The Whittaker functor is exact and fully faithful on the required projectives

The categorical nonstandard kernel. In the Fock $L\mathrm{/}D$ bases the normalized projection is a coordinate projection with coefficients $0,1$, so specialization at $1$ has precisely the off-standard span as kernel. Module-linearity makes this kernel stable under every $E_i,F_i$. Under [F3], expand $[f_{i}L(A)]$ for off-standard $A$ as ${\sum }_C{n}_C{L}_C(1)$, where $n_C\ge 0$ are actual composition multiplicities. Projecting gives ${\sum }_{C\in \mathrm{S}\mathrm{t}\mathrm{d}}{n}_C{D}_C(1)=0$, and independence forces every such $n_C=0$; the same holds for $e_i$. Exactness extends this to the entire off-standard Serre subcategory. For complex $c$, apply the product of the component projections with entry and residue shift $a\mapsto a-c_{\kappa }+n-n_{\kappa }$, $n_{\kappa }={\max }_{j\in \kappa }{q}_j$; its coordinate kernel consists of labels nonstandard in at least one component. A common downward origin shift includes any finitely many labels and residue moves in an admitted cut and does not change ${\mathcal{O}}_c$ or the standard predicate.

Maximal ground-column induction. Split off the last column, with two-block Levi ${\mathfrak{g}}^{\mathrm{\prime }}\oplus {\mathfrak{g}}^{\mathrm{\prime }\mathrm{\prime }}$ and upper radical $\mathfrak{r}$, and define $R(X)=U(\mathfrak{g}){\otimes }_{U(\mathfrak{q})}({\mathbb{C}}_{\beta }\otimes X)$, $\beta =(0^{N^{\mathrm{\prime }}},(N^{\mathrm{\prime }}{)}^{q_l})$, and right adjoint $H(Y)={\mathbb{C}}_{-\beta }\otimes Y^{\mathfrak{r}}$. PBW gives exact $R$ and $R(N(C^{\mathrm{\prime }})⊠N(C^{\mathrm{\prime }\mathrm{\prime }}))=N(C)$. The top Levi slice of a simple is its Levi simple: a proper submodule of that slice would generate a proper global submodule by PBW. There are no additional lower $\mathfrak{r}$-invariants, since moving opposite-radical PBW monomials across the nondegenerate contravariant form pairs a lower invariant to zero against every vector. Hence $H(L(C))=L(C^{\mathrm{\prime }})⊠L(C^{\mathrm{\prime }\mathrm{\prime }})$. For a costandard, invariant functionals vanish on all positive-degree opposite-radical monomials, giving $H(\mathrm{\nabla }(C))=\mathrm{\nabla }(C^{\mathrm{\prime }})⊠\mathrm{\nabla }(C^{\mathrm{\prime }\mathrm{\prime }})$. Taking the exact top eigenvalue slice of the first-block trace, when $B,C$ have the same degree split, now gives $[N(C):L(B)]=[N(C^{\mathrm{\prime }}):L(B^{\mathrm{\prime }})][N(C^{\mathrm{\prime }\mathrm{\prime }}):L(B^{\mathrm{\prime }\mathrm{\prime }})]$: smaller top eigenvalues contribute nothing and the top Levi simple determines its concatenated label uniquely.

Actual standard embeddings. Given $C$, shift columns $j\ge 2$ upward by integers $k_j\ge 0$, chosen successively so each shifted entry in column $j$ exceeds every earlier-column entry in its coset. The resulting $B$ is standard and Bruhat-minimal, hence $N(B)=L(B)$ by [F2]. Write column $j$ of $C$ as $a_{j,1}<\cdots <a_{j,q_j}$ and set ${\mathcal{E}}_j=(e_{a_{j,q_j}}\cdots e_{a_{j,q_j}+k_j-1})\cdots (e_{a_{j,1}}\cdots e_{a_{j,1}+k_j-1})$, composing rightmost first. Apply ${\mathcal{E}}_2$, then ${\mathcal{E}}_3$, through ${\mathcal{E}}_l$: the smallest entry is lowered first. In each elementary standard flag, alternative valid lowers occur in earlier columns, hence have strictly smaller prefix Bruhat label. Our desired section is largest and can be moved to the bottom: each adjacent swap splits by the Ext ordering in [F2]. Thus each move gives an actual monomorphism of standards, and exact subsequent residue functors preserve it. Composition gives $N(C)\hookrightarrow {\mathcal{E}}_l\cdots {\mathcal{E}}_{2}L(B)$. If $L(A)\hookrightarrow N(C)$, biadjunction gives a surjection from an iterated $f_i$ image of $L(A)$ onto the standard simple $L(B)$; step 1.1 forbids this for off-standard $A$. Every standard therefore has standard-labelled socle.

