Exact finite Whittaker coinvariants and Verma dual top homology

Put $R=U(n-)$ and I its augmentation ideal. Its root-height grading is generated in degree1 and $I^r$ is the sum of height degrees at least r: a height-j root vector is an iterated bracket of j simple generators. The Rees algebra sum $I^r{t}^r$ is generated by the finite Lie subspace $L=\mathrm{span}\{x_{\beta }{t}^s:0\le s\le \mathrm{ht}(\beta )\}$. Brackets close because heights add; distributing a t-exponent among PBW factors proves generation. It is therefore a quotient of U(L), hence Noetherian by F2. For a submodule A of a finite R-module B, the graded submodule sum(A intersect $I^{r}B)t^r$ of the finite Rees module sum $I^{r}B{t}^r$ has finitely many homogeneous generators of degrees at most c. Therefore I^r A subset A intersect I^rB subset I^(r−c)A for $r\ge c$. Translating x to x−eta(x) is an algebra automorphism, so this holds for $I_{\eta }$ as well. Each $R\mathrm{/}{I}_{\eta }^r$ is finite dimensional.

Take full algebraic duals using Chevalley transpose. The generalized Whittaker part is $Whgen(M^{\vee })={\bigcup }_r(M\mathrm{/}{I}_{\eta }^{r}M{)}^{\ast }$. For an exact sequence $0->A->B->C->0$ of finite R-modules with compatible g actions, this dual functor is exact: a functional on A killing $I_{\eta }^{r}A$ extends to $B\mathrm{/}{I}_{\eta }^{s}B$ once A intersect I_eta^sB subset I_eta^rA, supplied by step 1.1. Kernel and injection follow by factoring functionals vanishing on A. These generalized parts are g-stable: commuting a sufficiently long shifted n+ word through x in g gives finitely many adjoint words on the finite nilpotent n+ module g and shorter shifted words on the original functional, all eventually zero. Hence they belong to the exact Whittaker category of F1.

Applying the exact Wh functor of F1 to step 2.1 gives exactness of $M->Wh(M^{\vee })$. Its value is exactly $(M\mathrm{/}{I}_{\eta }M{)}^{\ast }$. These quotient spaces are finite dimensional by F2 and step 1.1. Dualizing again proves $Hba{r}_0$ exact on finite g category O. This argument does not assume ideal-adic completion is exact for arbitrary noncommutative rings; the needed cofinality was proved in step 1.1.

A Verma-filtered projective P is free over U(n−), and character translation preserves freeness, so $Hba{r}_i(P)=0$ for $i>0$ by the ordinary free-module Chevalley chain contraction. For arbitrary M choose $0->K->P->M->0$ by F2. Its long homology sequence and step 3.1 give $Hba{r}_1(M)=0$; for $i\ge 2$ they give $Hba{r}_i(M)=Hba{r}_{(i-1)}(K)$, so induction on i proves all higher vanishing. All kernels are still category-O objects.

For a Verma module, PBW identifies the twisted n− module with the regular module. Its character coinvariants are one-dimensional, and its center acts as ${\gamma }_{\lambda }$ by F4. Thus $Hba{r}_0(M_g(\lambda ))=C_{{\gamma }_{\lambda }}$, with the higher vanishing already proved.

If the finite Verma is simple this computes the simple value. Otherwise its maximal submodule contains an embedded Verma $M_g(mu)$ by F3. Exactness makes its one-dimensional $Hba{r}_0$ inject into the one-dimensional $Hba{r}_0$ of $M_g(\lambda )$, so the quotient has zero $Hba{r}_0$. Its simple quotient $L_g(\lambda )$ therefore has zero $Hba{r}_0$. F3 identifies exactly these two cases by the no-positive-integral-pairing condition.

Restricted duality fixes simples and the Verma and dual Verma have the same formal character, hence the same finite composition multiplicities: subtract their finitely many highest simple terms successively. Exact $Hba{r}_0$ consequently gives the dual Verma the same Grothendieck class as $C_{{\gamma }_{\lambda }}$. A finite-dimensional Z-module of that class is exactly the one-dimensional $C_{{\gamma }_{\lambda }}$. Higher homology is zero by step 4.1. This proves every asserted value and the Statement.