The finite W and degenerate cyclotomic Hecke actions satisfy a double centralizer

The graded unital images. Write $E=V^{\otimes d}$, $C=W_d$, and $D$ for the right Hecke image viewed as an algebra of operators on $E$. The supplied actual tensor model in F1 gives commuting filtered operator algebras, with the graded symbols containing the unital images $A^0$ of $U({\mathfrak{g}}^e)$ and $D^0$ of the truncated polynomial/permutation algebra. The character automorphism in F1 gives exactly the same $A^0$ as the untwisted representation; no equality of the fixed-chart twisted and untwisted modules is inferred. Since $A^0,D^0$ are mutual centralizers, taking leading homogeneous terms of commuting operators gives $A^0\subseteq \mathrm{gr}C\subseteq \mathrm{gr}Z(D)\subseteq Z(\mathrm{gr}D)\subseteq Z(D^0)=A^0$, where $Z$ is the commutant in $\mathrm{End}(E)$. Interchanging $C,D$ gives $D^0\subseteq \mathrm{gr}D\subseteq \mathrm{gr}Z(C)\subseteq Z(\mathrm{gr}C)\subseteq Z(A^0)=D^0$. Thus every inclusion is equality, establishing strict equality of both graded images as well as their commutants.

The exact ordinary-dual adjoint. For a finite rational $\mathfrak{g}$-module $E$, Skryabin transports ordinary evaluation $E^{\ast }\otimes E\to \mathbb{C}$ and coevaluation $\mathbb{C}\to E\otimes E^{\ast }$ to the tensor functors $F_E(M)=M⊛E$. The ordinary zigzag identities give both adjunctions between $F_E$ and $F_{E^{\ast }}$. These functors are exact: tensoring is exact, it preserves generalized Whittaker modules by a finite Engel-flag nilpotence bound, and Skryabin is an exact equivalence. They preserve ${\mathcal{R}}_c$: a faithful finite image module embeds the regular algebra into a finite direct sum of copies of itself by evaluation on a basis, so every finite-image module is a subquotient of a finite sum of tensor modules; exact tensor transport sends degree $d$ to degree $d+1$. For $N>0$, also $V^{\ast }={\bigwedge }^{N-1}V\otimes {\det }^{-1}$, and the exterior power is a summand of a tensor power by the antisymmetrizer. Determinant shifts therefore give closure under $V^{\ast }$ too. Consequently $F_V$ preserves projectives in the entire rational category through its exact right adjoint $F_{V^{\ast }}$. Here $V^{\ast }$ is the ordinary contragredient natural module, distinct from $V$; the transpose contravariant duality of category $\mathcal{O}$ is not this adjoint.

The filtered centralizers. For an operator in $Z(D)$, equality of the induced graded spaces in step 1.1 supplies a member of $C$ with the same leading symbol. Subtracting lowers its integer filtration degree. The finite grading of $E$ bounds the induced endomorphism filtration below, so this terminates, proving $Z(D)=C$. The same procedure proves $Z(C)=D$. Returning to the right-action opposite convention gives both mutual-centralizer identities and the stated associated graded image algebras. At degree zero both images are scalar, while for an empty pyramid and positive degree $E=0$ and both are zero; these are already covered by the actual tensor model of F1. This proves clause (1) for every origin, without a Morita assumption.

Stable injectivity. For a nonempty pyramid with $q_l\ge d$, 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 with those distinct strings permuted by $w$ and each box moved $a_i$ places left. These $l^{d}d!$ outputs are distinct basis tensors, so the graded PBW word images are independent. A nonzero filtered Hecke element has a nonzero leading PBW symbol, hence a nonzero leading operator and a nonzero actual operator. Thus the raw action is faithful. For the empty pyramid, degree zero is the scalar identity and positive-degree tensor space and cyclotomic algebra are zero, so the action is faithful there too without using an undefined $q_l$.