An actual cover isomorphism. Append the ground last column to $B^{\mathrm{\prime }}$. Every $C\ge B$ retains that ground column: a same-coset Bruhat move toward a larger label increases earlier-column sums, so the last-column sum cannot exceed its ground sum, whereas polynomiality makes each last-column entry at least its ground value. Equality forces all of them to be ground; admissibility excludes cross-coset swaps. Thus all $C\ge B$ have degree split $(d^{\mathrm{\prime }},0)$. Adjointness and step 1.2 show that $X=R(P(B^{\mathrm{\prime }})⊠{\mathbb{C}}_{c_l+q_l-n})$ has simple head $L(B)$. Lift the cover map to obtain $P(B)\twoheadrightarrow X$, whose image is all of $X$ because it surjects onto its head. By BGG and step 1.2 their standard-flag multiplicities agree in the fixed split and vanish elsewhere. The kernel of this existing epimorphism therefore has every composition multiplicity zero and is zero. Consequently $X\cong P(B)$; an equality of characters has only been used to kill the kernel of an already constructed map.

The detector and the injective hulls it contains. The ground simple is projective-injective. Tensoring by $V$ preserves both projectives and injectives because the distinct tensor functor $-\otimes V^{\ast }$ is exact and adjoint on both sides. The tensor product contravariant form gives $T\cong DT$. Every simple submodule of a standard-filtered object embeds in one of its successive standard sections, by taking the first section meeting it; hence step 2.1 proves standard socle for every projective and for $T$. Duality gives standard head for $T$. Every standard label of positive degree is ${\tilde{f}}_{i}B$ for a lower-degree standard label by [F3]; induction using the actual crystal head of $f_{i}L(B)$ gives $T\twoheadrightarrow L(A)$. Lift this epimorphism through $P(A)\twoheadrightarrow L(A)$; essentiality makes the lift surjective and projectivity splits it. Thus the indecomposable summands of $T$ are exactly the standard-labelled covers, all with positive multiplicity and all injective. Moreover $I(A)=DP(A)\in \mathrm{add}(DT)=\mathrm{add}(T)$ for every standard $A$. This hull membership does not presume individual summand self-duality.

Cumulative-column projectives. Put $V_{\le j}=E_1\oplus \cdots \oplus E_j$ and $Z_A={⨂}_i{\mathrm{Sym}}^{a_i}(V_{\le \mathrm{col}(i)})$, with degree $d={\sum }_i{a}_i$, and $Z(A,c)={\mathrm{Ind}}_{\mathfrak{p}}^{\mathfrak{g}}({\mathbb{C}}_{-\rho }\otimes Z_{A,c})$. Induct on the number of columns. One column is completely reducible. Removing the last column and using tensor identity gives $Z(A,c)=R(Z(A^{\mathrm{\prime }},c^{\mathrm{\prime }})⊠{\mathbb{C}}_{c_l+q_l-n})\otimes {⨂}_{i=N-q_l+1}^N{\mathrm{Sym}}^{a_i}(V)$. By induction and step 2.2 the first factor is a sum of actual projective covers. The remaining finite tensor product preserves projectives by its exact distinct-dual adjoint. Polynomial labels form whole blocks: their shared entry multiset cannot admit a below-ground entry, since column strictness would put an entry below the common coset minimum in the bottom row. Thus these ambient projectives lie in the required finite polynomial summand. Every $Z(A,c)$ is genuinely projective.

Every cover occurs and the quotient has a standard flag. For a label $B$, take $A=B-A_c$. Project each prefix onto its last-column Levi quotient. The tensor product of the resulting row symmetric powers surjects onto $V(B)$: row symmetrization followed by column antisymmetrization of the row-filled tensor produces a nonzero polynomial highest vector, and Levi complete reducibility gives the simple Schur quotient. Exact induction gives $Z(A,c)\twoheadrightarrow N(B)\twoheadrightarrow L(B)$, and projectivity plus essentiality of the cover makes $P(B)$ a split summand of $Z(A,c)$. Averaging symmetrizers embed $Z_A$ as a $\mathfrak{p}$-submodule of $V^{\otimes d}$; PBW preserves this injection and identifies the ambient induction with $T$. Its finite $\mathfrak{p}$-quotient has polynomial Levi simple sections: the radical acts nilpotently by the column filtration, and its invariant subspace in each simple section is Levi-stable and therefore the whole section. Inducing its composition filtration exactly gives a standard flag of $T\mathrm{/}Z(A,c)$.

