Dead end 1.8 · \(B_{w_\gamma}\) for \(M_\theta\) over \(\mathbb F_q((t))\): two routes, missing lemmas

Proof of (i). \(\mathbf 1_{X(O)}\) is \(K\)-invariant, so \(S_\chi(\mathbf 1_{X(O)})=Z_F(\chi)\varphi_{K,\chi}\). The rank-one computation of Proposition 1.6 step 2 gives \(F_w\varphi_{K,\chi}=c'_w(\chi)\varphi_{K,w\chi}\). Apply \(F_wS_\chi=b_w(\chi)S_{w\chi}\) (Proposition 1.6(c), \(w\in W_X'\)) to \(\mathbf 1_{X(O)}\). Then \(Z_F(\chi)c'_w(\chi)=b_w(\chi)Z_F(w\chi)\). Here \(Z_F\not\equiv0\), since on the region \(R\) it is an integral of a positive function for real \(\chi\).

Proof of (ii). By Proposition 1.5 step 2, for \(x\in X̊(F)\), \(\lambda(x)\) is determined by \((\operatorname{ord}f_1(x),\operatorname{ord}f_6(x))\), because \(\eta_1,\eta_6=\omega_1,\omega_6\) form a basis of \(\mathcal X(X)\) (S4-3). On \(\{\lambda(x)=\mu\}\) the integrand of (3.2) is constant times \(|\omega_X|\). \(X\) is smooth over \(O\) (S4-4/S4-11: \(\theta_0\) and the quotient are defined over \(\mathbb Z\), smooth after inverting 2). So the fibres of \(X(O)\to X(O/\varpi^N)\) are balls of volume \(q^{-N\dim X}\) for the residually non-vanishing invariant form. The condition \(\operatorname{ord}f_i=m_i<N\) depends only on \(x\bmod\varpi^N\). If \(E\) is totally ramified of degree \(e\ge N\), then \(p\in\varpi_E^N O_E\), so \(O_E/\varpi_E^N=\mathbb F_q[\varpi_E]/(\varpi_E^N)\).

Route A: rerun §6.19 in characteristic p (blocked). \(\gamma_1\) has support \(D_5\) (nodes 1, 3, 4, 2, 5 in the roles \(\alpha_1',\dots,\alpha_5'\) of §6.19, with \(n=5\)). The Brion path is \(X̊\xrightarrow{\alpha_1,U}Y_1\xrightarrow{\alpha_3,U}Y_2\xrightarrow{\alpha_4,U}Y_3\), with the orthogonal pair \((\alpha_2,\alpha_5)\) at the bottom. It is glued from \(PGL_2\backslash PGL_2^2\) and \(P_\alpha\backslash SL_3\), and (6.18) gives exponent \(n-1=4\). The rank-one factors (Theorem 5.2.1, case U) are characteristic-free (Strategy 1.1 rows on §5.3–5.4), and the gluing is algebra. Missing:

• M1. For each orbit \(Y_i\) of the path, over \(F_q((t))\): \(Y_i(F)\) is one \(B(F)\)-orbit; \(Z\times U_\alpha\cong Y\) along each edge; the integral \(S^{Y_i}\) converges on a non-empty open set and continues rationally; and the Fubini region for \(T_\alpha S^Z=S^Y\) is non-empty. The method of Proposition 1.5 applies to \(Y_i\) once generic uniqueness is shown for the family \(D_{Y_i}\) on \(X\setminus(\overline{Y_i}\setminus Y_i)\). The dimension count of Proposition 1.5 step 7 must then be redone for orbits with the same transported involution.
• M2. The \(B_{\bar F}\)-orbit structure along this path over \(\bar F\) of characteristic \(p\) (the types U, and the bottom pair lowering \(Y_3\) to one orbit). This is linear algebra on the split affine quadric \(\{Q=1\}\subset\mathbb A^{10}\) for \(\operatorname{Spin}_9\backslash\operatorname{Spin}_{10}\), char \(\neq2\). Not done here.

Route B: close fields (blocked). By (ii), \(Z_F=Z_E\) coefficientwise, provided \(Z_E\) is one rational function \(R_q\) for all totally ramified \(E\) with residue field \(\mathbb F_q\). Both are Laurent expansions on the same cone \(R\) of Proposition 1.5(a), so the identity of rational functions follows. Missing:

• M4. \(Z_E=R_q\), i.e. (1.4) \(\Omega_\chi(x_0)=c\beta(\tilde\chi)\) together with (4.5), for \(M_\theta\) over every totally ramified \(E\), with \(\psi\) of conductor \(O_E\) and \(dx(O_E)=1\).

The analytic steps for \(E\) are the characteristic-free ones of Proposition 1.5–Computation 1.7, so §3.8 and §5.3 need no modification once \(\psi\) has conductor \(O_E\), which is not the trace of a character of \(\mathbb Q_p\). But Sakellaridis's Statement 6.3.1 over \(E\) comes from Theorem 6.3.2, whose rank-one constants presuppose the integral models of Axiom 2.2.1Sakellaridis's Axiom 2.2.1, relabelled (A3a): rank-one models over O. over \(O_E\). That axiom is open for \(M_\theta\) even in characteristic 0 (Proposition VI page; (Q2)). Route B therefore also depends on (Q2).

Theorem 6.3.2 in general (blocked). Deriving (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5.(iii) from (A4)(A4) Spherical roots and colours: no pair of maximal rank of type N; classical Levis L_γ; Statements 6.3.1 and 7.1.5.(i), (ii) for an arbitrary symmetric variety in characteristic \(p\) needs Lemma 6.2.2 and Proposition 6.2.1. These rest on the classification of rank \(\le2\) wonderful varieties [Was96], [Ak83], available only in characteristic 0.