The Whittaker functor from the relevant parabolic category-O summand

Let U=U(m), I=I_chi. Replace x by y=x−chi(x). These shifted generators have the original brackets, because chi vanishes on brackets. Choose a basis adapted to the lower-central filtration, with weight w_i the largest lower-central index containing its basis vector. PBW reordering and the commutator formula show that I^n is spanned by ordered monomials of total lower-central weight≥n: one inclusion follows from reordering n factors, while the reverse follows because each basis vector of weight w lies in I^w, being a linear combination of iterated brackets of length w.

The Rees algebra R=⊕_(n≥0) I^n t^n is generated by y_i t^r, 0≤r≤w_i. Their span is a finite-dimensional Lie algebra: a nonzero bracket component y_k has w_k≥w_i+w_j, so r+s≤w_k. PBW gives its enveloping algebra a polynomial associated graded. For any left ideal, Hilbert basis gives finitely many generators of its homogeneous leading ideal; lift them and subtract matching highest-degree terms until the nonnegative degree decreases to zero. The lifts generate the original ideal. Thus the enveloping algebra is left noetherian, and its surjection onto R makes R left noetherian. For a left ideal J⊂U, ⊕(J∩I^n)t^n is a graded left ideal of R and has finitely many homogeneous generators. For fixed k, choosing n greater than k plus their largest degrees gives J∩I^n⊂I^k J. This is the precise Artin–Rees consequence needed here.

Set E_chi=∪_(n≥0)(U/I^n)^*, with left action (x f)(r)=f(r x). To prove injectivity, take a U-linear map phi:J→E_chi. U is noetherian, so choose finitely many generators of J. A common I^k kills their images and hence all images, since I is two-sided. The scalar functional lambda(j)=phi(j)(1) kills I^k J. Choose n with J∩I^n⊂I^k J, extend lambda from J/(J∩I^n) to U/I^n, and call the extension a∈E_chi. The map U→E_chi, u↦u a, extends phi, because at r its value on j is a(rj)=phi(rj)(1)=phi(j)(r). Baer's criterion gives injectivity. This supplies the injective-continuous-dual assertion cited without proof in BK Lemma8.20.

For a U(m)-finitely generated g-module X, evaluation at1 identifies Hom_m(X,E_chi) with the scalar functionals lambda:X→C that kill I^n X for some n. Indeed, reconstruct the map by F_lambda(v)(r)=lambda(rv); finitely many generators give a common n. The right U(g)-action is (lambda·u)(v)=lambda(uv). On E-valued maps its formula is therefore

(F·u)(v)(r)=lambda(u r v),

not the generally non-m-linear expression F(uv)(r)=lambda(r u v). This distinguishes the compressed scalar-functional notation in the source from a literal E-valued composition.

The right action preserves continuity. For a fixed u∈U(g), the finite PBW subspace generated by all iterated ad m on u is finite-dimensional and its negative-column action is nilpotent; sufficiently long products of ad m vanish. Moving u past a product of sufficiently many shifted m generators therefore leaves at least n of them on the left. Equivalently u I^(n+C)⊂I^n U(g) for some C=C(u). Thus lambda u kills I^(n+C)X. On this continuous-dual module the right shifted m-action is locally nilpotent. Injectivity of E_chi makes Hom_m(−,E_chi) exact, and right Skryabin makes its Whittaker vectors exact. These Whittaker vectors are precisely Hom_m(X,C_chi). Dualizing this finite-dimensional vector space gives an exact covariant functor.

The restricted dual uses the row-sum torus a=span{Σ_(row(j)=r)E_jj}, not the full Cartan and not the column-origin vector c. Every e-adjacent pair lies in one row, so chi has a-weight0. If X is generated over U(m) by finitely many a-weight spaces, an m-linear functional into C_chi vanishes outside those finitely many weights: lambda(uv)=chi(u)lambda(v), and chi(u)=0 in every nonzero a-weight. Thus the restricted-dual definition agrees with the entire finite-dimensional Hom space. Generalized parabolic Vermas are U(m)-free on a finite-dimensional Levi module by PBW; their finite projective flags/presentations from the repaired parabolic highest-weight supplier make every genuine parabolic O object finitely generated over U(m). This establishes exactness on precisely that category.

