Tor with the trivial module is computed by the weak BGG resolution

We compute the weight multiset of $B_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda )$ from the Verma filtration of [F1]. Fix a filtration $0=N_0\subseteq N_1\subseteq \cdots \subseteq N_n=B_k(\lambda )$ with $N_j\mathrm{/}{N}_{j-1}\cong M(w_j\circ \lambda )$, where $w_1,\dots ,w_n$ list $W_k$. For each $j$, the vanishing ${\mathrm{Tor}}_1(\mathbb{C},M(w_j\circ \lambda ))=0$ from [F3] and its long exact sequence give a short exact sequence $0\to N_{j-1}\mathrm{/}{\mathfrak{n}}^{-}{N}_{j-1}\to N_j\mathrm{/}{\mathfrak{n}}^{-}{N}_j\to (N_j\mathrm{/}{N}_{j-1})\mathrm{/}{\mathfrak{n}}^{-}(N_j\mathrm{/}{N}_{j-1})\to 0$; the last term is $\mathbb{C}{v}_{w_j}$ by [F2], Choose a weight-vector lift in $N_j$ of the highest weight vector in $N_j\mathrm{/}{N}_{j-1}$; such a lift exists by $\mathfrak{h}$-semisimplicity. Its coinvariant class ${\bar{v}}_{w_j}$ has weight $w_j\circ \lambda$ and maps to the generator of the last term. Together with the embedded earlier classes it spans $N_j\mathrm{/}{\mathfrak{n}}^{-}{N}_j$; induction on $j$ gives a spanning set for $B_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda )$.

The induced map ${\bar{d}}_k\text{\hspace{0.17em}⁣}:B_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda )\to B_{k-1}(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_{k-1}(\lambda )$ is $\mathfrak{h}$-equivariant: the differential $d_k$ is a $\mathfrak{g}$-homomorphism, ${\mathfrak{n}}^{-}{B}_k(\lambda )$ and ${\mathfrak{n}}^{-}{B}_{k-1}(\lambda )$ are $\mathfrak{h}$-stable, and the induced map on quotients commutes with the action of $\mathfrak{h}$.

The homology of $(B_{\bullet }(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_{\bullet }(\lambda ),{\bar{d}}_{\bullet })$ computes Tor: by [F1] and [F3] the complex $B_{\bullet }(\lambda )$ is an $F$-acyclic resolution of ${\mathrm{\Pi }}_{\lambda }$ with all terms and all syzygies in $\mathcal{O}$, so the acyclic-resolution theorem of [F4] gives ${\mathrm{Tor}}_n^{U({\mathfrak{n}}^{-})}(\mathbb{C},{\mathrm{\Pi }}_{\lambda })\cong H_n(B_{\bullet }(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_{\bullet }(\lambda ))$ for every $n$.

These classes have weights $w_j\circ \lambda$, which are pairwise distinct by [F2], and $B_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda )$ is $\mathfrak{h}$-semisimple by [F5]; each newly lifted class has nonzero image in the corresponding one-dimensional quotient of the short exact sequence in step 1.1, and the earlier classes embed injectively. Induction therefore proves these classes nonzero, linearly independent and a basis. Therefore $\mathrm{Wt}(B_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda ))=\{w\circ \lambda :\mathrm{\ell }(w)=k\}$ with each weight occurring once, and ${\dim }_{\mathbb{C}}{B}_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda )=\mathrm{\mid }{W}_k\mathrm{\mid }$.

For every $k\ge 1$ the map ${\bar{d}}_k$ is zero. Indeed, its source has weights $\{w\circ \lambda :\mathrm{\ell }(w)=k\}$ and its target has weights $\{w^{\mathrm{\prime }}\circ \lambda :\mathrm{\ell }(w^{\mathrm{\prime }})=k-1\}$ by step 2.1, and these two sets are disjoint by the pairwise distinctness in [F2]; an $\mathfrak{h}$-equivariant map sends a weight vector of weight $\mu$ into the weight-$\mu$ space of the target, so every basis vector of the source maps to $0$.

Consequently, for $k\ge 1$, the degree-$k$ homology of $B_{\bullet }(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_{\bullet }(\lambda )$ equals $B_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda )$ (all incoming and outgoing induced differentials at degree $k$ vanish by step 3.1), so ${\mathrm{Tor}}_k^{U({\mathfrak{n}}^{-})}(\mathbb{C},{\mathrm{\Pi }}_{\lambda })\cong B_k(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{B}_k(\lambda )\cong {\mathbb{C}}^{\mathrm{\mid }{W}_k\mathrm{\mid }}$ by steps 1.3 and 2.1. The claims about the vanishing induced differential and the weight multiset are steps 3.1 and 2.1, and for $k>\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ the module is zero by [F5].