§9 · Salvage 1 · Leading coefficients · 2026-10-18T06:44:44Z

Lemma S1.1 · (7.5) for every \(\Theta\): H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1). is equivalent to one colour set per simple root

proved correction

Correction. Corrects Theorem 1.9 · Theorem 8.0.2 over \(\mathbb F_q((t))\), \(p\) odd, symmetric \(X\): hypotheses, Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses.

For any \(W_X\)-stable \(\Theta=\Theta^+\sqcup-\Theta^+\) the inversion sums satisfy \(|N_\Theta(w)|-|N_{\check\Phi}(w)|=(1-w^{-1})\Xi\) with \(\Xi=\tfrac12|\Theta^+|-\check\rho\). Hence the triangularity hypothesis H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1). of Objection 5.20 is equivalent to (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1).: the virtual colours belonging to each simple spherical root \(\gamma\) sum to \(\check\gamma\); under (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., (7.5) holds with the cone \(\operatorname{cone}(\Theta^+)\) for every \(\Theta\), repeated or non-coroot weights included.

Lemma S1.1.

Setting. \(\Phi_X\) the normalized spherical roots, \(\Delta_X\) the simple ones, \(\check\Phi_X\subset\Lambda_X\) the coroots, \(W_X\) the little Weyl group, \(\check\rho=\frac12\sum_{\check\Phi_X^+}\check\gamma\), \(\mathcal R=\operatorname{cone}(\check\Phi_X^+)\). \(\mathcal T_{col}\) is the cone spanned by the valuations of all colours (strictly convex, \(X\) quasi-affine). \(\Theta\) is a finite \(W_X\)-stable multiset of triples \((\check\theta,\sigma,r)\), \(\check\theta\neq0\), with \(W_X\) acting on \(\check\theta\) only and closed under \((\check\theta,\sigma,r)\mapsto(-\check\theta,\sigma,r)\); \(\Theta^+=\{\theta:\check\theta\in\mathcal T_{col}\}\). For \(w\in W_X\) put $$N_\Theta(w)=\{\theta\in\Theta^+:\ w\check\theta\in-\mathcal T_{col}\},\qquad N_{\check\Phi}(w)=\{\check\gamma\in\check\Phi_X^+:\ w\check\gamma<0\},\qquad |S|=\textstyle\sum_{\theta\in S}\check\theta .$$ Assume Statement 7.1.5: \(\Theta=\Theta^+\sqcup-\Theta^+\), and \(D_\gamma:=N_\Theta(w_\gamma)\) is the multiset of virtual colours belonging to \(\gamma\in\Delta_X\).

Lemma. (a) For every \(w\in W_X\): \(\ |N_\Theta(w)|-|N_{\check\Phi}(w)|=(1-w^{-1})\,\Xi\), where \(\Xi:=\tfrac12|\Theta^+|-\check\rho\). (b) For \(\gamma\in\Delta_X\): \(|D_\gamma|=m_\gamma\check\gamma\) with \(m_\gamma\ge1\), and \(\langle\Xi,\gamma\rangle=m_\gamma-1\). (c) The following are equivalent: (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1). \(m_\gamma=1\) for every \(\gamma\in\Delta_X\); that is, \(D_\gamma\) is exactly one copy of \((\check\gamma,+,\langle\check\gamma,\rho_{P(X)}\rangle)\) for type \(G\), and exactly one pair \(\{\check\theta,\check\gamma-\check\theta\}\) for type \(T\) (the uniqueness half of Lemma 7.1.4); (H1) there is a strictly convex cone \(\mathcal T\supseteq\mathcal R\) with \(\check\rho-w\check\rho+w|I|\in\mathcal T\) for all \(w\in W_X\), \(I\subseteq\Theta^+\). (d) Under (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1).: \(|N_\Theta(w)|=|N_{\check\Phi}(w)|\) and \(|\Theta^+|-w|\Theta^+|=2(\check\rho-w\check\rho)\) for all \(w\); and for all \(w\), \(I\), with \(I_\pm=\{\theta\in I: w\check\theta\in\pm\mathcal T_{col}\}\), $$\check\rho-w\check\rho+w|I|=\big|N_\Theta(w^{-1})\smallsetminus(-wI_-)\big|+|wI_+|\ \in\ \operatorname{cone}(\Theta^+).$$ So (7.5) holds with \(\mathcal T=\operatorname{cone}(\Theta^+)\subseteq\mathcal T_{col}\), and \(\mathcal R\subseteq\operatorname{cone}(\Theta^+)\).