For the natural box module define $X_{ij}=(-1{)}^{a-b}{I}_{a-1}(T_{\mathrm{row}(j),\mathrm{row}(i)}^{(a-b)})$ when $a=\mathrm{col}(i)>b=\mathrm{col}(j)$, and $X_{ij}={\delta }_{ij}$ otherwise. These are the actual prefix coefficients, not reindexed series. They give a unitriangular matrix. For $x=E_{rs}$ with $\mathrm{col}(r)=\mathrm{col}(s)+1=A$, establish both $[x,{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}{X}_{ij}]+{\delta }_{ir}{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}{X}_{sj}\in U(\mathfrak{g})I$ and $[x,{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}{X}_{ij}]+{\delta }_{ir}{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}{X}_{sj}\in IU(\mathfrak{g})$. If $a<A$, disjoint supports give zero. If $a>A$, the original left/right normalizer theorems for prefix $a-1$ give both containments; a whole-pyramid shift differs from the corresponding prefix shift only by a uniform trace character, which fixes $\mathfrak{m}$, $\chi$ and both ideals. For $a=A$, expand the actual prefix last-column recursion [F6]. Its embedded prefix-$a-2$ terms and its final commutator with the positive previous-to-last matrix commute with $E_{rs}$. Only the column-Levi sum ${\sum }_{h}I(T_{u,h}^{(d-1)})E_{\mathrm{box}(h,a-1),\mathrm{box}(v,a-1)}$ contributes. Its bracket is ${\delta }_{s,\mathrm{box}(h,a-1)}\eta (I{T}_{u,h}^{(d-1)})E_{r,\mathrm{box}(v,a-1)}$. The negative bridge commutes with the prefix-$a-2$ coefficient, so its left and right reductions both give ${\delta }_{ir}\eta (I{T}_{u,\mathrm{row}(s)}^{(d-1)})$. The two consecutive displacement signs differ by minus one, giving precisely $-{\delta }_{ir}\eta X_{sj}$. Diagonal shifts add commuting scalars, so this boundary calculation works for both head and tail.

Finite-Levi Weyl gives, on row variables z_(n−q+1),…,z_n,

ch L(nu)=det(z_r^(nu_s+q−s))/det(z_r^(q−s)) =(Π_(r=n−q+1)^n z_r^(n−1)) det(z_r^(a_s))/det(z_r^(q−s)).

The numerator exponent is exactly a_s+n−1; the column height and reading offset cancel. The prefactor and denominator are independent of the entries. Products over columns therefore identify the row-torus character of V(N(A)) with a fixed factor times products of nested bottom-row flag minors, evaluated at y_(r,a)=z_r^a. For the ground column a_s=c+1−(n−q+s), the determinant cancels the Vandermonde and gives precisely Π z_r^c, verifying the origin normalization.

All exponential expressions lie in the group algebra of the finitely generated additive subgroup generated by their occurring complex weights and the integral row weights. That subgroup is finitely generated and torsion free, hence free abelian; its group algebra is a Laurent polynomial domain. The nonzero Vandermonde may therefore be cancelled. A zero character of an actual nonzero finite-dimensional module is impossible: its coefficients are nonnegative weight dimensions, and evaluation of the group algebra at1 is its positive dimension.

Fix a finite entry alphabet U, and introduce independent variables y_(r,a), with row space C^n and entry space U. Antisymmetrizing each column of a tensor gives its bottom-row minor, and multiplying gives a GL(U)-linear map from the column-wedge tensor F to Sym(C^n⊗U). Each minor is killed by the lower-row raising operators: replacing a present row by a later row duplicates a row inside its bottom-row set. The product has row weight (p_1,…,p_n).

Classical Cauchy decomposition follows from the actual published Schur–Weyl decomposition: identify the degree-N symmetric power with diagonal S_N invariants of (C^n)^tensor N⊗U^tensor N, then use the two Specht decompositions. The needed self-duality is complex linear: polytabloids have real coefficients, so their span is conjugation-stable; the bilinear tabloid form B(v,w)=sum v_T w_T is invariant and B(v,conjugate(v)) is the positive Hermitian norm. It is therefore nondegenerate on the Specht span. Schur's lemma with this explicit bilinear duality leaves one multiplicity space for each matching partition. The lower-row highest-weight line at (p_1,…,p_n) selects the partition (p_n,…,p_1), so the entire flag-minor image lies in the irreducible entry module of highest weight Lambda=Σ_j omega_(q_j). The ground product of minors is a nonzero polynomial. Hence the image is exactly this irreducible module. Actual16 proves that this constituent occurs once in F; therefore this nonzero normalized map has exactly the kernel of actual16's highest-component projection pi. No quantum flag algebra or quantum Peter–Weyl theorem is imported.

For different complex cosets, first apply the companion typed category equivalence and use one independent set of variables for each coset. The same Cauchy proof applies to each coset's columns, with their retained sorted heights. Multiplication across the disjoint variable sets is injective on the tensor product. Thus the kernel is the sum of the component projection kernels, exactly the off-standard lower-basis kernel of the product of actual16 models. The shift a↦a−c_kappa+n−n_kappa and inf I_kappa=1−n_kappa is explicit in that companion proof.

