The degenerate cyclotomic Hecke tower and its residue functors

Per-position polynomial projectors and integer spectra. At rank zero, $H_0(m)=\mathbb{C}$ has no polynomial generators, the sole empty-word projector and content projector are $1$, and the sector is the single ground-field block with its usual vector duality. Thus all rank-zero assertions follow directly; the spectral argument below concerns $d\ge 1$. The finite-dimensional algebra has a faithful regular representation. For adjacent dots put $\delta =x_r-x_{r+1}$ and ${\varphi }_r=s_r\delta +1$. The native cross relations give ${\varphi }_r{x}_r=x_{r+1}{\varphi }_r$, ${\varphi }_r{x}_{r+1}=x_r{\varphi }_r$ and ${\varphi }_r^2=1-{\delta }^2$. On a generalized joint weight whose adjacent entries differ by neither $1$ nor $-1$, $1-{\delta }^2$ is invertible by a finite nilpotent geometric series; hence the swapped weight is nonzero. If any entry of a nonzero weight is not integral, move its leftmost nonintegral entry left across all preceding integral entries. The swaps are invertible, but the resulting first entry contradicts ${\prod }_j(x_1-m_j)=0$. Thus all spectra are integral. For each position separately choose an annihilator $p_r(z)={\prod }_{a\in S_r}(z-a{)}^{N_r}$ from its finite regular spectrum, with $N_r=\dim H_d(m)$ sufficient. Chinese remainder interpolation gives mutually orthogonal polynomial projectors $E_{r,a}(x_r)$ summing to one. Their products $e(i)={\prod }_r{E}_{r,i_r}(x_r)$ give exactly the simultaneous generalized-weight projectors on every finite module; algebra identities follow from regular faithfulness. There is no assertion that the first-position cyclotomic polynomial annihilates other dots.

The content-block centre comparison. In an admissible stable realization the actual endomorphism theorem identifies $H:=H_{\alpha }^{\mathrm{\Lambda }}$ with ${\mathrm{End}}_{\mathcal{O}}(T_{\alpha }{)}^{op}$. Choose $Q=T_{\alpha }\oplus Q_0$, where $Q_0$ is a projective generator of its genuine category-O block, and put $A={\mathrm{End}}_{\mathcal{O}}(Q{)}^{op}$ and $M=\pi Q$. Full faithfulness on projectives identifies ${\mathrm{End}}_{\mathcal{O}}(Q)$ with ${\mathrm{End}}_H(M)$, and $M$ contains the regular left module $\pi T_{\alpha }=H$. No projectivity of all of $M$ over $H$ is assumed.

The exact centre calculation. A central endomorphism $z$ of $M$ commutes with the projection onto its direct summand $H$, so it restricts on $H$ to right multiplication $R_h$. Commuting with every $R_b$ forces $h\in Z(H)$. For each $m\in M$, extend the map $f_m:H\to M$, $a\mapsto am$, by zero on the chosen complement of $H$ in $M$. Commutation with this endomorphism gives $z{f}_m=f_m{R}_h$; evaluation at one gives $z(m)=hm$. Conversely multiplication by central $h$ commutes with every $H$-map. These constructions are inverse, proving $Z({\mathrm{End}}_{H}M)=Z(H)$, and passage to opposite rings preserves centres. The stated Choice assumption is inherited only for the category-$\mathcal{O}$ suppliers.

Central content interpolation. Symmetric polynomials in the dots commute with crossings by the native cross relations, because $s_r$ commutes with $x_r+x_{r+1}$ and $x_r{x}_{r+1}$ and with all other dots; these generate the elementary symmetric polynomials. Their elementary symmetric values distinguish the finitely many different multisets of generalized joint eigenvalues. Finite Chinese remainder interpolation in these symmetric operators therefore gives a central multiset projector. Its action on every regular generalized weight is one precisely on that multiset, so it equals the orbit sum $e_{\alpha }$ of the per-position projectors from step 1.1. This argument does not require the annihilating polynomials for different positions to be equal.

A single linkage block has no split central idempotent. ${\mathcal{O}}_{\alpha }$ is a single linkage block by Type-A tableau central summands are single linkage blocks: it admits no decomposition into two nonzero orthogonal factors. A central idempotent $f=f^2\mathrm{\ne }0$ of $A$ with $f\mathrm{\ne }1$ would split the projective generator $Q=Qf\oplus Q(1-f)$ into two nonzero summands whose Hom-spaces in both directions vanish, because every morphism between them is annihilated by composing with $f$ on either side in the two possible orders; this exhibits ${\mathcal{O}}_{\alpha }$ as the product of the two nonzero full subcategories generated by these summands, contradicting single-linkage. Hence $Z(A)$ has no nontrivial idempotent, and by step 1.2 the same holds for $Z(H)=Z(H_{\alpha }^{\mathrm{\Lambda }})$: the stable content component is a single block.

