Audit X4.2 · How Theorem 8.0.2 is proved, where each hypothesis enters, what the pages drop

Audit X4.2.

(I) Proof skeleton of Theorem 8.0.2 (labels (S1)–(S10) as in Theorem (cited) X4.1).

1. Mackey morphisms (§3). The open \(B\)-orbit gives \(S_\chi:C_c^\infty(X)\to I(\chi)\) and adjoints \(\Delta_\chi\); the eigenfunctions are \(\Omega_\chi=\Delta_\chi(\Phi_K)\). Uses (S1)–(S4): eigenmeasures are absolute values of forms over \(\mathfrak o\) on the \(\mathfrak o\)-schemes of maximal-rank orbits.
2. Sum formula (Theorem 4.2.2, p. 26; (6.2), p. 44). For generic \(\chi\) and \(x\in A_X^+\): \(\Omega_{\delta(X)^{1/2}\tilde\chi}(x)=\delta_{P(X)}^{-1/2}(x)\sum_{w\in W_X}B_w(\tilde\chi)\,{}^w\tilde\chi(x)\). Uses (S5) to know that \(A_X^+\) meets every \(K\)-orbit and (S6) to pass from the Iwahori-fixed vector to \(B(\mathfrak o)\)-orbits.
3. Simple spherical reflections (§6). (S8) evaluates \(B_{w_\gamma}\); it holds under (S7) by Theorem 6.3.2 (p. 45).
4. The formula with \(L\)-values (Theorem 7.2.1, p. 60). Under (S8) and (S9), \(B_w(\tilde\chi)=\beta(\tilde\chi)/\beta({}^w\tilde\chi)\) with \(\beta=\prod_{\check\gamma\in\check\Phi_X^+}(1-e^{\check\gamma})\big/\prod_{\Theta^+}(1-\sigma_{\check\theta}q^{-r_{\check\theta}}e^{\check\theta})\), so \(\Omega(x_{\check\lambda})/\beta\) is the \(W_X\)-invariant Laurent polynomial (7.4); for affine \(X\), \(\Omega(x_0)=c\,\beta\).
5. Theorem 8.0.2 (p. 64). \(P_{\check\lambda}:=\Omega(x_{\check\lambda})/L_X^{1/2}\in\mathbb C[\delta(X)^{1/2}A_X^*]^{W_X}\), \(P_0=1\); \(\mathbf 1_{x_{\check\lambda}K}\mapsto P_{\check\lambda}\) is \(\mathcal H(G,K)\)-equivariant through Satake; by the Remark after Theorem 7.2.1 (p. 62) \(\check\lambda\) is the lowest weight of \(P_{\check\lambda}\) for the order defined by the cone \(\mathcal T\) of colour valuations, and \(\mathcal T\cap\mathcal V=\{0\}\) for affine \(X\); triangularity makes the \(P_{\check\lambda}\) a basis. Uses (S10) for \(X/K\simeq\Lambda_X^+\).
6. The criterion. \(\mathcal H(G,K)\) acts on the target by multiplication through \(r\) and \(\mathbf 1_{X(\mathfrak o)}\mapsto1\), so \(\mathcal H(G,K)\cdot\mathbf 1_{X(\mathfrak o)}\) corresponds to \(\operatorname{im}r\): the module is generated by \(\mathbf 1_{X(\mathfrak o)}\) iff \(r\) is onto (\(\Leftarrow\) is Corollary 8.0.4(1)).

(II) Correspondence with the pages' list.

(III) Verdict on the summary. Accurate as a list, with these losses:

• L1. The simplification "\(k/\mathbb Q_p\) unramified" (p. 2) is dropped; the paper calls its removal trivial (not checked here).
• L2. No \(\mathcal O\)-model is fixed. Sakellaridis's \(X\) is an \(\mathfrak o\)-scheme with a chosen \(x_0\in\mathring X(\mathfrak o)\); \(K\backslash X(F)\), the axioms and \(\mathbf 1_{X(\mathfrak o)}\) depend on it, while the pages' \(X=G/G^\theta\) is a variety over \(\mathbb C\). Every verdict must name the model.
• L3. The paraphrase of 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). drops its second clause (complement of \(YP_\alpha\) = closure of the complement of \((YP_\alpha)_k\)), the \(\mathfrak o\)-scheme structure on \(Y\), and that \(T\) may be a non-split torus over \(\mathfrak o\).
• L4. In the requirement "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., "Statement 6.3.1" is not independent: Theorem 6.3.2 derives it from "no type \(N\), \(L_\gamma\) classical". Conversely those two conditions are what make "type of a colour" and \(\Theta\) well defined (Lemma 7.1.4, p. 59, proved by inspection of Theorem 6.11.1); the hypotheses of Theorem 7.2.1 proper are only 6.3.1 and 7.1.5.
• L5. Theorem E's "(SG) iff \(r\) onto" combines Theorem 8.0.2 (\(\Rightarrow\)) with Corollary 8.0.4(1) (\(\Leftarrow\)); correct. Its item on Example 8.0.1 is correct as stated under the hypotheses; the paper does not verify them for that example.
• L6. The sentence "none of them is verified on these pages for any family" is now outdated for \(E_6/F_4\): see Proposition X4.3, Proposition X4.4, Proposition X4.5.