The higher-level Schur-Weyl projective generator and Schur functor

Exactness of ${\pi }_{\alpha }$. $T_{\alpha }$ is a direct summand of the projective object $T=P_c\otimes V^{\otimes d}$ by [F1]; ${\mathrm{Hom}}_{\mathfrak{g}}(P,-)$ is exact for projective $P$, and passing to a direct summand preserves exactness, so ${\pi }_{\alpha }$ is exact.

The left adjoint. The tensor-Hom adjunction gives ${\pi }_{\alpha }^{\ast }=T_{\alpha }{\otimes }_{B_{\alpha }}-$ as left adjoint of ${\pi }_{\alpha }$; tensoring with a right module is right exact but not left exact in general, and the rank-one witness exhibits the failure of exactness. Free $B_{\alpha }$-modules map to direct sums of copies of $T_{\alpha }$, which are projective, and direct summands of projective objects are projective, so ${\pi }_{\alpha }^{\ast }$ sends $B_{\alpha }$-projectives to projectives.

The whole-summand generator. By [F4], the finite sum $G_d$ is projective and contains every Col-labelled cover; exact central projection gives the projective generator $G_{\alpha }$ of the whole block. The same supplier gives actual inclusions of each prefix summand in $T_d$ with standard-filtered quotients, providing the projective-copresentation prerequisite without identifying $G_{\alpha }$ with the detector $T_{\alpha }$.

Recovery on $\mathrm{add}(T_{\alpha })$. The evaluation ${\pi }_{\alpha }^{\ast }{\pi }_{\alpha }(T_{\alpha })=T_{\alpha }{\otimes }_{B_{\alpha }}{\mathrm{End}}_{\mathfrak{g}}(T_{\alpha })\to T_{\alpha }$ is an isomorphism by the endomorphism-algebra description [F2] (the standard evaluation of a free rank-one module), and for direct summands it follows by restriction of idempotents; hence ${\pi }_{\alpha }^{\ast }{\pi }_{\alpha }$ recovers every object of $\mathrm{add}(T_{\alpha })$, in particular every standard-labelled prinjective contained in $T_{\alpha }$.

The rank-one boundary. Take $q=(1,1)$, $c=(0,0)$ and $d=1$. The ground ${\mathfrak{s}\mathfrak{l}}_2$ highest weight is $-1$, and its natural tensor gives the regular-block detector $T=P(-2)$. The rank-one Verma formula $e{f}^{k}v=k(\lambda -k+1)f^{k-1}v$ gives the nonsplit sequence $0\to L(-2)\to \mathrm{\Delta }(0)\to L(0)\to 0$ and $\mathrm{\Delta }(-2)=L(-2)$. BGG reciprocity in [F1] gives $P(0)=\mathrm{\Delta }(0)$ and the nonsplit flag $0\to \mathrm{\Delta }(0)\to T\to L(-2)\to 0$; the flag cannot have $\mathrm{\Delta }(0)$ on top because the cover has head $L(-2)$. Thus $T$ has head and socle $L(-2)$ and middle factor $L(0)$. Let $x$ be its head-to-socle endomorphism. Then $x^2=0$, and the level-two $d=1$ endomorphism algebra is $R=\mathbb{C}[x]\mathrm{/}{x}^2$ by the supplied Hecke identification. Any map $T\to P(0)$ has image in $\mathrm{soc}P(0)=L(-2)$: its composite to $L(0)$ is zero. Hence $\pi P(0)=R\mathrm{/}(x)$, and ${\pi }^{\ast }\pi P(0)=T\mathrm{/}xT$ has head $L(-2)$, whereas $P(0)$ has head $L(0)$, so recovery fails. Moreover tensoring the injection $(x)\hookrightarrow R$ with $T$ gives $T\mathrm{/}xT\stackrel{x}{\to }T$, with nonzero kernel $\ker x\mathrm{/}xT\cong L(0)$. Thus ${\pi }^{\ast }$ is not exact. All modules lie in this fixed regular block and retain its scalar ${\mathfrak{g}\mathfrak{l}}_2$ centre.

Assembly. Since the charges decompose $T$ into finitely many central components in each degree and the categories are locally unital in the infinite case, the functors ${\pi }_{\alpha }$ assemble to the locally unital sum; the ambient category retains all labels, including those killed by the functors.