The extremal weight cochain of a Weyl element is closed and unique

The dual $\mathfrak{h}$-weights of $({\mathfrak{n}}^{+}{)}^{\ast }$ are $-\alpha$, $\alpha \in {\mathrm{\Phi }}^{+}$, each with multiplicity one: for $H\in \mathfrak{h}$ and $\epsilon \in ({\mathfrak{g}}_{\alpha }{)}^{\ast }$ one has $(H\cdot \epsilon )(x)=-\epsilon ([H,x])=-\alpha (H)\epsilon (x)$, and distinct root spaces give independent dual summands by [F1]; since $V$ is finite dimensional, the identification $C^q({\mathfrak{n}}^{+},V)={\mathrm{Hom}}_k({\mathrm{\Lambda }}^q{\mathfrak{n}}^{+},V)\cong {\mathrm{\Lambda }}^q({\mathfrak{n}}^{+}{)}^{\ast }\otimes V$ of [F7] is therefore spanned by the weight subspaces $C^q({\mathfrak{n}}^{+},V{)}_{\mu }={\sum }_{S\subseteq {\mathrm{\Phi }}^{+},\mathrm{\mid }S\mathrm{\mid }=q}{\epsilon }_S\otimes V_{\mu +⟨S⟩}$, where ${\epsilon }_S={\bigwedge }_{\alpha \in S}{\epsilon }_{\alpha }$ and $⟨S⟩={\sum }_{\alpha \in S}\alpha$, and a summand is zero when $\mu +⟨S⟩$ is not a weight of $V$.

The differential preserves the $\mathfrak{h}$-weight: in both sums of [F7] the loss of the weight $-{\alpha }_i$ from an exterior slot is exactly compensated by the action $x_i$ raising the weight by ${\alpha }_i$, or by the bracket $[x_i,x_j]\in {\mathfrak{g}}_{{\alpha }_i+{\alpha }_j}$, so $d(C^q({\mathfrak{n}}^{+},V{)}_{\mu })\subseteq C^{q+1}({\mathfrak{n}}^{+},V{)}_{\mu }$.

For every $u\in W$ and every $S\subseteq {\mathrm{\Phi }}^{+}$ there is a subset $S^{\mathrm{\prime }}\subseteq {\mathrm{\Phi }}^{+}$ with $u(\rho -⟨S⟩)=\rho -⟨S^{\mathrm{\prime }}⟩$. It suffices to prove this for a simple reflection $u=s_i$, because the claim is stable under composition of the successive simple reflections in a reduced word. For $s_i$: if ${\alpha }_i\in S$ then $s_i(\rho -⟨S⟩)=(\rho -{\alpha }_i)-(s_i⟨S\setminus \{{\alpha }_i\}⟩-{\alpha }_i)=\rho -⟨s_i(S\setminus \{{\alpha }_i\})⟩$ with $s_i(S\setminus \{{\alpha }_i\})\subseteq {\mathrm{\Phi }}^{+}$; if ${\alpha }_i\mathrm{\notin }S$ then $s_i(\rho -⟨S⟩)=\rho -{\alpha }_i-s_i⟨S⟩=\rho -⟨s_i(S)\cup \{{\alpha }_i\}⟩$ with $s_i(S)\cup \{{\alpha }_i\}\subseteq {\mathrm{\Phi }}^{+}$, and ${\alpha }_i\mathrm{\notin }{s}_i(S)$ because $s_i$ permutes ${\mathrm{\Phi }}^{+}\setminus \{{\alpha }_i\}$. Both cases use $s_i\rho =\rho -{\alpha }_i$ and $s_i{\alpha }_i=-{\alpha }_i$ from [F6].

If $S\subseteq {\mathrm{\Phi }}^{+}$ and $⟨S⟩=\rho -w\rho$, then $S={\mathrm{\Phi }}_w$. In the group ring $\mathbb{Z}[P]$ put $\xi ={\sum }_{S\subseteq {\mathrm{\Phi }}^{+}}{e}^{\rho -⟨S⟩}$. The two-case transformation of step 1.3 for each simple reflection is an involutive bijection of the subsets of ${\mathrm{\Phi }}^{+}$: it toggles the simple root and permutes all other roots. Hence $\xi$ is $W$-invariant, with coefficients recording the number of subsets giving each exponent. The coefficient at $e^{\rho }$ is one, since a nonempty sum of positive roots cannot be zero. Invariance implies the coefficient at $e^{w\rho }$ is also one. Thus exactly one subset has sum $\rho -w\rho$; [F6] identifies that subset as ${\mathrm{\Phi }}_w$. No half-root exponent or inverse-Weyl convention is introduced.

