${\mathbb{C}}[{\mathfrak{h}}^*]$-module.

We will use frequently and without comment the fact that ${\mathbb{C}}[{\mathfrak{h}}^*]$ is a free ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$-module. Moreover, as ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$ is a polynomial ring, any projective ${\mathbb{C}}[{\mathfrak{h}}^*]^{{W}}$ is free by the Quillen-Suslin Theorem.

\(1) By induction, we may assume that $i=j+1$. The inclusion $B_{ij}=eH_{c+i}\delta e\subseteq U_{c+i}$ is immediate. If $p\in H_{c+i}$ then, by , $$epe_-\delta\ =\ e\delta^{-1}\delta p
e_- \delta\ =\ \delta^{-1} e_- \delta p e_-\delta\ \in\ \delta^{-1}
e_-H_{c+i}e_-\delta\ =\ U_{c+j}.$$

\(2) By the PBW Theorem \[PBW\], each $H_d$ is free as a left ${\mathbb{C}}[{\mathfrak{h}}]$-module and as a right ${\mathbb{C}}[{\mathfrak{h}}^*]$-module. Therefore, $H_d$ is a free left ${\mathbb{C}}[{\mathfrak{h}}]^W$-module as is its summand $H_de$. Under the left action of ${{W}}$, $(H_de)^{{W}}=eH_de$ since, if $fe\in (H_{d}e)^W$, then $fe =|{{W}}|^{-1}\sum_{w\in {{W}}} wfe = efe$. But $ (H_{d}e)^W$ is a ${{W}}$-module summand of $H_{d}e$, while the actions of ${{W}}$ and ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$ commute. Thus $U_{d}=(H_{d}e)^{{W}}$ is a ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module summand of $H_{d}e$ and hence is free. By Corollary \[morrat-cor\], $N(k)\cong B_{k0}\otimes_{U_c}eH_c$ is a projective left $U_{c+k}$-module and hence a free left ${\mathbb{C}}[{\mathfrak{h}}]^{{W}}$-module.

On the other hand, $N(k)$ is a projective right $H_c$-module and hence a projective right ${\mathbb{C}}[{\mathfrak{h}}^*]$-module.

\(3) Set $X=H_{c}\otimes_{{\mathbb{C}}[{\mathfrak{h}}^*]} {\mathbb{C}}$. Clearly $X\in \mathcal{O}_{c}$ in the sense of and, by , $X  \cong {\mathbb{C}}[{\mathfrak{h}}]\otimes_{{\mathbb{C}}} {\mathbb{C}}{{W}}$ as left ${\mathbb{C}}[{\mathfrak{h}}]\ast {{W}}$-modules. Thus $X$ is a finitely generated free left ${\mathbb{C}}[{\mathfrak{h}}]$-module and so, by [@GGOR Proposition 2.21], $X$ has a filtration whose factors are standard modules.

By def