Weak BGG resolution

Tensoring the exact complex of [F1] with the finite-dimensional module ${\mathrm{\Pi }}_{\lambda }$ gives, by exactness of $-\otimes {\mathrm{\Pi }}_{\lambda }$, an exact complex $0\to B_{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}^{{\chi }_0}\otimes {\mathrm{\Pi }}_{\lambda }\to \cdots \to B_1^{{\chi }_0}\otimes {\mathrm{\Pi }}_{\lambda }\to B_0^{{\chi }_0}\otimes {\mathrm{\Pi }}_{\lambda }\to \mathbb{C}\otimes {\mathrm{\Pi }}_{\lambda }\to 0$ whose terms lie in $\mathcal{O}$, with $\mathbb{C}\otimes {\mathrm{\Pi }}_{\lambda }\cong {\mathrm{\Pi }}_{\lambda }$ as $\mathfrak{g}$-modules.

We classify the surviving pairs. Suppose $w\circ 0+\nu =u\circ \lambda$ with $\mathrm{\ell }(w)=k$ and $\nu \in \mathrm{Wt}{\mathrm{\Pi }}_{\lambda }$. Since $w\circ 0=w\rho -\rho$, this reads $\nu =u(\lambda +\rho )-w\rho$; applying $u^{-1}$ and putting $v=u^{-1}w$ gives $u^{-1}\nu =\lambda +\rho -v\rho =\lambda +{\sum }_{\alpha \in {\mathrm{\Pi }}_v}\alpha$ by [F6]. Now $u^{-1}\nu \in \mathrm{Wt}{\mathrm{\Pi }}_{\lambda }$ because the weight multiset is $W$-stable, so $\lambda -u^{-1}\nu \in Q^{+}$ by [F4]; on the other hand $\lambda -u^{-1}\nu =-{\sum }_{\alpha \in {\mathrm{\Pi }}_v}\alpha$ lies in $-Q^{+}$. A nonnegative integral combination of positive roots lying in $-Q^{+}$ is zero, so ${\sum }_{\alpha \in {\mathrm{\Pi }}_v}\alpha =0$, which forces ${\mathrm{\Pi }}_v=\mathrm{\varnothing }$ and hence $v=1$ by [F6]; thus $u=w$ and $\nu =w\lambda$.

Applying the exact projection functor $(-{)}_{{\chi }_{\lambda }}$ to the complex of step 1.1 gives the exact complex $0\to B_{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}(\lambda )\to \cdots \to B_1(\lambda )\to B_0(\lambda )\to ({\mathrm{\Pi }}_{\lambda }{)}_{{\chi }_{\lambda }}\to 0$ with all terms in $\mathcal{O}$, and $({\mathrm{\Pi }}_{\lambda }{)}_{{\chi }_{\lambda }}={\mathrm{\Pi }}_{\lambda }$ by [F5]. Hence the displayed sequence is a resolution of ${\mathrm{\Pi }}_{\lambda }$ by objects of $\mathcal{O}$.

By [F1] each $B_k^{{\chi }_0}$ is Verma-filtered with type $\{w\circ 0:\mathrm{\ell }(w)=k\}$ (each weight once), so [F2] gives that $B_k^{{\chi }_0}\otimes {\mathrm{\Pi }}_{\lambda }$ is Verma-filtered with type the multiset $\{w\circ 0+\nu :\mathrm{\ell }(w)=k,\nu \in \mathrm{Wt}{\mathrm{\Pi }}_{\lambda }\}$. Cutting by ${\chi }_{\lambda }$ and using [F3], the type of $B_k(\lambda )$ consists exactly of those $w\circ 0+\nu$ with $\mathrm{\ell }(w)=k$, $\nu \in \mathrm{Wt}{\mathrm{\Pi }}_{\lambda }$ and $w\circ 0+\nu \in W\circ \lambda$, the multiplicities being inherited from the multiset above.

Conversely, for every $w\in W$ of length $k$ the weight $w\lambda$ occurs in ${\mathrm{\Pi }}_{\lambda }$ by [F4], and $w\circ 0+w\lambda =w(\lambda +\rho )-\rho =w\circ \lambda$, so the pair $(w,w\lambda )$ is a surviving pair contributing the weight $w\circ \lambda$; by step 1.2 these are all the surviving pairs. For fixed $w$ the multiplicity of $w\circ \lambda$ in $\mathrm{Typ}{B}_k(\lambda )$ is therefore the product of the multiplicity of $w\circ 0$ in $\mathrm{Typ}{B}_k^{{\chi }_0}$, which is $1$ by [F1], and the multiplicity of $w\lambda$ in $\mathrm{Wt}{\mathrm{\Pi }}_{\lambda }$, which is $1$ by [F4]. Distinct $w$ give distinct weights by [F7]. Hence $\mathrm{Typ}{B}_k(\lambda )=\{w\circ \lambda :\mathrm{\ell }(w)=k\}$ with each weight occurring once.