Computation 1.7 · \(M_\theta\): \(W_X'\cong S_3\), coboundary \(\beta\), formula (7.4), triangularity

This is the combinatorial half of §§6–7 for \(M_\theta\), done without colours, Knop's criterion [Kn91, §6], or the case analysis of Lemmas 7.1.1–7.1.4. All of it is characteristic-free algebra. The only analytic input is \((\mathrm{S6.3.1})_p\), the char-\(p\) form of the hypothesis (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) for \(M_\theta\) (see Dead end 1.8).

Proof.

1. (a), computation. I generated \(W(E_6)\) (51840 elements) as integer matrices on the root lattice. I formed \(w_L\) (longest element of \(W(D_4)\), nodes 2–5) and \(\theta=-w_L\) on \(X(T)\). This gives \(\gamma_1=2\alpha_1+\alpha_2+2\alpha_3+2\alpha_4+\alpha_5\) and \(\gamma_6=\alpha_2+\alpha_3+2\alpha_4+2\alpha_5+2\alpha_6\), as in S4-3. The elements preserving \(\mathfrak a=\operatorname{span}(\gamma_1,\gamma_6)\) number \(1152=6\cdot192\). Those that are minimal in their \(W_L\)-coset (\(w\alpha_i>0\) for \(i=2,\dots,5\)) number 6 and are closed under products. Their actions on \((\gamma_1,\gamma_6)\) are: \(\mathrm{id}\); \((-\gamma_1,\gamma_1+\gamma_6)\); \((\gamma_1+\gamma_6,-\gamma_6)\); \((\gamma_6,-\gamma_1-\gamma_6)\); \((-\gamma_1-\gamma_6,\gamma_1)\); \((-\gamma_6,-\gamma_1)\). These are exactly the six elements of \(W(A_2)\) for the simple system \(\{\gamma_1,\gamma_6\}\). They are distinct, so the action is faithful and \(N_W(\mathfrak a)/W_L\cong S_3\), with no extra element. So the generic degree \(g\) of \(D/W_X'\to A^*/W\) equals 1, consistent with Proposition 1.5(b). The action on \(\Lambda_X\) is the contragredient one, with simple coroots \(e_1,e_6\) because \(\langle e_i,\gamma_j\rangle\) is the Cartan matrix.

2. (b), cocycle. \(w\mapsto\prod_{\check\alpha>0,w\check\alpha<0}(-e^{\check\alpha})=\sigma(w)e^{\check\rho-w^{-1}\check\rho}\) is a cocycle and \(b_w\) is a cocycle (Proposition 1.6(c)), so \(B\) is a cocycle. So is \(w\mapsto\beta/(\beta\circ w)\). For a simple \(\gamma\), \(w_\gamma\) permutes \(\check\Phi_X^+\setminus\{\check\gamma\}\) and negates \(\check\gamma\). Hence \(\beta/(\beta\circ w_\gamma)=\frac{(1-e^{\check\gamma})(1-q^{-4}e^{-\check\gamma})}{(1-e^{-\check\gamma})(1-q^{-4}e^{\check\gamma})}=-e^{\check\gamma}\frac{1-q^{-4}e^{-\check\gamma}}{1-q^{-4}e^{\check\gamma}}=B_{w_\gamma}\). Two cocycles that agree on generators agree everywhere.

3. (b), symmetrization. By Proposition 1.6(d), \(\Omega_\chi(x_\lambda)/\beta=\delta_{P(X)}^{-1/2}(x_\lambda)\sum_{w}{}^w\!\big(e^{\lambda}/\beta\big)\), and \(1/\beta=\prod(1-q^{-4}e^{\check\gamma})/\prod(1-e^{\check\gamma})\). Expanding the numerator and using \(\sum_w{}^w(e^\nu/\prod_{\check\gamma>0}(1-e^{\check\gamma}))=s_\nu:=\sum_w(-1)^we^{\check\rho-w\check\rho+w\nu}/\prod(1-e^{\check\gamma})\) (Weyl's formula in lowest-weight form, \(\check\rho\) the half-sum) gives the displayed product, with \(T_{\check\gamma}s_\nu=s_{\nu+\check\gamma}\).