V is exact on the genuine parabolic category and sends N(A) to its finite-Levi standard V(A). Actual16 identifies [L_O(A)] with its source positive-parameter dual-canonical vector L_A at parameter1. Its projection pi L_A is zero exactly off Std. The preceding flag-minor factorization consequently gives zero row-torus character of V(L_O(A)) for every off-standard A. Positivity of actual module characters makes V(L_O(A))=0.

The kernel of the Grothendieck map of V is stable under every residue e/f operator, because the actual tensor identity and Casimir/flip compatibility give V e=e V and V f=f Wh. Thus after quotienting by the already-known off-standard kernel, any additional kernel would be a submodule of actual16's irreducible highest component (or the irreducible tensor product for the direct sum of coset root algebras). The ground class survives: its V image is the one-dimensional C_c by the induced-module computation. Hence no additional kernel exists. Every Std simple therefore has a nonzero V image. This argument does not presume that image simple, does not rely on injectivity of the exponential evaluation on the entire abstract polynomial module, and does not require a classification of all finite-dimensional W simples.

Take finite alphabets containing the relevant finite labels and tensor words, with the proved boundary block projections; actual16 gives compatibility under enlarging the cuts. The argument applies on each cut and hence to every object in the selected finite degree/category. All comparisons use integer order within a coset, not an order on C.

The actual highest-component projection $\pi$ is module-linear and, in its integral $L\mathrm{/}D$ bases, is the coordinate projection with $\pi L_A=0$ off $\mathrm{S}\mathrm{t}\mathrm{d}$ and $\pi L_A=D_A$ on $\mathrm{S}\mathrm{t}\mathrm{d}$. Its kernel is therefore a direct summand, and specialization at parameter one gives exactly $K={⨁}_{A\mathrm{\notin }\mathrm{S}\mathrm{t}\mathrm{d}}\mathbb{Z}{L}_A(1)$, with no extra specialization kernel. For a nonstandard simple, exact residue translation and the source $K_0$ dictionary give $[f_i{L}_O(A)]\in K$ and $[e_i{L}_O(A)]\in K$. Expanding these actual modules in their simple classes and projecting gives a sum of independent $D_B(1)$ with nonnegative composition multiplicities; each standard-label coefficient must vanish. Exactness then proves that the Serre subcategory of objects with only off-standard composition factors is stable under both residue functors. In several complex cosets apply the product of component coordinate projections and the typed residue shift $i^{\mathrm{i}\mathrm{n}\mathrm{t}}=i-c_{\kappa }+n-n_{\kappa }$. To treat finitely many objects or elementary moves outside an initial admitted cut, shift every origin down by a sufficiently large common integer; the category, cosets and standard predicate are unchanged and all those labels lie above the new ground. No order on different complex cosets is used.

Every standard $N(C)$ embeds in an iterated lowering translate of a standard simple. To construct the monomorphism, shift the entries of column $j$ upwards by an integer $k_j\ge 0$, choosing these successively so that every shifted entry in that column is greater than every earlier-column entry in the same coset. The resulting $B$ is standard and minimal in its fixed-multiset Bruhat orbit; the triangular standard/simple multiplicity matrix gives $N(B)=L_O(B)$. For each column $j=2,\dots ,l$, list its original entries increasingly and lower its smallest entry from $a+k_j$ to $a$, then its next entry, and so on, using $e_{a+k_j-1},\dots ,e_a$ in that chronological order. At each step the distinguished change in column $j$ is valid: its smaller entries already have their original values, while its larger entries and the later columns remain above the required range. Every other valid change occurs in an earlier column and gives a strictly smaller label in the prefix Bruhat order. The distinguished standard is therefore the largest section of the multiplicity-free residue standard flag. Move it to the bottom by splitting adjacent extensions. Indeed the projective-cover kernel of $N(D)$ is filtered by standards $N(E)$ with $E>D$, while $\mathrm{Hom}(N(E),N(C))=0$ unless $E\le C$, since a nonzero image has its $L_O(E)$ head as a composition factor of $N(C)$. Thus for largest $D$ and smaller $C$, every map from that projective kernel to $N(C)$ vanishes; the projective-cover exact sequence gives ${\mathrm{Ext}}^1(N(D),N(C))=0$. This proves the precise splitting orientation locally from the highest-weight suppliers. This produces an actual monomorphism at each elementary step. Exactness preserves these monomorphisms under the remaining lowering functors, giving $N(C)\hookrightarrow e_{i_1}\cdots e_{i_r}{L}_O(B)$. For mixed cosets the same construction is performed only among comparable entries; the typed standard and tensor equivalence transports the inclusion.

