§3 · Agent 3 · Hypotheses in char p · 2026-10-18T03:01:08Z

Proposition 3.10 · (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. in char \(p\): (i), (ii), Statement 7.1.5, and the combinatorics of 6.3.1

proved audit: correct

Second audit: correct. Against the definitions on Sakellaridis pp. 43-46, 58-60, I verified the Brion path for w_gamma1: orbit dimensions 26, 25, 24, 23, 22, 22; every edge of type U at both ends; a2 and a5 lower Y_{s4s3s1} to the same orbit; l(w) = 8 = 2*3 + 2; (s2s4s3s1)^-1 s5s4s3s1 = w_gamma1. For the gluing, every vertex pair (a1,a3), (a3,a4), (a4,a2) and also (a4,a5) at Y_{s4s3s1} has one black pair +-g0, the other Levi roots moved out by theta, a 4-root maximal parabolic set and a rank-2 T_8 image, so P^2 = P_a\SL3, while the pair (a2,a5) has the two A1 factors swapped by theta, so PGL2\PGL2xPGL2 (Lie argument plus N(Delta) = Delta); both varieties are in Prop. 6.4.1's list, as in §6.19, and the text's 'three A2-vertices' covers four A2 pairs, all of which I checked. The first-edge argument (labels spell a reduced word of w_gamma1 in W_{1..5}; the first label is not in Phi_L) makes D1 the only colour belonging to gamma1, and Theta, Theta^+ = {e1, e6, e1+e6} and Statement 7.1.5 check out; (iv) relies on Proposition 3.7(2), which holds after that step's one-line repair.

Assembles (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. from Proposition 3.6 and Proposition 3.7: no maximal-rank pair has type \(N\) (nor \(T\)), the Levis \(L_{\gamma_1},L_{\gamma_6}\) are of type \(D_5\), and the relevant colours, virtual weighted colours, \(\Theta=\check\Phi_X\times\{(+,4)\}\) and Statement 7.1.5 are verified directly, replacing the case analysis of Lemma 7.1.2. For Statement 6.3.1 the combinatorial hypothesis of Sakellaridis's Prop. 6.4.1 is verified in characteristic \(p\): a Brion path for \(w_{\gamma_1}\) with labels \(\alpha_1,\alpha_3,\alpha_4,\{\alpha_2,\alpha_5\}\), glued from \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2^2\) and \(P_\alpha\backslash\mathrm{SL}_3\); only the analytic part remains.

Proposition 3.10.

Over an algebraic closure of \(\mathbb F_q(\!(t)\!)\), \(p\) odd (indeed over any algebraically closed field of characteristic \(\ne2\)):

$$\mathring X=Y_1\xrightarrow{\alpha_1}Y_{s_1}\xrightarrow{\alpha_3}Y_{s_3s_1}\xrightarrow{\alpha_4}Y_{s_4s_3s_1}\overset{\alpha_2,\ \alpha_5}{\rightrightarrows}Y_{s_2s_4s_3s_1}=Y_{s_5s_4s_3s_1}$$

is a Brion path for \(w_{\gamma_1}\) in the sense of §6.2 (all edges of type \(U\), orbit dimensions \(26,25,24,23,22\)); its codimension-one orbit has closure \(D_1\), so \(\check\theta=e_1=\check\gamma_1\) as Statement 6.3.1 requires for type \(G\); the rank-two varieties \(ZP_{\delta\epsilon}/R(P_{\delta\epsilon})\) at its vertices are \(\mathbb P^2=P_\alpha\backslash\mathrm{SL}_3\) at the three \(A_2\)-vertices and \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\times\mathrm{PGL}_2\) at the bottom, so the path satisfies the hypothesis of Prop. 6.4.1 (as in §6.19 with \(n=5\)). The same holds for \(\gamma_6\) by the diagram automorphism \(\delta\).