Symmetric polynomials in the $x_r$ are central. It suffices to show that the elementary symmetric functions in the $x_r$ are fixed by conjugation with each basic transposition $s_r$, since these generate the symmetric polynomials and commute with all $x_i$ with $i\mathrm{\notin }\{r,r+1\}$. The defining cross relations of The degenerate cyclotomic Hecke algebra give $x_{r+1}{s}_r=s_r{x}_r+1$ and $s_r{x}_{r+1}=x_r{s}_r+1$; multiplying the first by $s_r$ on the left yields $s_r{x}_{r+1}{s}_r=x_r+s_r$, and the second yields $s_r{x}_r{s}_r=x_{r+1}-s_r$. Consequently $s_r(x_r+x_{r+1})s_r=x_r+x_{r+1}$ and $s_r(x_r{x}_{r+1})s_r=(x_{r+1}-s_r)(x_r+s_r)=x_r{x}_{r+1}$, so the two elementary generators in the pair $\{x_r,x_{r+1}\}$ are fixed, and $s_r^2=1$ makes the same true for every polynomial in them; together with the commuting generators this fixes every symmetric polynomial. Comparing with step 2.1, $e_{\alpha }$ commutes with every $x_i$ and every $s_r$, hence is central in $H_d(m)$.

An intrinsic block in a stable realization. Fix $\alpha$ of height $d$ for any admitted consecutive $I$. Choose an integer $k$ below all its residue supports and all charges, with $q_l=m_l-k+1\ge d$ and $q_l>0$, and put $q_j=m_j-k+1$, $n_{\mathrm{r}\mathrm{o}\mathrm{w}}=q_1$, and $c_j=m_j-q_j+n_{\mathrm{r}\mathrm{o}\mathrm{w}}=m_1$. These heights are nonincreasing and the common origin is admissible; the cyclotomic roots of this stable realization are exactly $c_j+q_j-n_{\mathrm{r}\mathrm{o}\mathrm{w}}=m_j$. Its algebra is therefore the very same presentation $H_d(m)$, with the same named generators. The joint polynomial projectors of step 1.1 are uniquely determined by their regular generalized eigenspaces, so its projector for the fixed multiset $\alpha$ is exactly $e_{\alpha }$, independently of the auxiliary lower endpoint or the ambient residue interval. The stable endomorphism supplier thus identifies this exact algebra $e_{\alpha }{H}_d(m)$ with ${\mathrm{End}}_{\mathcal{O}}(T_{\alpha }{)}^{op}$, and step 2.2 proves (b). For any finite list of residue operations choose $k$ stable for every occurring degree and below every occurring residue; the same identity of presentations in adjacent degrees fixes $\iota$, the projectors $u_{\alpha ,i}$, and their induction/restriction bimodules. This uses no equivalence between different category-$\mathcal{O}$ cuts and no transport of Specht modules.

Simple-preserving duality. In the stable finite realization, the proof paragraph Simple images and standard heads of The Whittaker functor is exact and fully faithful on the required projectives identifies every Hecke simple with $\pi L(A)$ for a standard label. Restricted category-$\mathcal{O}$ duality fixes $L(A)$, as recorded in The integral parabolic category-O blocks for higher-level Schur-Weyl duality, and the paragraph The actual duality comparison of the Whittaker supplier proves $\pi D\cong (\pi -{)}^{⊛}$. Hence $(\pi L(A){)}^{⊛}\cong \pi DL(A)\cong \pi L(A)$. The intrinsic block identity of step 3.2 preserves the generator-fixing anti-involution, so the same conclusion holds for arbitrary admitted $I$. Commutation with residue functors alone would not establish this simple-preserving assertion.

Zero components and local units. Centrality does not exclude $e_{\alpha }=0$: steps 2.2 and 3.2 prove primitivity only for nonzero components and do not assert that every content occurs. Suppose $I=[a,b]$. For any nonzero supported component choose the stable realization of step 3.2 with $k<a$. The nonzero object $T_{\alpha }$ has a simple constituent, hence a strict-column label of weight $\mathrm{\Lambda }-\alpha$ by The integral parabolic category-O blocks for higher-level Schur-Weyl duality. In this realization the ground column $j$ consists of the letters $k,\dots ,m_j$. Since $\alpha$ is supported on $[a,b]$ and ${\alpha }_i={\delta }_i-{\delta }_{i+1}$, the multiplicity of every letter $k,\dots ,a-1$ in $\mathrm{\Lambda }-\alpha$ is $l$, and that of every letter above $b+1$ is zero. Strictness allows each letter at most once per column, so every lower letter occurs in all columns and no upper letter occurs. Delete the common lower letters. The remaining columns have the fixed heights $m_j-a+1\ge 0$ and entries in the finite alphabet $[a,b+1]$. Only finitely many such tableaux exist, independently of $k$ and $d$; their entry multisets determine $\alpha$ uniquely, since the simple roots are linearly independent. Consequently only finitely many supported components are nonzero, and the sum of their identities is a unit of the tower. For arbitrary $I$, finite sums of component identities are local units because every element of the direct sum has finite component support.

Level zero and the vanishing witness. At level $0$ the cyclotomic polynomial is the empty product $1$, so $H_d(m)=0$ for $d>0$ and only $H_0(m)=\mathbb{C}$ remains; this is the convention of (2). For the witness, $I=\{0\}$, level $1$, charge $(0)$, $d=2$: the level-one presentation gives $x_1=0$ and $x_2=s_1$, so $H_2(0)=\mathbb{C}[S_2]$ with eigenvalues $\pm 1$ for $x_2$; hence $E_0(x_2)=0$, $e(0,0)=0$, and the supported component $H_{2{\alpha }_0}^{\mathrm{\Lambda }}=e_{(0,0)}{H}_2(0)$ is $0$, while $H_2(0)\mathrm{\ne }0$. This completes the proof of the two claims (b) and the level-$0$ convention; the exactness and biadjointness clauses of (4) are the declared supplier statements of Hecke induction and restriction functors on degenerate cyclotomic blocks, not reproved here.