Prove the raw centralizers here before using any Morita conclusion. Give $M$ its finite box grading, let $C$ be the image of $W$ and $D$ the right Hecke image regarded as operator algebras, and give them the induced integer filtrations in $\mathrm{End}(M)$. They commute by the actual tensor model. The loop/PBW symbol dictionary [F8] gives $A^0\subseteq \mathrm{gr}C$ for the unital image of the character-twisted $U({\mathfrak{g}}^e)$ representation, and the Hecke symbol dictionary gives $D^0\subseteq \mathrm{gr}D$ for the truncated polynomial/permutation image. The character ${\kappa }_c$ defines the automorphism $u\mapsto u+{\kappa }_c(u)1$ on primitive centralizer generators, with inverse given by $-{\kappa }_c$, so $A^0$ is EXACTLY the untwisted unital Vust image; the degree-zero scalar origin terms remain present in the full representation. Vust gives $Z(D^0)=A^0$ and $Z(A^0)=D^0$. Taking leading terms of commuting operators yields $A^0\subseteq \mathrm{gr}C\subseteq \mathrm{gr}Z(D)\subseteq Z(\mathrm{gr}D)\subseteq Z(D^0)=A^0$, and interchanging $C,D$ gives $D^0\subseteq \mathrm{gr}D\subseteq \mathrm{gr}Z(C)\subseteq Z(\mathrm{gr}C)\subseteq Z(A^0)=D^0$. All inclusions are equality. This uses the declared raw loop/PBW/Vust suppliers and has no dependency on the later double-centralizer theorem's Morita clause.

Put $D_{\chi }N={\bigcup }_{r\ge 1}(N\mathrm{/}{I}^{r}N{)}^{\ast }$ with the RIGHT $U(\mathfrak{g})$ action proved above, and $H_N={\mathrm{Wh}}_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}(D_{\chi }N)={\mathrm{Hom}}_{\mathfrak{m}}(N,{\mathbb{C}}_{\chi })$. The original-head right-normalizer theorem [F6] gives $[{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w),x]\in IU(\mathfrak{g})$ for the SAME original $W\subset U(\mathfrak{p})$, not an unidentified reflected algebra. Thus $(fw)(v)=f({\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)v)$ preserves $H_N$, since its commutator error starts with a shifted $\mathfrak{m}$ generator killed by $f$. Finite duality now defines the genuine covariant left $W$ action on $V(N)=H_N^{\ast }$. Precomposition with a source morphism intertwines this action, proving functoriality. The numerical identity ${\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}-{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}=-{\mathrm{tr}}_{\mathfrak{m}}\mathrm{ad}$ is consistent with the two column counts, but the actual original-head normalizer theorem supplies the algebra identification.

The inverse $Y=X^{-1}$ is the finite sum of descending-column paths, with a length-$t$ path product multiplied by $(-1{)}^t$. Prove $[{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}{Y}_{ij},E_{rs}]+{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}{Y}_{ir}{\delta }_{sj}\in IU(\mathfrak{g})$ directly from this path sum. Before the unique possible column-$A$ vertex, every factor belongs to a larger prefix $W$ and therefore normalizes the right ideal; after it, the factors have disjoint support from $E_{rs}$. Reduction of the product commutator is consequently legitimate. The surviving bracket replaces a factor starting at $r$ by its factor starting at $s$. Pair the path which next visits $s$ with the path omitting that visit; the products coincide because $X_{ss}=1$ and their signs are opposite. Only the terminating visit $s=j$ remains, with sum $-{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}{Y}_{ir}{\delta }_{sj}$. This proves the right inverse equation without multiplying arbitrary right-ideal congruences on their left. Adjacent negative matrices generate $\mathfrak{m}$, so the commutator rule propagates both matrix equations to every negative generator.

The ordinary finite tensor pairing restricts to $D_{\chi }(N\otimes Z)\cong D_{\chi }N\otimes Z^{\vee }$ as right $U(\mathfrak{g})$ modules. Continuity is checked on both sides: take an Engel flag with $b$ successive factors on which $\mathfrak{m}$ acts trivially. If $I^r$ kills each coefficient functional on $N$, expanding the diagonal shifted action shows that $I^{r+b}$ kills the tensor functional. Conversely use the flag successively: on a successive quotient the diagonal shifted action is its first-factor shifted action; subtracting the already-controlled lower-flag coefficients gives a fixed larger first-factor exponent for each coefficient. Induction through the finite flag proves that all coefficient functionals lie in $D_{\chi }N$. This establishes equality of the continuous subspaces as well as the uncompleted tensor-dual equality.