  1. (i) is Proposition 3.6(3). The unramifiedness condition is on \(\tilde\chi\): in the proof of Theorem 8.0.2 only unramified characters occur (S4-10, item 4).
  2. (ii) is Proposition 3.7(4).
  3. (iv), relevant colours. \(D\) belongs to \(\gamma\) if \(\bar D\) is the closure of the codimension-one orbit of a Brion path for \(w_\gamma\) (p. 58). The labels of a Brion path for \(w_{\gamma_1}\) spell a reduced word of \(w_{\gamma_1}=s_1s_3s_4s_2s_5s_4s_3s_1\in W_{\{1,\dots,5\}}\) (Proposition 3.6(2)), so they avoid \(\alpha_6\). The first edge starts at \(\mathring X\) with a label \(\alpha\) of type \(U\) for \(\mathring X\), i.e. \(\alpha\notin\Phi_L\) by Proposition 3.6(3); so \(\alpha=\alpha_1\), and the codimension-one orbit is \(Y_{s_1}=\mathring XP_{\alpha_1}\smallsetminus\mathring X\), whose closure is the colour moved by \(P_{\alpha_1}\), namely \(D_1\) (Proposition 3.7(2)). The path in (iii) exists, so \(D_1\) belongs to \(\gamma_1\), and only \(D_1\); symmetrically \(D_6\) and \(\gamma_6\). \(D_1\) is defined over \(F\) (it is the component of \(X\smallsetminus\mathring X\) moved by the \(F\)-group \(P_{\alpha_1}\), so Galois-stable), hence \(\sigma=+\).
  4. (iv), \(\Theta\). \(\langle\rho_L,\alpha_j^\vee\rangle=(-3,1,1,1,1,-3)\) (computed) gives \(\langle\rho-\rho_L,\alpha_j^\vee\rangle=(4,0,0,0,0,4)\), so \(\rho_{P(X)}=4\omega_1+4\omega_6\) and \(r=\langle e_i,\rho_{P(X)}\rangle=4\). Both spherical roots are of type \(G\), so there are no additions (2), (3) of p. 59. With \(s_{\gamma_i}(v)=v-\langle v,\gamma_i\rangle e_i\): \(s_{\gamma_1}e_1=-e_1\), \(s_{\gamma_6}e_1=e_1+e_6\), \(s_{\gamma_1}(e_1+e_6)=e_6\); the orbit of \((e_1,+,4)\) is the six coroots of \(A_2\) and contains \((e_6,+,4)\), so \(\Theta\) is this orbit with multiplicity one. \(\Theta^+\) (inside the cone of the colour valuations \(e_1,e_6\)) is \(\{e_1,e_6,e_1+e_6\}\); every element of \(\Theta\) is \(>0\) or \(<0\); \(\{\check\theta\in\Theta^+:s_{\gamma_1}\check\theta<0\}=\{e_1\}\) since \(s_{\gamma_1}e_6=e_1+e_6\) and \(s_{\gamma_1}(e_1+e_6)=e_6\); this is the virtual colour of \(D_1\), the one belonging to \(\gamma_1\). Same for \(\gamma_6\). So Statement 7.1.5 holds.
  5. (iii), the path. Orbit dimensions from Proposition 3.6(1): \(\dim Y_v=42-\big(4+\#(R_v\cap\Phi_L)+\tfrac12\#(R_v\smallsetminus\Phi_L)\big)\), \(R_v=v^{-1}\Phi^+\cap\theta(v^{-1}\Phi^+)\). Computed: dimensions \(26,25,24,23,22,22\); every edge of type \(U\); \(Y_{s_2s_4s_3s_1}=Y_{s_5s_4s_3s_1}\) since \((s_2s_4s_3s_1)^{-1}s_5s_4s_3s_1=w\in W(X)\); \(\ell(w)=8=2\cdot3+2\) (reduced); \(w\) stabilizes \(\Phi_L^+\) and maps \(\omega_1\mapsto\omega_6-\omega_1\), \(\omega_6\mapsto\omega_6\), so it is the element \(w_{\gamma_1}\) of the canonical \(W_X\); this proves Sakellaridis's Prop. 6.2.1 for \(\mathcal M_\theta\) in characteristic \(p\).
  6. (iii), gluing. At \(Z=Y_v\) with adjacent edge labels \(\delta,\epsilon\), the Levi \(L^v_{\delta\epsilon}\) has roots \(\pm\delta',\pm\epsilon',\pm(\delta'+\epsilon')\) (\(\delta'=v^{-1}\delta\), \(\epsilon'=v^{-1}\epsilon\)). Computed at each of the three \(A_2\)-vertices: \(\theta\) fixes exactly one pair \(\pm\gamma_0\) of these (a black root, so \(U_{\pm\gamma_0}\subseteq H\)) and moves the others out of the Levi; the roots of \(P^v_{\delta\epsilon}\cap\theta P^v_{\delta\epsilon}\) in the Levi are \(\pm\gamma_0\) and two others, forming a maximal parabolic subset; the image of \(T^\theta\) in the adjoint torus has rank 2. As in Proposition 3.6 step 5, the stabilizer of the rank-two quotient is that maximal parabolic of \(\mathrm{PGL}_3\), so the quotient is \(\mathbb P^2\). At the bottom (\(\alpha_2,\alpha_5\) orthogonal), \(\theta\delta'=-\epsilon'\) and \(\theta\epsilon'=-\delta'\) (computed): the Levi \(A_1\times A_1\) is \(\theta\)-stable with the two factors exchanged, the stabilizer contains a twisted diagonal \(\mathrm{PGL}_2\), and it is proper with identity component of Lie algebra a submodule of \(\mathfrak{sl}_2\oplus\mathfrak{sl}_2\) under \(\Delta(\mathfrak{sl}_2)\) (two irreducible summands for \(p\ne2\)), hence equal to it: the quotient is \(\mathrm{PGL}_2\backslash\mathrm{PGL}_2\times\mathrm{PGL}_2\).
  7. Left for (iii): the analytic identities of Prop. 6.5.1 over \(\mathbb F_q(\!(t)\!)\) (Remark 3.11).

Builds on

  • Proposition 3.6 · Char \(\ne2\): 45 maximal-rank Borel orbits \(Bn_wx_0\), all pairs of type \(G\) or \(U\)
  • Proposition 3.7 · Char \(\ne2\): two colours, the valuation cone, spherical roots \(\gamma_1,\gamma_6\)

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