  1. Counting under \(v\in W_X\). An element \(\varphi\in\Theta^+\) lies in \(v\Theta^+\) iff \(v^{-1}\varphi\in\Theta^+\), i.e. iff \(\varphi\notin N_\Theta(v^{-1})\). The rest of \(v\Theta^+\) is \(v\Theta^+\cap(-\Theta^+)=vN_\Theta(v)=-N_\Theta(v^{-1})\). As multisets of vectors (all copies of one vector lie in \(N_\Theta\) together, and \(\Theta\) is \(W_X\)- and $(-1)$-stable), \(v|\Theta^+|=|\Theta^+|-2|N_\Theta(v^{-1})|\). The same count for \(\check\Phi_X^+\) gives \(v(2\check\rho)=2\check\rho-2|N_{\check\Phi}(v^{-1})|\). Subtract and put \(v=w^{-1}\): this is (a).
  2. (b). The vector multiset of \(N_\Theta(w_\gamma)\) is stable under \(\check\theta\mapsto-w_\gamma\check\theta\), so \(|D_\gamma|\) is \(w_\gamma\)-anti-invariant, i.e. \(|D_\gamma|=m_\gamma\check\gamma\). By Statements 6.3.1 and 7.1.5, \(D_\gamma\) contains \(\check\gamma\) (type \(G\)) or a pair with sum \(\check\gamma\) (type \(T\)), so \(m_\gamma\ge1\). Now (a) with \(w=w_\gamma\) reads \(|D_\gamma|-\check\gamma=(1-w_\gamma)\Xi=\langle\Xi,\gamma\rangle\check\gamma\).
  3. (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1). \(\Rightarrow\) (d). By (b), \(\langle\Xi,\gamma\rangle=0\) for all simple \(\gamma\), so \(\Xi\) is \(W_X\)-fixed and (a) gives \(|N_\Theta(w)|=|N_{\check\Phi}(w)|\); doubling gives the second identity. For \(w,I\): \(\check\rho-w\check\rho=|N_{\check\Phi}(w^{-1})|=|N_\Theta(w^{-1})|\). Also \(w|I_-|=-|{-wI_-}|\), and \(-wI_-\subseteq-wN_\Theta(w)=N_\Theta(w^{-1})\). Adding gives the display, a sum of elements of \(\Theta^+\). Finally \(\check\gamma=|D_\gamma|\in\operatorname{cone}(\Theta^+)\) for simple \(\gamma\).
  4. (H1) \(\Rightarrow\) (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1).. Take \(w=w_\gamma\), \(I=D_\gamma\): \(\check\rho-w_\gamma\check\rho+w_\gamma|D_\gamma|=\check\gamma-m_\gamma\check\gamma\in\mathcal T\). If \(m_\gamma>1\) then \(-\check\gamma\in\mathcal T\), while \(\check\gamma\in\mathcal R\subseteq\mathcal T\), contradicting strict convexity. So \(m_\gamma=1\) by (b). (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1). \(\Rightarrow\) (H1) is (d), with \(\mathcal T=\operatorname{cone}(\Theta^+)\), strictly convex because it lies in \(\mathcal T_{col}\).
  5. What this corrects. (i) On p. 62 Sakellaridis uses \(|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho\); the correct identity under (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1). is \(2(\check\rho-w\check\rho)\), the factor 2 found in Computation 1.7 and Audit 5.17; step 3 replaces his argument. (ii) Theorem 1.9 step 2 justifies (7.5) only by the inversion-set argument of Computation 1.7, valid for multiplicity-free \(\Theta^+\subseteq\check\Phi_X^+\); step 3 proves it for every \(\Theta\) satisfying (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1).. (iii) The hypothesis H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1). of Objection 5.20 is exactly (U)(U) · For every simple spherical root γ, the virtual colours belonging to γ sum to its coroot: one copy of the coroot for type G, one pair for type T. Equivalent to (7.5) (S1-1)., i.e. the uniqueness half of Lemma 7.1.4 (proved by Sakellaridis only by inspecting the characteristic-0 classification). A5-20's statement that A1's argument gives H1H1 · Sakellaridis's triangularity (7.5) for the actual Θ of X. Equivalent to (U): the virtual colours belonging to each simple spherical root sum to its coroot (S1-1). 'exactly when \(\Theta^+\subseteq\check\Phi_X^+\) without repetitions' is too narrow: for \(T\backslash PGL_2\) (split) \(\Theta^+=\{\check\gamma/2,\check\gamma/2\}\) is repeated and not a set of coroots, \(m_\gamma=1\), and (7.5) holds. A5-20's rank-one counterexample \(\Theta^+=\{\check\gamma,\check\gamma\}\) is the case \(m_\gamma=2\).
  6. Checks (exact arithmetic, script s1_aJ.py). \(|N_\Theta(w)|=|N_{\check\Phi}(w)|\) for all \(w\) in three cases: \(E_6/F_4\) (\(W_X=W(A_2)\), \(\Theta^+=\check\Phi_X^+\)); Sakellaridis's Example 7.2.4 (\(PGL_2\backslash PGL_2^3\), \(W_X=(\mathbb Z/2)^3\), type-\(T\) colours shared by two roots); and a \(B_2\) example with one type-\(G\) and one type-\(T\) root.

Builds on

  • Theorem 1.9 · Theorem 8.0.2 over \(\mathbb F_q((t))\), \(p\) odd, symmetric \(X\): hypotheses
  • Objection 5.20 · Audit of A1-9: the characteristic-p Theorem 8.0.2 and its hypotheses
  • Computation 1.7 · \(M_\theta\): \(W_X'\cong S_3\), coboundary \(\beta\), formula (7.4), triangularity
  • Audit 5.17 · Audit of A1-7: W_X′ ≅ S_3, the coboundary β, the constant c and triangularity for M_θ

Built on by

  • Proposition S1.2 · The coefficient \(a_J\) of \(m_\lambda\) in the right side of (7.4)
  • Lemma S1.3 · Positivity of \(\hat S\) forces \(c\neq0\) and \(a_J/c>0\) over every local field
  • Remark S1.4 · Sakellaridis's proof of Theorem 8.0.2 also needs \(a_J\neq0\); the p. 62 Remark misses it
  • Theorem S1.5 · Theorem 8.0.2 over \(\mathbb F_q((t))\) for symmetric \(X\): the complete hypothesis list
  • Summary S1.6 · Verdict inputs A5-20 and A5-21 re-derived; how R-4(b) and its open items read
  • Re-check V1 · Re-check by G1
  • Summary 7.5 · Verdict after the audit: unchanged for \(E_6/F_4\); the general direct form completed

Sources