Skryabin's equivalence for the type-A Whittaker pair

Unit, membership and counit. The shifted $\mathfrak{m}$ action on $Q_{\chi }$ lowers the bounded-below Kazhdan filtration: $(x-\chi (x))u{1}_{\chi }=[x,u]1_{\chi }$, and for negative doubled degree $-2r$ the commutator lowers filtration by $2r>0$. This proves local nilpotence on $Q_{\chi }{\otimes }_{W}M$ too, so the adjunction lands in $\mathcal{C}(\lambda )$. By [F1], its unit $M\to \mathrm{W}\mathrm{h}(Q_{\chi }{\otimes }_{W}M)$ is an isomorphism. For $N\in \mathcal{C}(\lambda )$, the triangle identity makes $\mathrm{W}\mathrm{h}(c_N)$ an isomorphism. Left exactness gives $\mathrm{W}\mathrm{h}(\ker c_N)=0$, hence $\ker c_N=0$ by detection. Now the short exact sequence with this injective counit has $H^1$ of its domain zero by [F1]; its cohomology sequence gives $\mathrm{W}\mathrm{h}(\mathrm{coker}{c}_N)=0$. Detection makes that cokernel zero. Thus the unit and counit are isomorphisms, proving the equivalence without presuming exactness of $\mathrm{W}\mathrm{h}$.

Exactness and detection. Exactness of $\mathrm{W}\mathrm{h}$ follows because it is inverse to the exact functor $Q_{\chi }{\otimes }_{W(\lambda )}-$ (tensoring with the right-free module $Q_{\chi }$ proved in the local cohomology supplier [F1]), and the detection statement is the case $H^0=0$ of [F1].

The transported tensor functor. For $M\in W(\lambda )$-mod and finite-dimensional $X$, the functor $M\mapsto \mathrm{W}\mathrm{h}((Q_{\chi }{\otimes }_{W(\lambda )}M)\otimes X)$ is exact in $M$ because the composed functors are exact; it is the transport of $?\otimes X$ across the equivalence.

Invertibility of the displayed tensor map. Let $M$ be a $\mathfrak{p}$-module and put $N=Q_{\chi }{\otimes }_{W}M$. PBW gives representatives $\eta (u)1_{\chi }$, and ${\psi }_M(\eta (u)1_{\chi }\otimes m)=um$ is balanced because right multiplication by $\eta (w)$ changes $u$ to $uw$. Hence ${\mu }_{M,X}=({\psi }_M\otimes 1){\mathrm{\mid }}_{\mathrm{W}\mathrm{h}}$ is a defined linear map. Use [F4]'s finite flag of $X$. Each quotient of $N\otimes X_j$ by $N\otimes X_{j-1}$ is a finite direct sum of $N$, with trivial $\mathfrak{m}$ action on its finite factor. By [F1] it has vanishing positive twisted cohomology and invariants $M\otimes (X_j\mathrm{/}{X}_{j-1})$. Induction and the long exact sequence give the same vanishing on each $N\otimes X_j$, and give short exact sequences on their invariant spaces. The map $\mu$ preserves these filtrations. On each successive quotient, the invariants are represented by $1_{\chi }\otimes m\otimes \bar{x}$ and ${\psi }_M$ sends them to $m\otimes \bar{x}$. Thus its quotient maps are identities; induction through the finite short exact sequences makes $\mu$ bijective. This argument holds for arbitrary $M$ and needs no cofree realization or completion of $M$.

Associativity. The associativity isomorphisms of the transported functor are obtained by inserting the canonical associativity isomorphisms of the tensor product of $\mathfrak{g}$-modules into the definition of $⊛$ and applying the inverse equivalence; they are compatible by the naturality of the unit and counit of the adjunction and by the coherence of the tensor associativity in $\mathcal{C}(\lambda )$.

Balancing and $W$-linearity. PBW gives unique representatives $\eta (u)1_{\chi }$, $u\in U(\mathfrak{p})$, in $Q_{\chi }$. Define ${\psi }_M(\eta (u)1_{\chi }\otimes m)=um$. Since the right action of $w\in W\subset U(\mathfrak{p})$ sends this representative to $\eta (uw)1_{\chi }$, the formula is right-$W$-balanced: $(uw)m=u(wm)$. Moreover ${\psi }_M(\eta (v)q\otimes m)=v{\psi }_M(q\otimes m)$ for $v\in U(\mathfrak{p})$. The shift $\eta$ adds a Lie character on $\mathfrak{p}$, so on its primitive generators, and hence multiplicatively on $U(\mathfrak{p})$, $(\eta \otimes \mathrm{i}\mathrm{d})\mathrm{\Delta }(v)=\mathrm{\Delta }\eta (v)$. The $W$ action on the source Whittaker subspace is therefore $\mathrm{\Delta }\eta (w)$, while the target restricted $\mathfrak{p}$-module $M\otimes X$ has action $\mathrm{\Delta }(w)$. Applying ${\psi }_M\otimes \mathrm{i}\mathrm{d}$ to the displayed coproduct identity proves ${\mu }_{M,X}(wz)=w{\mu }_{M,X}(z)$. Together with step 2.1 this proves the actual stated $W$-module isomorphism. No equality with cofree evaluation or citation-only character-equation argument is used.

Independence of the order. For $X=V^{\otimes d}$ the isomorphism ${\mu }_d$ is the composite of the single-step maps ${\mu }_{\bullet ,V}$; two different parenthesizations differ by the associativity isomorphisms of step 2.2, and both are restrictions of the balanced evaluation maps of step 3.1, applied with the same iterated coproduct on $U(\mathfrak{p})$; coassociativity and the displayed shift identity make these maps equal. Therefore the resulting isomorphism ${\mathbb{C}}_c⊛V^{⊛d}\to V_c^{\otimes d}$ is independent of the composed associativity isomorphisms.