If a simple $L_O(A)$ embeds in $N(C)$, compose with the monomorphism of step 1.21 and use repeated biadjunction to obtain a nonzero, hence surjective, map $f_{i_r}\cdots f_{i_1}{L}_O(A)\to L_O(B)$ with $B$ standard. Step 1.20 excludes this when $A$ is nonstandard. Thus the socle of every standard has only standard labels. For a simple submodule of a standard-filtered module, choose the first filtration term it intersects nontrivially: it lies in that term, has zero intersection with the preceding term and embeds in the corresponding standard section. Consequently EVERY standard-filtered module has standard-label socle, including all projectives and the tensor object $T$. This is a statement about actual submodules, not about its entire list of composition factors.

An operator in $Z(D)$ has, by step 1.22, the same largest homogeneous symbol as a member of $C$. Subtract that member to lower its integer degree. The finite grading of $M$ bounds the endomorphism filtration below, so this procedure terminates and gives $Z(D)=C$. Interchanging the two images gives $Z(C)=D$. Restoring the right-action opposite convention proves $W_d={\mathrm{End}}_B(M)$ and $B={\mathrm{End}}_W(M{)}^{\mathrm{o}\mathrm{p}}$, where $B$ is the cyclotomic IMAGE algebra. For tensor degree $d=0$ both images are scalar; for an empty pyramid and positive degree both images and both commutants in $\mathrm{End}(0)$ are zero. No admissibility, stable injectivity, projectivity or Morita assertion entered this raw filtered argument.

For $N={\mathrm{Ind}}_{\mathfrak{p}}^{\mathfrak{g}}D$ with finite-dimensional $D$, PBW gives $N=U(\mathfrak{m})\otimes D$. Restriction $r(f)(d)=f(1\otimes d)$ identifies $H_N$ with $D^{\ast }$; its inverse is $\widehat{f}(u\otimes d)=\chi (u)f(d)$. Moreover $r(fw)(d)=r(f)({\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)d)$. Hence $r^{\ast }:D\to V(N)$ is $W$-linear for the head-twisted action $w\circ d={\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}(w)d$. For every $\mathfrak{p}$-map $b:D\to D^{\mathrm{\prime }}$, $(\mathrm{Ind}b)(1\otimes d)=1\otimes b(d)$ makes the restriction square commute, so $V(\mathrm{Ind}b)$ is the ACTUAL map $b$ under these identifications. This proves induced quotient and antisymmetrizer naturality without using equality of characters or presuming right-Hecke linearity. In particular $V(N(A))$ is the finite-Levi standard $V(A)$; a source standard is not asserted simple.

The two matrix equations persist under direct sums and summands by an adapted basis and uniqueness of the triangular left lift. For tensor factors $Z,Z^{\mathrm{\prime }}$ put ${\delta }_Z=(\mathrm{i}\mathrm{d}\otimes {\rho }_Z)\mathrm{\Delta }$, $X_{Z\otimes Z^{\mathrm{\prime }}}={\delta }_Z(X_{Z^{\mathrm{\prime }}})(X_Z\otimes 1)$ and $Y_{Z\otimes Z^{\mathrm{\prime }}}=(Y_Z\otimes 1){\delta }_Z(Y_{Z^{\mathrm{\prime }}})$. Inserting the two successive left/right Whittaker lifts proves their equations, and these displayed products are inverse. Modules with trivial derived ${\mathfrak{s}\mathfrak{l}}_N$ action use identity matrices, including central Jordan multiplicity spaces. Complete reducibility of ${\mathfrak{s}\mathfrak{l}}_N$ writes each finite-dimensional module as a sum of a derived irreducible tensored with such a multiplicity space; finite-dimensional dominant-weight classification and exterior antisymmetrizers realize the derived irreducible as a summand of natural tensor powers after a scalar-character correction. Thus every finite-dimensional $\mathfrak{g}$-module $Z$ has these double-admissible matrices. This algebraic construction also allows central Jordan multiplicities; the category-$\mathcal{O}$ tensor compatibility below requires a Cartan-semisimple tensor factor. The usual left contragredient $Z^{\ast }$ and the right vector-space dual $Z^{\vee }$, defined by $(\varphi a)(z)=\varphi (az)$, keep their separate action conventions; no $Z^{\ast }=Z$ identification is used.