The finite-image row projectives. Assume $c$ admissible. A row occupancy $\nu =({\nu }_1,\dots ,{\nu }_n)$ determines a tuple ${\mathbf{j}}_{\nu }$ of terminal row boxes and the diagonal row-multiset projection $e_{\nu }$. Since the right tensor matrices preserve row multisets, $e_{\nu }\in W_d$ by step 2.1; these orthogonal idempotents sum to one. Here is the actual basis of their projective left ideals. For a commuting matrix $\sum b_{\mathbf{i},\mathbf{j}}{E}_{\mathbf{i},\mathbf{j}}$, place permutations identify diagonal $S_d$-orbits, and the $U(\mathfrak{p})$ image forces $b_{\mathbf{i},\mathbf{j}}=0$ when some source column is smaller than its output column. Comparing its coefficients with $x_k$ and replacing $\mathbf{j}$ by $R_k(\mathbf{j})$ expresses each nonterminal coefficient in coefficients with larger total source column sum: the first term is $b_{R_k(\mathbf{i}),R_k(\mathbf{j})}$, the diagonal term is $(c_{\mathrm{col}(i_k)}+q_{\mathrm{col}(i_k)}-c_{\mathrm{col}(j_k)+1}-q_{\mathrm{col}(j_k)+1})b_{\mathbf{i},R_k(\mathbf{j})}$, and the two remaining sums run over swaps $(hk)$ with respectively both columns at most, or both columns greater than, the $k$th columns. Their signs are respectively plus and minus. This is terminating elimination from terminal source tuples. The graded string-orbit matrices are independent and indexed by the same terminal orbits, so step 1.1 makes the terminal-coordinate map an isomorphism. Let ${\mathrm{\Xi }}_{\mathbf{i},{\mathbf{j}}_{\nu }}$ be the normalized preimages. Evaluation on $v_{{\mathbf{j}}_{\nu }}$ gives exactly the orbit sums under ${\prod }_r{S}_{{\nu }_r}$, which form the monomial basis of $Z_{\nu }={⨂}_r{\mathrm{Sym}}^{{\nu }_r}(V_{\le p_r})$ with its actual $c$-twist. Thus $W_d{e}_{\nu }\cong Z_{\nu }$ as actual modules, and the regular module is their sum. This proves these projectivities in the finite image, without invoking Morita.

Projectivity of the ground object. For $r\ge 0$, take degree $Nr$ and the one-dimensional determinant-shifted ground object $D_r={\mathbb{C}}_c⊛{\det }^r$, with tableau $A_{c+r\mathbf{1}}$. Its content reconstructs it uniquely among column-strict tableaux: in each charge coset the minimum appears in every column and hence in its bottom box; removing these boxes and repeating reconstructs all columns. Therefore its scalar central-character summand has exactly this one simple, by F3. The regular-module multiplicity of $D_r$ is one. Indeed, the Levi decomposition of $Z_{\nu }$ uses semistandard tableaux of shapes given by the column partitions of ground differences, with the alphabet in column $j$ consisting of the terminal row boxes whose rows reach that column. This follows by expanding the complete symmetric characters of its symmetric factors and using the Schur semistandard formula, or successive horizontal-strip multiplication; its finite radical filtration makes it an actual $\mathfrak{p}$-filtration by these Levi sections. For $D_r$ the shapes are rectangles $(r^{q_j})$ and the allowed alphabet has exactly $q_j$ letters. Each length-$q_j$ strictly increasing tableau column must contain every letter in order. There is exactly one tuple, and its row-$i$ multiplicity is $r{p}_i$. Thus the coefficient is one for ${\nu }_i=r{p}_i$ and zero for every other occupancy. Other standards cannot contain $D_r$ because their scalar central characters differ. The supported regular summand has length one and is $D_r$ itself, so $D_r$ is projective over $W_{Nr}$. For a surjection in the rational finite-W category ${\mathcal{R}}_c$, a common determinant twist puts its terms into finite sums of polynomial-image categories. The scalar degree operator splits these by degree. A map from ${\mathbb{C}}_c$ becomes a map from $D_r$ into the degree-$Nr$ summand, where it lifts by the projectivity just proved. The inverse determinant twist gives the original lift. Hence ${\mathbb{C}}_c$ is projective in ${\mathcal{R}}_c$, without claiming projectivity in arbitrary $W(\lambda )$-mod.

