§9 · Salvage 1 · Leading coefficients · 2026-10-18T06:45:58Z

Proposition S1.2 · The coefficient \(a_J\) of \(m_\lambda\) in the right side of (7.4)

proved audit: correct

Second audit: correct. I re-derived the inversion-sum formula: 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)., S1-1(d) gives |N_Θ(w)| = |N_Φ̌(w)|, and for u ∈ W_J the only I ⊆ Θ^+ with |I| = ρ̌ − uρ̌ is N_Θ(u^{-1}) (both leftover multisets lie in a strictly convex cone and sum to 0), so it equals my constant-term formula; the (P) recursion N_Θ(ww_γ) = w_γN_Θ(w) ⊔ D_γ is right. Before reading the scripts I computed, by exact Laurent division and by products of rank-one characters: E_6/F_4 gives 1, 1+t, c and Example 7.2.4 gives 1, 1−q^{-1}, 1−q^{-1}, 1−q^{-2} with Π_0 = 1−q^{-2} constant, matching (b), (d) and the agent's s1_aJ.py. Macdonald's product formula in (c) is cited and checked only numerically; the B_2 sample c = (1+t_1)(1+t_2)(1+t_1t_2) checks out, which is enough for the non-vanishing claim.

Under Statements 6.3.1, 7.1.5, (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). and (E), the coefficient of \(m_\lambda\) in the right side of (7.4) is \(a_J=\sum_{u\in W_J}(-1)^{\ell(u)}\prod_{\theta\in N_\Theta(u)}(-\sigma_\theta q^{-r_\theta})\), \(W_J=\mathrm{Stab}(\lambda)\), equal to review G6's constant-term formula. When every colour of a simple root is a positive multiple of its coroot (true for \(E_6/F_4\)), \(a_J\) is a Poincaré polynomial of \(W_J\) with parameters of absolute value $<1$, hence non-zero for every \(q\); for shared type-\(T\) colours it is not, and the general non-vanishing is left to Lemma S1.3.

Proposition S1.2.

Assume Statement 6.3.1 for every \(\gamma\in\Delta_X\), Statement 7.1.5, (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). of Lemma S1.1, and (E): \(\operatorname{cone}(\Theta^+)\cap C^-=\{0\}\). Let \(\lambda\in\Lambda_X\) be antidominant, \(W_J=\operatorname{Stab}_{W_X}(\lambda)\) (generated by the \(w_\gamma\) with \(\langle\lambda,\gamma\rangle=0\)), and $$\Pi_\lambda:=\prod_{\theta\in\Theta^+}\big(1-\sigma_\theta q^{-r_\theta}T_{\check\theta}\big)s_\lambda ,$$ the right side of (7.4) without the factor \(\delta_{P(X)}^{-1/2}(x_\lambda)\).

(a) All weights of \(\Pi_\lambda\) lie in \(\lambda+\operatorname{cone}(\Theta^+)\); \(\Pi_0\) is a constant \(c\) (the \(\omega\) of (7.1); its value (7.3), which rests on Lemma 4.2.3 and hence on (4.5), is not used); and the coefficient of \(m_\lambda\) (equivalently of \(e^\lambda\)) in \(\Pi_\lambda\) is $$a_J=\operatorname{CT}\Big(\prod_{\theta\in\Theta^+}(1-\sigma_\theta q^{-r_\theta}e^{\check\theta})\prod_{\check\gamma\in\check\Phi_J^+}(1-e^{-\check\gamma})\Big)=\sum_{u\in W_J}(-1)^{\ell(u)}\prod_{\theta\in N_\Theta(u)}\big(-\sigma_\theta q^{-r_\theta}\big).$$ It depends only on \(J\); \(a_{W_X}=c\).

(b) \(E_6/F_4\) (\(\Theta^+=\check\Phi_X^+\), \(W_X=W(A_2)\), \(t=q^{-4}\)): \(a_J=\sum_{u\in W_J}t^{\ell(u)}\), i.e. \(a_\emptyset=1\), \(a_{\{\gamma_1\}}=a_{\{\gamma_6\}}=1+t\), \(a_{W_X}=(1+t)(1+t+t^2)=c\).

(c) Assume (P): every virtual colour belonging to a simple \(\gamma\) is a positive multiple of \(\check\gamma\) (automatic for types \(G\) and \(T\) non-split; for type \(T\) split it says \(\check\theta=\check\gamma/2\); it holds whenever the colour cone equals \(\mathcal R\)). Then \(a_J=\sum_{u\in W_J}t_u\) is the Poincaré polynomial of \(W_J\) with parameters $$t_\gamma=q^{-r_\gamma}\ (G),\qquad t_\gamma=-q^{-(r+r')}\ (T\text{ split}),\qquad t_\gamma=q^{-(r+r')}\ (T\text{ non-split}),$$ and \(a_J=\prod_{\check\alpha\in\check\Phi_J^+}\frac{1-t_\alpha t^{\check\alpha}}{1-t^{\check\alpha}}\neq0\) for every \(q>1\).