The ground central-character multiset determines a unique column-strict label. Within each charge coset its smallest entry $c_{\kappa }+1-n$ occurs once per column; strictness permits that entry at most once in each column and only at its bottom, so its multiplicity forces every such bottom entry. Remove these entries and repeat with the next minimum in every remaining column. Induction fixes all ground entries. Different charge cosets are distinct and are never compared. In the source convention ${\lambda }_i=a_i+i-1$, the coordinates ${\lambda }_i+{\rho }_i=a_i+(N-1)\mathrm{/}2$ show that the central character is precisely this entry multiset. Hence the ground block has the sole column-strict label $A_c$. The generalized standard has diagonal simple multiplicity one and no other simple label in this block, so $N(A_c)=L_O(A_c)$. Actual BGG reciprocity and the projective standard flag give $(P(A_c):N(A_c))=1$ and no other section, so $P_c=P(A_c)=N(A_c)=L_O(A_c)$. Simple-preserving restricted duality makes this simple projective injective. The typed coset equivalence transports the same proof to every admissible complex origin. Tensoring has the exact ordinary-contragredient tensor adjoint on both sides, so $T=P_c\otimes V^{\otimes d}$ is projective-injective. Simple-preserving restricted transposition duality fixes $P_c$ and the natural module and commutes with this tensor product; hence $T$ is self-dual. Its standard flag follows from the actual parabolic tensor identity, so step 2.4 gives a standard-label socle and duality gives a standard-label head. Conversely every degree-$d$ standard label $A$ is an incoming crystal move ${\tilde{f}}_{i}B$ from a degree-$d-1$ standard label, except ground in degree zero. Inducting from the ground quotient, exact $f_i$ gives $f_i{T}_{d-1}\twoheadrightarrow f_i{L}_O(B)\twoheadrightarrow L_O(A)$ using the ACTUAL simple-head theorem; $f_i{T}_{d-1}$ is a summand of $T_d$, so $L_O(A)$ is a quotient of $T_d$. Lift any such quotient through its projective cover. Essentiality of the cover makes the lift $T_d\to P(A)$ surjective, and projectivity of $P(A)$ splits it. Therefore the projective summands of $T_d$ are exactly the covers of the standard-label simples, with positive multiplicities; they are injective as summands of $T_d$. It follows that every nonzero submodule $X\subseteq T_d$ has a nonzero standard-label simple socle. Exact $V$ and the nonvanishing already proved in step 1.18 then give $V(X)\mathrm{\ne }0$. This proves the required detector; it does not assert that every composition factor of $T_d$ is standard or that $T_d$ generates the whole parabolic category.

For a finite-dimensional left $W$ module $M$, the left Whittaker lift of $m\otimes z_j$ is ${\sum }_i(X_{ij}\cdot 1_{\chi }\otimes m)\otimes z_i$; the right lift of $f\otimes z^i$ is ${\sum }_j(f\otimes 1_{\chi }\cdot Y_{ij})\otimes z^j$. The matrix equations put these vectors in the appropriate Whittaker spaces. For the natural module each lift respects the finite column flag and has diagonal $1_{\chi }\otimes m\otimes z_j$; the tensor, direct-sum and summand constructions of step 3.2 retain a finite flag with trivial $\mathfrak{m}$ sections. The local cohomology supplier gives vanishing positive twisted cohomology on $Q_{\chi }{\otimes }_{W}M$. On its tensor product with this flag, the long exact sequence therefore makes invariants exact on successive sections, identifying each section with $M$ tensored with the corresponding trivial section. The lift induces the identity there. Finite filtration induction proves it is an isomorphism. The same argument for the reflected right pair proves invertibility of the right lift. This is exactly the cohomology/flag argument of the repaired Skryabin tensor proof, applied to the two displayed lifts. Set $A_Z(w)=Y_Z{\delta }_Z(w)X_Z$. Left Whittaker invariance makes its entries elements of the ORIGINAL $W$. The identity ${\delta }_Z(w)X_Z=X_Z{A}_Z(w)$ gives $w(m\otimes z_j)={\sum }_i{A}_Z(w{)}_{ij}m\otimes z_i$; $Y_Z{\delta }_Z(w)=A_Z(w)Y_Z$ gives $(f\otimes z^i)w={\sum }_{j}f{A}_Z(w{)}_{ij}\otimes z^j$. The head/tail shifts add Lie characters, so their coproduct identities reduce both computations to the same unshifted ${\delta }_Z$. Therefore the perfect finite pairing $⟨f\otimes \varphi ,m\otimes z⟩=f(m)\varphi (z)$ is a MODULE pairing: $(M⊛Z{)}^{\ast }\cong M^{\ast }{⊛}_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}{Z}^{\vee }$. This is the necessary algebra-action argument, rather than a vector-space duality substituted for it.