4. The constant. Since \(s_{w\star\mu}=(-1)^ws_\mu\) for \(w\star\mu=w(\mu-\check\rho)+\check\rho\), and \(s_0=1\), we get \(s_{|I|}=(-1)^{\ell(w)}\) when \(I\subseteq\check\Phi^+_X\) is the inversion set of \(w\). For the two remaining subsets \(\{\check\gamma_1+\check\gamma_6\}\) and \(\{\check\gamma_1,\check\gamma_6\}\) one has \(|I|=\check\rho\), so the numerator \(e^{\check\rho}\sum_w(-1)^w\) vanishes and \(s_{|I|}=0\). In \(A_2\) the inversion sets have sizes 0, 1, 1, 2, 2, 3. So \(c=\prod(1-q^{-4}T)s_0=1+2q^{-4}+2q^{-8}+q^{-12}=(1+q^{-4})(1+q^{-4}+q^{-8})\). Independently, \(e^{\check\gamma}(\delta_{P(X)}^{1/2})=q^{-\langle\check\gamma,\rho_{P(X)}\rangle}\) equals \(q^{-4}\) for the two simple coroots and \(q^{-8}\) for their sum. This gives \(\beta(\delta_{P(X)}^{1/2})=\frac{(1-q^{-4})^2}{(1-q^{-8})(1-q^{-12})}=c^{-1}\), in agreement with Lemma 4.2.3 (\(\Omega_{\delta^{1/2}}=1\)), whose proof uses only the invariant measure. A numerical check of \(\sum_{w}{}^w(1/\beta)=(1+t)(1+t+t^2)\) at random points, and of \(\beta/(\beta\circ w_{\gamma_1})=B_{w_{\gamma_1}}\), confirms steps 2–4.

5. (c), weights. The weights of the numerator of \(s_{\lambda+|I|}\) are \((\check\rho-w\check\rho+w|I|)+w\lambda\). First, \(w\lambda-\lambda\in\mathcal R\) for antidominant \(\lambda\). Second, write \(I=I_+\sqcup I_-\) according to the sign of \(w\check\theta\). The map \(\check\theta\mapsto-w\check\theta\) injects \(I_-\) into the inversion set \(\{\check\gamma>0:w^{-1}\check\gamma<0\}\), whose sum is \(\check\rho-w\check\rho\). So \(\check\rho-w\check\rho+w|I|\) is a sum of positive coroots, i.e. lies in \(\mathcal R\). (This corrects the intermediate identity \(|\Theta^+|-w|\Theta^+|=\check\rho-w\check\rho\) on p. 62, which is off by a factor 2 when \(\Theta^+=\check\Phi_X^+\); the inequality (7.5) itself holds.) Dividing by \(\prod(1-e^{\check\gamma})\) keeps the weights in \(\lambda+\mathcal R\).

6. (c), leading coefficient. The coefficient of \(m_\lambda\) in \(\prod(1-q^{-4}T)s_\lambda\) is \(1+\sum_{\emptyset\ne I}(\pm1)(-q^{-4})^{|I|}[s_{\lambda+|I|}=\pm s_\lambda]\). Its absolute value is at least \(1-((1+q^{-4})^3-1)>0\) for \(q\ge3\). The factors \(\delta_{P(X)}^{-1/2}(x_\lambda)\) and \(c^{-1}\) are non-zero constants. \(P_0=1\) by step 4.

7. Cone. If \(v=a\check\gamma_1+b\check\gamma_6\in\mathcal R\cap C^-\), then \(2a-b\le0\) and \(-a+2b\le0\), so \(a+b\le0\) and \(a=b=0\).