All-projective copresentations. For an indecomposable projective $P$, choose a splitting $Z(A,c)=P\oplus Q$ from step 4.1 and view $P$ as its actual submodule of $T$. The sequence $0\to Q\to T\mathrm{/}P\to T\mathrm{/}Z(A,c)\to 0$ gives a standard flag of $T\mathrm{/}P$, since $Q$ is projective and has its BGG flag. Its socle is standard by step 2.1, so its finite injective hull $I(T\mathrm{/}P)$ belongs to $\mathrm{add}(T)$ by step 3.1. Compose $T\twoheadrightarrow T\mathrm{/}P\hookrightarrow I(T\mathrm{/}P)$ to obtain $0\to P\to T\to I(T\mathrm{/}P)$, exact at $P,T$. Finite sums give every projective a two-step copresentation $0\to P\to T_0\to T_1$ with both $T_i\in \mathrm{add}(T)$. The last arrow need not be surjective.

The right-adjoint unit. Choose a full finite projective generator of the entire category-O summand containing $T$ as an explicit summand, with algebra $A$, and let $e$ be the projector onto this actual $T$. Thus $eAe={\mathrm{End}}_{\mathfrak{g}}(T{)}^{\mathrm{o}\mathrm{p}}=B$, retaining all its multiplicities. For $F(X)=eX$, $B=eAe$, the right adjoint is $G_e(M)={\mathrm{Hom}}_B(eA,M)$ with unit ${\eta }_X(x)(ea)=eax$. The restriction $e{\eta }_X$ is an isomorphism, so its kernel and cokernel are annihilated by $e$. If $X$ has standard socle then ${\eta }_X$ is monic. In particular ${\eta }_T$ is monic and splits since its source $T$ is injective. A complementary summand $C$ has $eC=0$, but adjunction gives ${\mathrm{Hom}}_A(C,G_{e}FT)={\mathrm{Hom}}_B(eC,FT)=0$, forcing its inclusion to vanish and $C=0$. Thus $\eta$ is an isomorphism on $\mathrm{add}(T)$. Apply exact $F$ and left-exact $G_e$ to step 5.1; the comparison of kernels gives ${\eta }_P$ an isomorphism for every projective. Adjunction therefore proves $F$ fully faithful on all projectives, while left-adjoint recovery has only been asserted on $\mathrm{add}(T)$.

Tilting full faithfulness. If $X$ has standard head and $Y$ standard socle, then ${\eta }_Y$ is monic and its cokernel is annihilated by $e$. Every map from $X$ to that cokernel vanishes: a nonzero image would have a standard simple head because it is a quotient of $X$, while all its factors would be off-standard. By adjunction each $B$-map $FX\to FY$ gives $X\to G_{e}FY$, which therefore factors uniquely through $Y$. This proves the Hom isomorphism. A tilting's standard flag gives standard socle by step 2.1, and its costandard flag gives standard head by duality. Thus the quotient is fully faithful on all tiltings.

The actual Whittaker comparison and quotient. By [F1] the exact Whittaker functor factors through $F$ as the actual equivalence $U{\otimes }_B-$, with $U$ a right $B$-progenerator and $W_d={\mathrm{End}}_B(U)$. Its inverse is ${\mathrm{Hom}}_{W_d}(U,-)$, and its unit on $FX$ is $m\mapsto (u\mapsto u\otimes m)$. Under the source action this is precisely ${\gamma }_X(f)=V_{\text{\hspace{0.17em}⁣}\mathrm{W}\mathrm{h}}(f)j^{-1}$, since the counit on $T{\otimes }_{B}FX$ is evaluation; thus the comparison is a natural isomorphism of actual maps and Hecke modules. In particular $U$ is a left $W_d$-progenerator, and $V_{\text{\hspace{0.17em}⁣}\mathrm{W}\mathrm{h}}$ is fully faithful on projectives by step 6.1. The corner functor $F$ is the Serre quotient with kernel consisting of objects with only off-standard factors; applying the genuine equivalence identifies the same quotient for $V_{\text{\hspace{0.17em}⁣}\mathrm{W}\mathrm{h}}$. Projectivity is preserved on $\mathrm{add}(T)$, whose image consists of the corresponding projective covers. This does not assert preservation for other ambient projectives.

