The type-A finite W-algebra and the Whittaker category

Write $c_j=n-q_{\mathrm{col}(j)}-\cdots -q_l$. The scalar functional $\kappa (e_{ij})={\delta }_{ij}{c}_j$ kills $[\mathfrak{p},\mathfrak{p}]$: a bracket can have a diagonal component only from $e_{ij},e_{ji}\in \mathfrak{p}$, forcing equal columns and equal $c_i,c_j$. Thus $x\mapsto x+\kappa (x)1$ preserves brackets and extends to $\eta$ by the enveloping universal property; the negative scalar shift is its inverse.

Put $J=U(\mathfrak{g})I_{\chi }$. For $v\in W$ and $x\in \mathfrak{m}$, $(x-\chi (x))\eta (v)=\eta (v)(x-\chi (x))+[x,\eta (v)]\in J$. Since these shifted generators generate $J$ as a left ideal, $J\eta (v)\subseteq J$. Hence for $u,v\in W$, $[x,\eta (uv)]=[x,\eta (u)]\eta (v)+\eta (u)[x,\eta (v)]\in J$, using this normalizer property for the first term and the left-ideal property for the second. Additivity, scalar closure and $[x,1]=0$ give a unital subalgebra.

Left multiplication makes $Q_{\chi }=U(\mathfrak{g})\mathrm{/}J$ a left module. Step 2.1 proves that $\bar{a}\cdot u=\overline{a\eta (u)}$ is well defined for $u\in W$. Since $\eta (uv)=\eta (u)\eta (v)$, $(\bar{a}\cdot u)\cdot v=\bar{a}\cdot (uv)$, and left multiplication commutes with this right action. Thus $Q_{\chi }$ is the stated bimodule, with right-action homomorphism into ${\mathrm{End}}_{U(\mathfrak{g})}(Q_{\chi }{)}^{\mathrm{o}\mathrm{p}}$.

$\mathrm{W}\mathrm{h}(M)$ is a $W(\lambda )$-module. Let $M\in \mathcal{C}(\lambda )$ and $v\in \mathrm{W}\mathrm{h}(M)$. First, $uv=\chi (u)v$ for every $u\in U(\mathfrak{m})$: this holds for $u\in \mathfrak{m}$ by definition and extends to products by induction, since if $yv=\chi (y)v$ then $(xy)v=x(yv)=\chi (y)xv=\chi (x)\chi (y)v$. Hence $I_{\chi }v=0$, because $I_{\chi }=\ker (\chi \text{\hspace{0.17em}⁣}:U(\mathfrak{m})\to \mathbb{C})$ and every element of $I_{\chi }$ acts on $v$ by its $\chi$-value, which is zero. Consequently $U(\mathfrak{g})I_{\chi }\cdot v=0$, since a general element of $U(\mathfrak{g})I_{\chi }$ is a sum of products $ai$ with $a\in U(\mathfrak{g})$, $i\in I_{\chi }$, and $(ai)v=a(iv)=0$. Now take $u\in W(\lambda )$, $x\in \mathfrak{m}$. Then $x(\eta (u)v)=\eta (u)xv+[x,\eta (u)]v=\chi (x)\eta (u)v+0$, because $[x,\eta (u)]\in U(\mathfrak{g})I_{\chi }$ by the definition of $W(\lambda )$. Hence $\eta (u)v\in \mathrm{W}\mathrm{h}(M)$, and the map $(u,v)\mapsto \eta (u)v$ is bilinear and associative by the algebra structure of $U(\mathfrak{p})$ and step 1.1. Here the character $\chi$ and the parabolic data are those of [F1]. This verifies the module structure; the vanishing statement $\mathrm{W}\mathrm{h}(M)=0\Rightarrow M=0$ is the cohomological detection theorem proved in Gan-Ginzburg and Skryabin inputs for the type-A pyramid good grading and Skryabin's equivalence for the type-A Whittaker pair, and is recorded here as part of the categorical interface.