(d) Without (P), \(u\mapsto(-1)^{\ell(u)}\prod_{N_\Theta(u)}(-\sigma q^{-r})\) need not be multiplicative. For Sakellaridis's Example 7.2.4 (\(W_X=(\mathbb Z/2)^3\), \(\Theta^+\) the four weights \((\check\alpha_1+\check\alpha_2-\check\alpha_3)/2\), \((\check\alpha_1-\check\alpha_2+\check\alpha_3)/2\), \((-\check\alpha_1+\check\alpha_2+\check\alpha_3)/2\), \((\check\alpha_1+\check\alpha_2+\check\alpha_3)/2\), all with \(\sigma=+\), \(r=\frac12\)): \(a_J=1,\ 1-q^{-1},\ 1-q^{-1},\ 1-q^{-2}\) for \(|J|=0,1,2,3\), whereas the Poincaré polynomial would be \((1-q^{-1})^{|J|}\). These are non-zero; the general case is Lemma S1.3.

  1. Expansion. \(\Pi_\lambda=\sum_{I\subseteq\Theta^+}c_I\,s_{\lambda+|I|}\) with \(c_I=\prod_{\theta\in I}(-\sigma_\theta q^{-r_\theta})\), where \(s_\nu=\sum_{w}(-1)^we^{\check\rho-w\check\rho+w\nu}/\prod_{\check\Phi_X^+}(1-e^{\check\gamma})\). Write \(u\star\mu=u(\mu-\check\rho)+\check\rho\). Then \(s_\nu=0\) if \(\nu-\check\rho\) is singular; otherwise \(s_\nu=(-1)^us_\mu\) for the unique antidominant \(\mu\) and \(u\) with \(\nu=u\star\mu\), and \(s_\mu\) is the character with lowest weight \(\mu\).
  2. Weights. The numerator exponents of \(s_{\lambda+|I|}\) are \((\check\rho-w\check\rho+w|I|)+w\lambda\). They lie in \(\lambda+\operatorname{cone}(\Theta^+)\), by Lemma S1.1(d), because \(w\lambda-\lambda\in\mathcal R\) for antidominant \(\lambda\), and because \(\mathcal R\subseteq\operatorname{cone}(\Theta^+)\). The exponent for \(w=u^{-1}\) is \(\mu\) itself, so \(\mu\in\lambda+\operatorname{cone}(\Theta^+)\), and the weights of \(s_\mu\) lie in \(\mu+\mathcal R\). Hence all weights of \(\Pi_\lambda\) lie in \(\lambda+\operatorname{cone}(\Theta^+)\). For \(\lambda=0\), \(\Pi_0\) is \(W_X\)-invariant and its antidominant weights lie in \(\operatorname{cone}(\Theta^+)\cap C^-=\{0\}\) by (E), so \(\Pi_0\) is a constant; call it \(c\). Then \(\Omega_\chi(x_0)=c\,\beta(\tilde\chi)\) by (7.4).
  3. Only \(u\in W_J\) contributes. The weight \(\lambda\) occurs in \(s_\mu\) only if \(\lambda\in\mu+\mathcal R\). Together with \(\mu\in\lambda+\operatorname{cone}(\Theta^+)\) and strict convexity, this forces \(\mu=\lambda\), which has multiplicity 1 in \(s_\lambda\). So exactly the \(I\) with \(\lambda+|I|=u\star\lambda\) contribute, each with sign \((-1)^u\). For those, \(|I|=(u\lambda-\lambda)+(\check\rho-u\check\rho)\). Now (7.5) with \(w=u^{-1}\) gives \(\check\rho-u^{-1}\check\rho+u^{-1}|I|=\lambda-u^{-1}\lambda\in\operatorname{cone}(\Theta^+)\cap(-\mathcal R)=\{0\}\). Hence \(u\in W_J\) and \(|I|=\check\rho-u\check\rho=|N_{\check\Phi}(u^{-1})|\).
  4. Only one \(I\) for each \(u\). By Lemma S1.1(d), \(|I|=|N_\Theta(u^{-1})|\). Split \(I\) by the sign of \(u^{-1}\check\theta\): \(u^{-1}|I|=|u^{-1}I_+|-|{-u^{-1}I_-}|\), where \(-u^{-1}I_-\subseteq N_\Theta(u)\), while \(u^{-1}|N_\Theta(u^{-1})|=-|N_\Theta(u)|\). Hence \(|u^{-1}I_+|+|N_\Theta(u)\smallsetminus(-u^{-1}I_-)|=0\). This is a sum of non-zero vectors of a strictly convex cone, so both multisets are empty and \(I=N_\Theta(u^{-1})\). Since \(\sigma\) and \(r\) are constant on \(W_X\)-orbits and under negation, \(\prod_{N_\Theta(u^{-1})}=\prod_{N_\Theta(u)}\). This gives the sum in (a). The constant-term form is the same sum: \(\check\rho-u\check\rho=\check\rho_J-u\check\rho_J\) for \(u\in W_J\), and \(\sum_{u\in W_J}(-1)^ue^{u\check\rho_J-\check\rho_J}=\prod_{\check\Phi_J^+}(1-e^{-\check\gamma})\).
  5. (b). For \(E_6/F_4\), \(\Theta^+=\check\Phi_X^+\) with multiplicity one and \(\sigma=+\). The exponent is \(r=\langle\check\gamma,\rho_{P(X)}\rangle=4\) on the simple coroots and is constant on \(W_X\)-orbits (Computation 1.7). So \(N_\Theta(u)=N_{\check\Phi}(u)\) has \(\ell(u)\) elements, and each term is \((-1)^{\ell(u)}(-t)^{\ell(u)}=t^{\ell(u)}\). Exact check (script s1_aJ.py): three independent computations agree for \(\lambda=0,(-1,-2),(-2,-1),(-1,-1),(-3,-3),(-2,-4)\) in the basis \(\check\gamma_1,\check\gamma_6\), with values \(c,\ 1+t,\ 1+t,\ 1,\ 1,\ 1+t\). The three computations are: the direct expansion of \(\Pi_\lambda\) through the Kostant partition function, the constant term, and the inversion sum.
  6. (c). Under (P), if \(w\check\gamma>0\) then \(wD_\gamma\) consists of positive multiples of \(w\check\gamma\in\mathcal R\), hence lies in \(\Theta^+\). Therefore \(N_\Theta(ww_\gamma)=w_\gamma N_\Theta(w)\sqcup D_\gamma\). Along a reduced word \(u=w_{\gamma_1}\cdots w_{\gamma_k}\) this gives \(N_\Theta(u)=\bigsqcup_j(w_{\gamma_k}\cdots w_{\gamma_{j+1}})D_{\gamma_j}\). So the term of \(u\) is \(\prod_jt_{\gamma_j}\) with \(t_\gamma=-\prod_{D_\gamma}(-\sigma q^{-r})\), which gives the listed values.

