and that δ(X)^{-1/2}χ̃ is unramified. Theorem 6.3.2 is proved under this assumption, which is stronger in form than (S7)'s 'no type-N spherical roots'. It holds for E6/F4 by S4-5(3), so no claim changes. Minor: virtual weighted colours are built from relevant colours, which for affine X are all colours (Lemma 7.1.1).

States Sakellaridis's description of the unramified Hecke module \(C_c^\infty(X)^K\) and his principal-module criterion exactly as in arXiv:0905.4244v3, with page numbers, every standing assumption and the invariants \(A_X,\Lambda_X,W_X,\delta(X)\) and the restriction map \(r\).

Theorem (cited) X4.1.

Page numbers are the printed (= PDF) pages of arXiv:0905.4244v3.

Standing setting (§1.1 p. 2, §1.7 pp. 6–7, §2.1 p. 8).

Theorem 7.2.1 (p. 60) assumes exactly (S8) for all simple spherical roots and (S9); inside §7, i.e. under (S1)–(S7).

Invariants (pp. 3, 7–9, 41–44, 59–60, 63).

Theorem 8.0.2 (p. 64). Under (S1)–(S10): there is a canonical isomorphism \(C_c^\infty(X)^K\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), compatible with the \(\mathcal H(G,K)\)-structure and the Satake isomorphism, under which \(\mathbf 1_{X(\mathfrak o)}\mapsto1\).

Corollary 8.0.4 (p. 65). Under (S1)–(S10), if \(r\) is surjective (equivalently \(\mathcal H_X\simeq\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\)), then (1) \(C_c^\infty(X)^K\) is a principal \(\mathcal H(G,K)\)-module generated by \(\mathbf 1_{X(\mathfrak o)}\); (2) every Hecke eigenspace in \(C^\infty(X)^K\) with character in \(\operatorname{specM}\mathcal H_X\) is one-dimensional and \(\dim\operatorname{Hom}_G(\pi,C^\infty(X))\le1\) for unramified \(\pi\); (3) every non-zero \(H\)-invariant functional on an unramified \(\pi\) is non-zero on \(\pi^K\). The proof is "straightforward and will be omitted".

Example 8.0.1 (p. 64). For \(X=\mathrm{SL}_2^3\backslash\mathrm{Sp}_4\times\mathrm{Sp}_4\) one has \(W_X=N_W(A_X^*)\) and \(\delta(X)=1\), but \(W_X\)-nonconjugate points of \(A_X^*\) in the diagonal \(\mathrm{SO}_4\) are \(W\)-conjugate, so \(r\) is not surjective: the hypothesis of Corollary 8.0.4 is not automatic.

  1. Where each item is stated. (S1): §1.1, p. 2, which also states that \(k\) has characteristic zero. (S2), (S3): §1.7, pp. 6–7, which also requires (S4)–(S6) ("We also assume that \(X\) satisfies Axioms 2.2.1, 2.4.1 and 2.4.2, which is always the case at almost every place if \(X\) and \(G\) are defined over a global field"); \(x_0\) and \(H\): §2.1, p. 8. (S4): p. 9, where the author adds that it "is clearly the case at almost every place if \(X\) is defined globally". (S5), (S6): pp. 10–11, proved in [Sa12] "under assumptions which hold at almost every place", with Benoist–Oh and Delorme–Sécherre cited for "similar statements" for symmetric varieties. (S7): opening of §7, p. 58. (S8): pp. 44–45; by Theorem 6.3.2 (p. 45) it holds for every normalized spherical root and every Brion path as soon as (S7) holds. (S9): p. 60, conjectured by the author for affine homogeneous \(X\) and "very easy to check in each particular case". (S10): p. 63, which adds "In particular, the strong form of the Cartan decomposition holds: \(X/K\simeq\Lambda_X^+\) and \(A_X^*\subseteq A^*\)".
  2. The invariants. \(A_X\), \(\Lambda_X\), \(\Lambda_X^+\), \(A_X^+\), \(A_X^*\): §2.1, p. 8. \(W(X)=W_X\ltimes W_{P(X)}\) and the reflection action: §2.2, p. 9. \(\delta(X)\): §1.7, p. 7. The translate and its \(W_X\)-action: p. 3, where the map \(\delta(X)^{1/2}A_X^*\to\operatorname{specM}\mathcal H(G,K)\) is said to factor through \(\delta(X)^{1/2}A_X^*/W_X\). Spherical roots, normalization and types: §6.1–6.2, pp. 41–44. Virtual colours and \(\Theta\): §7.1, pp. 59–60. \(r\) and \(\mathcal H_X\): §8, pp. 63–65.
  3. Role of Example 8.0.1. It is the only place in the paper where the hypothesis of Corollary 8.0.4 is shown to fail. The mechanism is a gluing: two points of \(\delta(X)^{1/2}A_X^*\) that are not \(W_X\)-conjugate become \(W\)-conjugate, so every element of \(\operatorname{im}r\) takes equal values at them while \(\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\) separates them. Its Remark on p. 65 records that it forces a correction to [Sa08, Theorem 6.3.2]. The author does not verify (S1)–(S10) for this example.
  4. Status. Everything above is quoted from the paper; nothing is re-proved here (cited).

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