Fix an admissible origin: c_j−c_k integral implies c_j=c_k. The original arbitrary-origin deformed tensor algebra remains separate and unrestricted. In a column of height q whose global row numbers are n−q+1,…,n, write its strict entries a_1,…,a_q. The genuine head-model action proved in step 3.1 makes its finite-Levi highest weight nu_s=a_s+n−q+s−1. This follows directly from the global highest weight lambda_i=a_i+i−1 and eta_head diagonal shift n−q_1−…−q_col: the column-reading position i=q_before+(global_row−n+q) gives lambda_i+eta_head=a_i+global_row−1.

Let $Z$ be a finite-dimensional Cartan-semisimple $\mathfrak{g}$-module. Then $N\otimes Z$ is again a genuine parabolic object: its weights are sums of weights of $N$ and $Z$, and local finiteness for $\mathfrak{p}$ follows from $U(\mathfrak{p})n\otimes Z$ being finite-dimensional for each $n$. It is finitely generated over $U(\mathfrak{m})$ by tensors of generators of $N$ with a basis of $Z$: induction on word length uses $xn\otimes z=x(n\otimes z)-n\otimes xz$. Thus its $V$ image is defined on the stated domain. Apply the original-head right Skryabin counit $H_N{\otimes }_W{Q}_{\chi ,\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}\to D_{\chi }N$, $f\otimes \bar{a}\mapsto fa$. Tensor it with $Z^{\vee }$, apply right Wh and use step 2.3 to obtain $H_N{⊛}_{\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}{Z}^{\vee }\cong H_{N\otimes Z}$. Combine this actual counit with the module pairing of step 4.1 and take finite duals. This gives the natural $W$ isomorphism ${\nu }_{N,Z}:V(N\otimes Z)\to V(N)⊛Z$ for EVERY genuine parabolic object $N$, including objects without an induced-module realization. Coassociativity, counit associativity and the inverse product formulas of step 3.2 make its two iterated tensor constructions agree under transported associativity. Naturality in a map of finite tensor factors follows from its ordinary dual map and the basis-independent evaluation pairing; hence adjacent flips are intertwined.

Compute the actual Casimir coefficients in ordinary box matrix units. On the LEFT lift, the coefficient at $v_k$ after $\mathrm{\Omega }=\sum E_{ab}\otimes E_{ba}$ is ${\sum }_h{E}_{h,k}{\eta }_{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}{X}_{hj}$. Positive matrices give $(E_{h,k}-{\delta }_{hk}{\rho }_k^{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}})X_{hj}$ after undoing the shift. Negative matrices give $\chi (E_{h,k})X_{hj}-X_{kj}$ by step 1.9. Their total scalar is $-{\rho }_k^{\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{l}}-{\sum }_{b>\mathrm{col}(k)}{q}_b=q_{\mathrm{col}(k)}-n$. On the RIGHT lift the coefficient at $v^h$ is ${\sum }_k{\eta }_{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}{Y}_{ik}{E}_{h,k}$; step 2.2 moves negative matrices left, giving $\chi (E_{h,k})Y_{ik}-Y_{ih}$, and the total scalar is $-{\rho }_h^{\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{d}}-{\sum }_{b<\mathrm{col}(h)}{q}_b=q_{\mathrm{col}(h)}-n$. Multiplying by $Y$ on the left or $X$ on the right therefore gives the SAME entries $C_{j,i}={\sum }_{\mathrm{col}(h)\le \mathrm{col}(k)}{Y}_{i,k}{E}_{h,k}{X}_{h,j}+{\sum }_k(q_{\mathrm{col}(k)}-n)Y_{i,k}{X}_{k,j}+{\sum }_{\mathrm{row}(h)=\mathrm{row}(k),\mathrm{col}(h)=\mathrm{col}(k)-1}{Y}_{i,h}{X}_{k,j}$. There is no extra displacement sign on the ordinary $E_{h,k}$; signs are already in $X,Y$. The entries lie in $W$ because the Casimir preserves the actual Whittaker subspace. The module pairing now intertwines the two transported Casimirs, while the continuous tensor pairing transposes the source Casimir term by term. Hence $\nu$ intertwines the source and target Casimir maps. With flip compatibility this also intertwines their polynomial and generalized-eigenspace residue projections, with the usual contragredient retained for reverse translation.