(Equality case.) Let $\eta$ be a weight of $V$ and $S\subseteq {\mathrm{\Phi }}^{+}$ with $\mathrm{\parallel }\eta -⟨S⟩+\rho \mathrm{\parallel }=\mathrm{\parallel }\lambda +\rho \mathrm{\parallel }$. Choose $u\in W$ with $\nu :=u^{-1}(\eta +\rho -⟨S⟩)$ dominant, possible by [F5]; then $u^{-1}\eta$ is a weight of $V$ by [F2] and equals $\lambda -\beta$ with $\beta \in Q_{+}$ by [F3], and $u^{-1}(\rho -⟨S⟩)=\rho -⟨S^{\mathrm{\prime }}⟩$ for some subset $S^{\mathrm{\prime }}\subseteq {\mathrm{\Phi }}^{+}$ by step 1.3, so $\nu =\lambda +\rho -(\beta +⟨S^{\mathrm{\prime }}⟩)$. Since $\nu$ is dominant and $\lambda +\rho$ is strictly dominant, $(\nu ,\alpha )\ge 0$ and $(\lambda +\rho ,\alpha )>0$ for every $\alpha \in {\mathrm{\Phi }}^{+}$ by [F5], so, using $W$-invariance of the form for the first equality, $\mathrm{\parallel }\lambda +\rho {\mathrm{\parallel }}^2=(\nu ,\nu )=(\nu ,\lambda +\rho )-(\nu ,\beta +⟨S^{\mathrm{\prime }}⟩)\le (\nu ,\lambda +\rho )=(\lambda +\rho -\beta -⟨S^{\mathrm{\prime }}⟩,\lambda +\rho )=\mathrm{\parallel }\lambda +\rho {\mathrm{\parallel }}^2-(\beta +⟨S^{\mathrm{\prime }}⟩,\lambda +\rho )\le \mathrm{\parallel }\lambda +\rho {\mathrm{\parallel }}^2$. Equality holds throughout, so $(\beta +⟨S^{\mathrm{\prime }}⟩,\lambda +\rho )=0$; as $\beta +⟨S^{\mathrm{\prime }}⟩\in Q_{+}$ and $(\lambda +\rho ,\alpha )>0$ for every positive root, $\beta =0$ and $S^{\mathrm{\prime }}=\mathrm{\varnothing }$. Hence $u^{-1}\eta =\lambda$, so $\eta =u\lambda$, and $\rho -⟨S⟩=u\rho$, so $⟨S⟩=\rho -u\rho$; step 2.1 then gives $S={\mathrm{\Phi }}_u$. If $u_1,u_2$ both satisfy these conclusions for the same $(\eta ,S)$, then $\rho -u_1\rho =⟨S⟩=\rho -u_2\rho$, so $u_1\rho =u_2\rho$ and $u_1=u_2$ because $\rho$ has trivial stabilizer by [F5].

Fix $w\in W$ and $q\ge 0$; a cochain in $C^q({\mathfrak{n}}^{+},V{)}_{w\cdot \lambda }$ is a sum of terms ${\epsilon }_S\otimes v$ with $\mathrm{\mid }S\mathrm{\mid }=q$ and $v\in V$ satisfying $-⟨S⟩+\eta =w\cdot \lambda$ for the weight $\eta$ of $v$, that is $\eta -⟨S⟩=w\cdot \lambda$. Then $\mathrm{\parallel }(\eta -⟨S⟩)+\rho \mathrm{\parallel }=\mathrm{\parallel }w\cdot \lambda +\rho \mathrm{\parallel }=\mathrm{\parallel }w(\lambda +\rho )\mathrm{\parallel }=\mathrm{\parallel }\lambda +\rho \mathrm{\parallel }$ by $W$-invariance of the form, so the equality case of step 3.1 applies and forces $S={\mathrm{\Phi }}_w$ and $\eta =w\lambda$; in particular $\mathrm{\mid }S\mathrm{\mid }=\mathrm{\ell }(w)$. Hence $C^q({\mathfrak{n}}^{+},V{)}_{w\cdot \lambda }=0$ for $q\mathrm{\ne }\mathrm{\ell }(w)$, while for $q=\mathrm{\ell }(w)$ the only contributing subset is $S={\mathrm{\Phi }}_w$ with $\eta =w\lambda$, and $V_{w\lambda }$ is one dimensional by [F2], so $C^{\mathrm{\ell }(w)}({\mathfrak{n}}^{+},V{)}_{w\cdot \lambda }=\mathbb{C}{\gamma }_w$ with ${\gamma }_w$ as in the Statement.

The cochain ${\gamma }_w$ is nonzero, since the ${\epsilon }_{\alpha }$ are nonzero on the one-dimensional root lines and $v_{w\lambda }\mathrm{\ne }0$, and its weight is $-⟨{\mathrm{\Phi }}_w⟩+w\lambda =(w\rho -\rho )+w\lambda =w(\lambda +\rho )-\rho =w\cdot \lambda$ by [F6]. By steps 1.2 and 4.1 the cochain $d{\gamma }_w$ lies in $C^{\mathrm{\ell }(w)+1}({\mathfrak{n}}^{+},V{)}_{w\cdot \lambda }=0$, so ${\gamma }_w$ is a cocycle, and it is not a coboundary because $C^{\mathrm{\ell }(w)-1}({\mathfrak{n}}^{+},V{)}_{w\cdot \lambda }=0$ by step 4.1; hence $[{\gamma }_w]\mathrm{\ne }0$ spans $H^{\mathrm{\ell }(w)}({\mathfrak{n}}^{+},V{)}_{w\cdot \lambda }$, which is therefore one dimensional. Changing any ${\epsilon }_{\alpha }$ or $v_{w\lambda }$ by a nonzero scalar, or changing the order of the wedge, multiplies ${\gamma }_w$ by a nonzero scalar, so the nonzero class may be rescaled, but its one-dimensional cohomology line depends only on $w$ and not on those choices.