Macdonald's product formula for the Poincaré polynomial of a finite Weyl group with parameters (Macdonald, The Poincaré series of a Coxeter group, Math. Ann. 199, 1972) is \(W(t)=\prod_{\check\alpha>0}(1-t_\alpha t^{\check\alpha})/(1-t^{\check\alpha})\), with \(t^{\check\alpha}=\prod_it_i^{c_i}\) in simple-coroot coordinates. It is a polynomial identity. I checked it exactly at random rational points, including negative values, for \(A_3,B_2,B_3,C_3,D_4,G_2,F_4\) (script s1_poincare.py). Since \(0<|t_\gamma|<1\) (\(r,r'>0\)), every factor is non-zero. (P) holds when the colour cone is \(\mathcal R\). Indeed \(\check\theta,\check\gamma-\check\theta\in\mathcal R\) forces \(\check\theta\in[0,1]\check\gamma\) (simple coroots form a basis of the span of \(\mathcal R\)), and \(\langle\check\theta,\gamma\rangle=1\) then gives \(\check\theta=\check\gamma/2\). Sample: in a \(B_2\) case with \(\gamma_1\) of type \(G\) (\(r=3\)) and \(\gamma_2\) of type \(T\) split (\(r=r'=\frac12\)), all three computations give \(c=(1+q^{-3})(1-q^{-1})(1-q^{-4})\).

  1. (d). In Example 7.2.4 the reflection \(w_{\alpha_2}\) keeps \(\check\alpha_1\) positive. It sends the colour \((\check\alpha_1+\check\alpha_2-\check\alpha_3)/2\), which belongs to \(\alpha_1\), to \(-(-\check\alpha_1+\check\alpha_2+\check\alpha_3)/2\in-\Theta^+\). So the inductive step of 6 fails, and the term of \(u=w_{\alpha_1}w_{\alpha_2}\) is \(+q^{-1}\), not \(q^{-2}\). The three computations agree on the stated \(a_J\).

I also ran an exhaustive search over abstract configurations of this kind (\(W_X=(\mathbb Z/2)^n\), \(n\le4\), colour coordinates in \(\{0,\pm1\}\); 7.1.5, (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)., (E) and 'belongs iff \(\langle\check\theta,\gamma\rangle>0\)' imposed; \(r=\frac12\) for type \(T\) and \(r=1\) for type \(G\)). It found 9 (\(n=3\)) and 24 (\(n=4\)) admissible \(\Theta\), and no \(a_J\) vanishing at any \(q^{-1/2}\in(0,3^{-1/2}]\) (script s1_search.py). This is evidence only; the proof for general \(\Theta\) is the analytic argument of Lemma S1.3.

Builds on

Built on by

  • 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
  • Re-check V2 · Re-check by G6

Sources