Simple images and standard heads. The exact corner functor kills precisely off-standard simples and identifies its other simple images with the distinct exhaustive $B$-simples; the genuine equivalence of step 8.1 transports these to the corresponding finite-W simples $L_W(A)$. For standard $A$, $P(A)\in \mathrm{add}(T)$, so $\pi P(A)$ is an indecomposable $B$-projective with head $\pi L(A)$: its Hom to each simple is the corresponding Hom from $P(A)$ by the elementary add$(T)$ equivalence. Lift $P(A)\to L(A)$ through $N(A)\to L(A)$; the image surjects onto the standard's head and is therefore the whole standard. Applying exact $\pi$ makes $\pi N(A)$ a nonzero quotient of this indecomposable projective, so it has the same unique simple head. Through Morita, the actual finite-Levi standard $V(A)=V_{\text{\hspace{0.17em}⁣}\mathrm{W}\mathrm{h}}N(A)$ likewise has unique simple head $L_W(A)$. This does not assert that a specified finite-Levi highest vector generates it.

Independent row compression. Add $t$ full ground rows until the enlarged smallest column has height at least $d$; all roots $a_j=c_j+q_j-n$ remain fixed. For each row $r$, the normalized coefficient $D_r^{(1)}$ is its row-Cartan operator, with ground scalar ${\kappa }_r={\sum }_{j:r\text{ lies in column }j}{c}_j$. On the actual tensor space and every finite-Levi standard of degree $d$ its eigenvalues are ${\kappa }_r+h$, $0\le h\le d$, since the remaining weight is polynomial row occupancy. Thus the SAME finite-image element $e={\prod }_{r\text{ deleted}}{\prod }_{h=1}^d(1-(D_r^{(1)}-{\kappa }_r)\mathrm{/}h)$ is the zero-deleted-occupancy projector on both tensor space and all these standards. In $U^{+}$ it selects tensors using only retained rows. The actual dot formula preserves row multisets, so $e$ commutes with dots and crossings and belongs to the left image by the independent raw first centralizer of [F1]. Each right matrix restricts identically under the retained-box embedding: left neighbours and row/column indices agree and its scalar $a_j$ is unchanged. Therefore $e{U}^{+}=U$ as right Hecke modules. Extending a commuting endomorphism of $e{U}^{+}$ by zero shows $e{A}^{+}e=A$, where $A^{+},A$ are the large and small left image algebras. For the finite-Levi standard, the zero-deleted-occupancy part of ${\mathbb{C}}_{c_j}\otimes S_{{\mu }^{(j)}}({\mathbb{C}}^{q_j+t})$ is ${\mathbb{C}}_{c_j}\otimes S_{{\mu }^{(j)}}({\mathbb{C}}^{q_j})$ by its tensor Young-symmetrizer realization, and vanishes exactly when a deleted-row partition entry is nonzero. In the original path expansion, Levi projection kills off-column factors and each surviving strictly-forward path has at most one factor in each column. A deleted intermediate row changes removed-coordinate occupancy in a column with no later factor there to undo it; compression therefore kills that monomial. Remaining paths are exactly the small paths. The head shift in column $j$ changes by $-(j-1)t$, cancelling the retained source-reading shift $+(j-1)t$; its normalized highest weight remains the entry plus row index minus one. The corner action and its Gauss coefficients consequently give $eV(A^{+})=V(A)$ for contractible labels and zero for noncontractible labels, as actual module identifications.

Simple contraction without highest-vector assumptions. Every noncontractible standard simple is killed by $e$, because it is a quotient of the standard annihilated in step 9.2. For a finite algebra and an idempotent $e$, restriction gives a bijection between its simple modules surviving $e$ and the simples of $eAe$: any nonzero vector in $eL$ generates $eL=eAev$, and for a small simple $M$ the module $Ae{\otimes }_{eAe}M$ has a unique simple quotient surviving $e$. Indeed its $e$-part is $M$; every proper submodule has zero $e$-part, since a submodule containing that part generates the whole induced module. The sum of all such proper submodules is still annihilated by $e$ and is the unique maximal submodule. Apply this to the independent corner in step 9.2. By step 9.1 both simple lists are complete, and small standard labels are in bijection with contractible large standard labels by adjoining full ground rows. Thus the survivors form a subset of the contractible list with exactly the same finite cardinality, so every contractible simple survives. Its restriction is a simple quotient of $eV(A^{+})=V(A)$, which has the unique head from step 9.1, hence $e{L}_W(A^{+})=L_W(A)$.

