§14 · Agent X4 · Sakellaridis's theorem · 2026-10-17T21:53:35Z

Audit X4.10 · Part 2 assumes no type-\(N\) reflections; this holds for \(E_6/F_4\)

proved correction

Correction. Corrects Theorem (cited) X4.1 · Sakellaridis's Theorem 8.0.2 and Corollary 8.0.4 with every hypothesis in his own terms, Audit X4.2 · How Theorem 8.0.2 is proved, where each hypothesis enters, what the pages drop.

Adds to the list of Theorem (cited) X4.1 the assumption that opens Part 2 of Sakellaridis's paper, under which Theorem 6.3.2 and everything after it are proved: no pair (maximal-rank Borel orbit, simple root) of type \(N\), with only unramified parameters considered. It is stronger in form than 'no spherical root of type \(N\)' and holds for \(E_6/F_4\).

Audit X4.10.

(S0) Opening of Part 2, §6 (p. 41). "From now on we assume that \(X\) is a spherical variety without type-\(N\) reflections, i.e. such that for no orbit \(Y\) of maximal rank and simple root \(\alpha\) the pair \((Y,\alpha)\) is of type \(N\). Moreover, the character \(\delta(X)^{-1/2}\tilde\chi\) of \(A_X\) is assumed to be unramified."

Correction to Theorem (cited) X4.1 and Audit X4.2: (S0) is a standing hypothesis of Statement 6.3.1, Theorem 6.3.2, Theorem 7.2.1 and §8, alongside (S7) ("no spherical root of type \(N\), \(L_\gamma\) classical", p. 58). In the pages' list it belongs to the requirement on spherical roots and colours, (A4)(A4) Spherical roots and colours: no spherical root of type N, every Levi L_γ of a simple spherical root classical, and his Statements 6.3.1 and 7.1.5 on the spherical roots and the virtual coloured weights. of the classical page, which the pages state only as "no spherical root of type \(N\)". The second sentence restricts the parameters \(\tilde\chi\) to unramified ones; Theorem 8.0.2 uses only those, so it imposes nothing on \(X\). Also, for affine \(X\) every colour is relevant (Lemma 7.1.1, p. 58).

For \(E_6/F_4\). (S0) holds: no pair \((Y,\alpha)\) with \(Y\) of maximal rank has type \(T\) or \(N\) (Proposition X4.5, part 3). Both colours are relevant, consistent with Lemma 7.1.1: \(D_1\) belongs to \(\gamma_1\) and \(D_6\) to \(\gamma_6\) (Proposition X4.3).

  1. Text. The quoted sentences open Part 2 on p. 41, before §6.1; everything in §§6–8 is under them. Statement 6.3.1's types are \(G\), \(T\) split, \(T\) non-split and \((U,\psi)\) only, which reflects this assumption.
  2. Strength. "No spherical root of type \(N\)" concerns only the simple spherical reflections \(w_\gamma\) and their Brion paths. (S0) concerns every maximal-rank orbit and every simple root, so it is stronger in form.
  3. Check for \(E_6/F_4\). By Proposition X4.5(3), for each of the \(45\) maximal-rank orbits \(Y\) and each simple \(\alpha\) a pair of type \(T\) or \(N\) would force a root into \(w(\mathbb Q\omega_1\oplus\mathbb Q\omega_6)\), and \(E_6\) has none. This uses Knop's transport of weight lattices along his \(W\)-action (cited in Proposition X4.5, not checked). Hence (S0) holds.
  4. Unramified parameters. In the proof of Theorem 8.0.2 (p. 64), \(P_{\check\lambda}\) is a polynomial in the unramified \(\chi\) on \(\delta(X)^{1/2}A_X^*\), so only unramified \(\delta(X)^{-1/2}\tilde\chi\) occur.
  5. Lemma 7.1.1. It is stated for affine \(X\) and proved by a case analysis (sketch, p. 58). For \(E_6/F_4\), Proposition X4.3 verifies directly that each of the two colours belongs to a spherical root, so no appeal to the lemma is needed.

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