The higher-level Schur-Weyl endomorphism isomorphism

The Hecke action lies in the g-endomorphisms. For the empty pyramid, the empty tensor in degree zero and its ground object give $T_0=U_0=H_0=\mathbb{C}$. In positive degree $V=0$ gives $T_d=U_d=0$, and the level-zero cyclotomic relation is the empty product $1$, so $H_d=0$. The stated endomorphism identities and faithfulness follow directly, with the unique zero-algebra map injective; the remaining argument assumes a nonempty pyramid. Each generator $x_j$ of the truncated tensor action is built from the trace Casimir and place permutations and commutes with the diagonal $\mathfrak{g}$-action; the place permutations are natural in the module variable, so the resulting right $H_d(\mathbf{a})$-action on $T$ consists of $\mathfrak{g}$-endomorphisms, and its image is $B_d$ by definition. Hence there is a surjection $H_d(\mathbf{a})\twoheadrightarrow B_d$ and an inclusion $B_d\subseteq {\mathrm{End}}_{\mathfrak{g}}(T{)}^{\mathrm{o}\mathrm{p}}$, with the opposite convention fixed by the right action.

The Whittaker comparison is injective. Let $\phi :T\to T$ be nonzero and put $X=\mathrm{im}\phi$, a nonzero submodule of the finite-length object $T$. Choose a simple submodule $L(A)\subseteq X$; because $\mathrm{soc}X\subseteq \mathrm{soc}T$, [F2] gives $A\in \mathrm{S}\mathrm{t}\mathrm{d}$. Exactness gives an injection $V(L(A))\hookrightarrow V(X)$, and the first module is nonzero, so $V(X)\mathrm{\ne }0$. Exactness also identifies $V(X)$ with the image of $V(\phi )$, hence $V(\phi )\mathrm{\ne }0$. This proves injectivity of ${\mathrm{End}}_{\mathfrak{g}}(T{)}^{\mathrm{o}\mathrm{p}}\to {\mathrm{End}}_{W_d}(V_c^{\otimes d}{)}^{\mathrm{o}\mathrm{p}}$. No restriction on all composition factors of $T$ is used.

The image of the comparison. On $V(T)=V_c^{\otimes d}$ the induced map lands in the centralizer of the finite-W action, ${\mathrm{End}}_{W_d}(V_c^{\otimes d}{)}^{\mathrm{o}\mathrm{p}}$; by the filtered double centralizer [F3] this centralizer equals $B_d$; and the induced map contains $B_d$ because every Hecke generator on $V_c^{\otimes d}$ arises from the corresponding endomorphism of $T$ by [F1]. Since $B_d$ is the whole target, the injective map of step 1.2 is also surjective, so ${\mathrm{End}}_{\mathfrak{g}}(T{)}^{\mathrm{o}\mathrm{p}}=B_d$.

Stable range. For a nonempty pyramid, if $q_l\ge d$ then the action of $H_d(\mathbf{a})$ is faithful by [F3] and the image is the full algebra $B_d=H_d(\mathbf{a})$; a single central component is correspondingly a block $H_{\alpha }(\mathbf{a})$, not the whole algebra. Outside the stable range the corner identification of [F4] applies for admissible origins, while the map $H_d(\mathbf{a})\twoheadrightarrow B_d$ remains surjective by step 1.1.

Central components. The central idempotents $f_{\alpha }$ of the finite-dimensional algebra $B_d$ decompose $T={⨁}_{\alpha }{T}_{\alpha }$ and $B_d={⨁}_{\alpha }{B}_{\alpha }$; applying the isomorphism of step 2.1 componentwise gives ${\mathrm{End}}_{\mathfrak{g}}(T_{\alpha }{)}^{\mathrm{o}\mathrm{p}}=f_{\alpha }{B}_d{f}_{\alpha }=B_{\alpha }$. The component $B_{\alpha }$ sits inside the cyclotomic block as the corner $f_{\alpha }{H}_{\alpha }(\mathbf{a})f_{\alpha }$ by [F4]; it is not the entire degree-$d$ algebra.

Failure of raw faithfulness below the stable range. For a nonempty pyramid, suppose $d>q_l$ and use an enlargement by full ground rows with smallest height at least $d$. In its last column, raise the first $q_l+1$ entries above their ground values by $1$; raise its top entry further by $d-q_l-1$, and leave every other entry ground. This is column strict, has total degree $d$, and is standard: for columns with the same charge, the changed rightmost entry is at least every unchanged entry to its left in that row; different charge cosets are incomparable. The first $q_l$ rows of this column are retained, but its next row is deleted, so this standard label is noncontractible. Its nonzero large finite-W simple belongs to a complementary central block. Hence the supporting idempotent $f$ of [F4] is proper; the faithful large tensor progenerator makes $1-f$ nonzero also in the large Hecke algebra. The small raw action has the nonzero kernel $(1-f)H_d(\mathbf{a})$ and is not faithful. Together with step 2.2 this proves the asserted precise stable-range criterion.