Fix the head-normalized ground map $i_0:V(P_c)\to {\mathbb{C}}_c$ obtained from step 3.1 and the finite-Levi character calculation below. Iterate $\nu$ on $P_c\otimes V^{\otimes (d-1)}$ to obtain $i_d:V(P_c\otimes V^{\otimes d})\to {\mathbb{C}}_c⊛V^{⊛d}$, and compose with the proved ACTUAL balanced tensor map ${\mu }_d$ to get $j_d$. Casimir and flip compatibility prove $(W,H_d)$-bimodule linearity. Any source endomorphism $b$ is transported as $j_{d}V(b)j_d^{-1}$; in particular a native Casimir/place word representing right multiplication by $h$ becomes the actual map $m\mapsto mh$. This identifies maps in the cyclotomic IMAGE algebra, without asserting a faithful abstract Hecke action outside its allowed range. The construction uses no origin integrality. Typed category-O/coset transport is used only for the selected admissible-origin source labels and their character dictionaries, whereas the unrestricted deformed tensor model retains its arbitrary origins.

$H_d$ acts on $T$ by the universal tensor Casimir and places. At the first tensor step the actual finite $\mathfrak{p}$ flag of the natural module has one standard section per column. On the $j$th section, the ordinary quadratic-Casimir identity $\mathrm{\Omega }=(C_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-C_{\mathrm{s}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{c}\mathrm{e}}-C_V)\mathrm{/}2$ gives eigenvalue $c_j+q_j-n$. Successively applying the factors in the column-order flag makes ${\prod }_j(x_1-(c_j+q_j-n))$ annihilate $T$; subsequent tensor factors leave this first-step relation intact. Hence the source action factors through the genuine cyclotomic algebra, not only its $V$ image. Step 7.1 transports it to $M=V_c^{\otimes d}$ as the actual $(W,H_d)$ bimodule action.

$V$ maps ${\mathrm{End}}_{\mathfrak{g}}(T{)}^{\mathrm{o}\mathrm{p}}$ into ${\mathrm{End}}_W(M{)}^{\mathrm{o}\mathrm{p}}=B$ by step 2.5, and its image contains $B$ because every target Hecke operator is the transport of its actual source Casimir/place operator. A nonzero source endomorphism has a nonzero image $X\subseteq T$. Step 3.3 proves $V(X)\mathrm{\ne }0$, and exactness identifies $V(X)$ with the image of its transported endomorphism. Hence the endomorphism map is injective as well as surjective, giving ${\mathrm{End}}_{\mathfrak{g}}(T{)}^{\mathrm{o}\mathrm{p}}=B$. Detection uses the standard-label SOCLE criterion, never a claim that all composition factors of $T$ are standard.

Choose a full finite projective generator Q of the genuine O summand containing T as an explicit summand and write its algebra as A; the projector onto T is e and B=eAe. The exact Schur functor is e(−). It kills precisely the offStd simples. Its left adjoint Ae⊗_B− is right exact and satisfies e(Ae⊗X)=X. The counit Ae⊗eN→N has kernel and cokernel killed by e. Applying V sends it to an isomorphism. Therefore V factors through e(−), and its factor F on B-mod is naturally M⊗_B− by finite free presentations. This factor is exact: the only failure of exactness after the left adjoint is killed by e and hence by Wh. It is faithful, since every B-simple is eL_O(A) for a Std label and its V image is nonzero; exactness then detects every nonzero image of a morphism.

A finite-dimensional right B-module whose tensor functor is exact is projective. Explicitly, take its finite projective cover P→M with kernel K⊂P rad B. Tensor the inclusion K→P with B/rad B; flatness makes K/K rad B inject into P/P rad B, but this map is zero, so Nakayama gives K=0. Faithfulness forces every right indecomposable projective to occur in M: otherwise tensor with the corresponding left simple would vanish. Thus M_B is a progenerator. The first independently proved raw centralizer identity in step 2.5 gives Wd=End_B(M). The usual generator presentation/evaluation argument now gives the genuine Morita equivalence Wd-mod≃B-mod. In particular F sends B-simples to Wd-simples, proving the full relevant V simple-image claim without assuming it earlier.

Tensor compatibility and the actual Casimir/flip intertwiners are proved in steps 1.9–1.18, using the original head/right normalizer and the actual balanced Skryabin map. Thus $V(P_c)={\mathbb{C}}_c$ and $V(P_c\otimes V^{\otimes d})=V_c^{\otimes d}$. The preceding kernel/Morita construction proves $V(L_{\mathcal{O}}(A))=L_W(A)$ for standard labels and zero otherwise; it never asserts that the image of a source standard $N(A)$ is simple. The normalized comparison is determined on the ground and its tensor words, hence on the detecting tensor bimodule and on finite free quotient presentations through the established exact Morita factorization. Its ground normalization removes scalar ambiguity. This proves the stated properties of the genuine Whittaker functor, while keeping it distinct from the Hecke Hom quotient.