The finite-W projective covers. Write $V_{\mathrm{W}\mathrm{h}}$ for the genuine continuous-dual functor $V$ of The Whittaker functor from the relevant parabolic category-O summand, and put $Q(A)=V_{\mathrm{W}\mathrm{h}}(P(A))$ for standard labels. At degree zero this is the ground object, projective by step 4.1. For a positive-degree standard $A$, the actual standard-label crystal supplies $A={\tilde{f}}_{i}B$ and a quotient $L_O(B)\otimes V\to L_O(A)$. Lifting along the projective cover and splitting proves $P(A)$ is a summand of $P(B)\otimes V$. The genuine tensor comparison sends this to a summand of $Q(B)⊛V$, projective by step 1.2. To identify its head, write $P(B)\otimes V={⨁}_{C}P(C{)}^{m_C}$ with standard $C$, using the independently proved detector head/socle property in F3. Exact $V_{\mathrm{W}\mathrm{h}}$ gives $\dim {\mathrm{Hom}}_W(Q(C),L_W(D))\ge {\delta }_{CD}$. On the other hand ordinary-dual adjunction, the induction hypothesis and the genuine simple/tensor comparison give $\dim {\mathrm{Hom}}_W(Q(B)⊛V,L_W(D))=[L_W(D)⊛V^{\ast }:L_W(B)]=[L_O(D)\otimes V^{\ast }:L_O(B)]=m_D$. Comparing this equality with the sum over $C$ forces $\dim {\mathrm{Hom}}_W(Q(C),L_W(D))={\delta }_{CD}$ for every summand with $m_C>0$. Thus each $Q(A)$ is the genuine projective cover of $L_W(A)$. Exhaustion of finite-image simples and occurrence of every standard cover in the detecting tensor object now show that $U_d=V_{\mathrm{W}\mathrm{h}}(P_c\otimes V^{\otimes d})$ is a projective generator of $W_d$-mod. This uses the actual simple-image and tensor maps of F3, rather than inferring covers from characters or mutual centralizers.

Morita for the finite images. Since $U_d$ is a finite projective generator of $W_d$-mod, every module admits a presentation by finite sums of $U_d$. The evaluation maps from the tensor-Hom adjunction are isomorphisms on $U_d$ and its direct sums and summands, hence on every module by passing to the cokernels of such presentations. Conversely the same argument for presentations of modules over ${\mathrm{End}}_{W_d}(U_d{)}^{\mathrm{o}\mathrm{p}}$ gives the inverse evaluation. Thus ${\mathrm{Hom}}_{W_d}(U_d,-)$ is an equivalence with inverse $U_d{\otimes }_{B_d}-$, and step 2.1 identifies its target algebra with the genuine Hecke image $B_d$. This proves clause (3) under admissibility independently of any supporting-block Morita assumption. In the empty-pyramid case the positive-degree tensor and both image algebras are zero, so their module categories are zero and the equivalence is immediate; degree zero is the scalar equivalence.

The nonadmissible boundary. For $c=(0,1)$, $q=(1,1)$, $d=2$, use the basis $b=(v_1\otimes v_1,v_1\otimes v_2,v_2\otimes v_1,v_2\otimes v_2)$. The raw formula gives $x_1{b}_1=x_1{b}_2=0$, $x_1{b}_3=b_1-b_2+b_3$, $x_1{b}_4=b_2+b_4$, and $s_1$ swaps $b_2,b_3$ and fixes $b_1,b_4$. Put $w=b_1+b_2+b_3+2b_4$, let $f(b_4)=w\mathrm{/}2$ and $f(b_1)=f(b_2)=f(b_3)=0$, and let $N(b_2)=N(b_3)=b_1$, $N(b_4)=-b_1$, $N(b_1)=0$. Solving the two linear commutation equations with $x_1,s_1$ gives $Z(x_1,s_1)=\mathrm{span}\{f,1-f,N\}$. Indeed $f^2=f$, $fN=Nf=N^2=0$, and direct coefficient comparison gives precisely these three free parameters. By step 2.1 this is $W_d\cong \mathbb{C}\times R$, $R=\mathbb{C}[ϵ]\mathrm{/}{ϵ}^2$. Its module decomposes as $\mathbb{C}w\oplus U$, $U=\mathrm{span}\{b_1,b_2,b_3\}$, where $ϵ$ acts by $N$. The basis $b_2,b_1,b_2-b_3$ identifies $U\cong R\oplus R\mathrm{/}(ϵ)$. The last summand is not projective: a section of $R\twoheadrightarrow R\mathrm{/}(ϵ)$ would send $1$ to an element killed by $ϵ$, hence to $(ϵ)$, whose quotient is zero. Thus the tensor module is nonprojective over the left finite-W image, and the Morita conclusion fails, although the mutual centralizers still hold.