Fullness and the central Hecke corner. If a contractible $A^{+}$ and another label have the same entry-multiset central character, admissibility separates charge cosets and identifies all charges within each coset. Its minimal bottom entry $c_j+1-(n+t)$ appears in each column of that coset; strictness and nonnegative ground differences force the same entries into the bottom row of the other label. Remove that row and repeat for every deleted row. Thus the contractible labels form a union of whole central-character blocks. Let $f$ be their central support in $A^{+}$; then $e=ef=fe$, since $e$ kills all complementary simples. Every simple of $A_f=f{A}^{+}f$ survives $e$ by step 10.1. If $A_{f}e{A}_f$ were proper, its nonzero quotient would have a simple killed by $e$, a contradiction. Hence $f={\sum }_k{a}_{k}e{b}_k$. Multiplication $A_{f}e{\otimes }_{e{A}_{f}e}eM\to M$ has inverse $m\mapsto {\sum }_k{a}_{k}e\otimes e{b}_{k}m$; balancing and this identity verify both composites. This gives a faithful Morita equivalence on all supported-block modules. The operator $f$ belongs to the right centralizer and is central there, so restriction identifies $f{B}^{+}f={\mathrm{End}}_{A_f}(f{U}^{+}{)}^{\mathrm{o}\mathrm{p}}$ with $B={\mathrm{End}}_{e{A}_{f}e}(e{U}^{+}{)}^{\mathrm{o}\mathrm{p}}$. To verify enlarged stable faithfulness directly, choose $d$ distinct terminal boxes on length-$l$ Jordan strings. In the associated graded tensor action each bounded PBW word $x_1^{a_1}\cdots x_d^{a_d}w$, $0\le a_i<l$, sends their ordered terminal tensor to the tensor in which those distinct strings have been permuted by $w$ and the box on string $i$ moved $a_i$ places left. These $l^{d}d!$ outputs are distinct basis tensors, so the graded PBW word images are linearly independent. A nonzero filtered Hecke element has a nonzero leading PBW symbol; graded injectivity makes its leading operator nonzero, hence its actual operator is nonzero. The enlarged stable action is therefore faithful, giving $B=f{H}_d(\mathbf{a})f$. Its trace is therefore symmetric and nondegenerate by [F4]. This local proof precedes, and does not depend on, a later stable-range definition.

The actual duality comparison. Let $\beta :T\to DT$ be the tensor contravariant form. Put $\delta M={\mathrm{Hom}}_{\mathfrak{g}}(M,T)$ with $(bf)(m)=f(m)b^{\ast }$, and $X^{\mathrm{\#}}={\mathrm{Hom}}_B(X,B)$ with the same twist. Transposition gives the natural isomorphism $\delta M\to \pi DM$, $f\mapsto Df\circ \beta$, whose inverse is $g\mapsto {\beta }^{-1}\circ Dg\circ {\mathrm{bidual}}_M$. This uses only $T\cong DT$, not individual projective self-duality. Tensor-Hom gives $\delta {\pi }^{\ast }\cong \mathrm{\#}$, and the symmetric trace of step 11.1 gives $\mathrm{\#}\cong D_B$ by $h\mapsto \mathrm{tr}\circ h$. Define ${\theta }_M:M\to D{\pi }^{\ast }\delta M$ by ${\theta }_M(m)(t\otimes f)=(t,f(m))$. The form identity $(xtb,t^{\mathrm{\prime }})=(t,x^{\mathrm{t}\mathrm{r}}{t}^{\mathrm{\prime }}{b}^{\ast })$ makes this balanced and $\mathfrak{g}$-linear. Applying $\pi$ gives a natural map $\pi \to \pi D{\pi }^{\ast }\delta \cong \mathrm{\#}\pi D$ between exact functors; exactness of $\mathrm{\#}$ follows from the symmetric algebra's injective regular module. On $T$ it is the regular trace-duality isomorphism, and hence is invertible on add$(T)$. Use the copresentations of step 5.1 and kernels to prove it invertible on every projective; use finite projective presentations and cokernels to prove it invertible on every object. Dualizing gives the genuine module isomorphism $\pi D\cong D_B\pi$.

Individual standard prinjectives are self-dual. For standard $A$, $I(A)=DP(A)\in \mathrm{add}(T)$ by step 3.1. Step 12.1 gives $\pi I(A)=D_B\pi P(A)$ and fixes its simple label because $DL(A)=L(A)$. In a symmetric algebra the primitive projective $B{e}_A$ is also the injective hull of its corresponding simple: the perfect trace identifies its linear dual with the corresponding right ideal, and the $\ast$ twist sends the primitive to the primitive carrying the dual simple. Since that simple is fixed here, $D_B\pi P(A)\cong \pi P(A)$. Both $I(A)$ and $P(A)$ are projectives, so full faithfulness of step 6.1 lifts this isomorphism and its inverse; their composites are identities by faithfulness. Hence $I(A)\cong P(A)$, so each standard projective-injective is individually self-dual. This last conclusion has been derived after the copresentation, central-corner and duality constructions, and was not used to prove them.