Universal principal Whittaker equivalence and arbitrary central coefficients

The invariant restrictions of F1 are polynomial coordinates on S. Indeed their invertible linear part has nonzero entries only between equal positive weights. Invert those finite linear blocks. Each resulting coordinate has the form $y_j+P_j$, where every monomial of $P_j$ has degree at least2 and total weight $m_j+1$. All its variables therefore have strictly smaller weights. At the smallest weight recover $y_j$ directly; at each subsequent weight recover it by subtracting the polynomial in already recovered lower-weight variables. This finite recursion is a polynomial inverse. Hence restriction $S(g{)}^g\to$ C[S] is an isomorphism, with the stated positive weights.

Give x in $g_j$ Kazhdan degree1−j. Order n on the right in PBW. The actual defining relations of Y then set each rightmost n generator to chi(x), giving its PBW vector-space model U(b−), with gr $Y=C[f+b+]$. Every surviving generator has strictly positive Kazhdan degree, so each bounded filtered piece of Y is finite dimensional. On Y the shifted n action is ad x: (x−chi(x))u1=[x,u]1. The defining ideal is ad n-stable because $\chi ([n,n])=0$. A homogeneous x in $n_j$ lowers Kazhdan degree by $j\ge 1$. Thus sufficiently long shifted words kill each vector of Y.

Form $C^r=Y$ tensor ${\mathrm{\Lambda }}^r(n\ast )$ with the CE differential of F4, giving the ghost dual to $n_j$ degree j. The differential preserves the Kazhdan filtration. The associated graded complex is exactly the classical CE complex for N on f+b+, by taking leading brackets in the formula of F4. By F2 this is C[S] tensor the polynomial de Rham complex of N. Left-invariant vector fields and their dual frame identify the CE bracket term with the Maurer–Cartan term; in exponential affine coordinates the de Rham contraction on a form of coefficient degree a and exterior degree b with $a+b>0$ is contraction with the Euler field divided by a+b. It satisfies $dh+hd=1$ on $a+b>0$ and kills constants. Thus graded CE cohomology is C[S] in degree0 and zero in higher degree.

Every bounded Kazhdan piece of C is a finite-dimensional filtered complex, because all coefficient and ghost variable weights are positive. If a leading cocycle is a graded boundary, subtract its lifted primitive and lower its filtration degree. Only finitely many such corrections occur in that bounded piece. This proves $H^i(C)=0$ for $i>0$ and gr $\mathrm{Wh}(Y)=C[S]$, and proves strictness of its induced filtration. The center map $Z\to \mathrm{Wh}(Y),z\to z1$, is filtered; its leading map is the invariant restriction from step 1.1, using the adjoint-weight0 statement of F3. Finite-degree lifting gives surjectivity and the leading-symbol argument injectivity. Thus Z identifies Wh(Y) with Z; its products agree through central endomorphisms of Y.

The gauge coordinates give grY=C[N] tensor C[S] over grZ=C[S]. Lift the homogeneous monomial basis of C[N] to filtered Y. Successive leading-degree subtraction expresses every vector as a finite Z-linear combination of these lifts. A nonzero finite relation has a nonzero largest leading symbol in that free graded basis, contradiction. Thus the lifts form an actual right Z-basis of Y.

The finite CE complex C is a bounded complex of free right Z-modules. By step 3.1 its only cohomology is Z in degree0. It splits as a right Z complex: at the largest positive degree, exactness surjects onto a free term; lift its basis to split that surjection. Its kernel is projective. Repeat backwards finitely many times, and then split the remaining augmentation onto the projective module Z. A free module is projective by lifting each of its basis vectors, and direct summands remain projective, so every specified splitting is justified under A1. This constructs a Z-linear homotopy equivalence C to Z in degree0. Tensoring the actual contraction with any E preserves it. Since n acts on the Y factor, the tensored complex is exactly the CE complex of Y tensor_Z E. Its cohomology is E in degree0 and zero higher.

Every $Y{\otimes }_{Z}E$ belongs to $C_{\chi }$: a vector uses finitely many Y factors, and step 1.2 bounds the lengths of shifted words on all of them. Submodules and quotients inherit this property. Any nonzero P in $C_{\chi }$ has a nonzero Whittaker vector: starting with a nonzero v killed by the r-th ideal power, take a nonzero vector in the last nonzero shifted-word space on v. Every shifted x kills it. The finite-dimensionality of U(n) modulo that ideal power follows from finite PBW and the character-translation automorphism, though only the word-length argument is needed for the invariant. Hence Wh detects zero objects.

For P in $C_{\chi }$, put $E=\mathrm{Wh}(P)$, with the central Z action inherited from P. The evaluation map $Y{\otimes }_{Z}E\to P$ is well defined since the two central actions on Y agree, and sends 1 tensor e to e. By step 5.1 its map on Wh is the identity $E\to E$. The kernel has $\mathrm{Wh}=0$ by left exactness of invariants, so is zero by step 6.1. In the resulting short exact sequence from the induced module to P, the CE long exact sequence and $H^1$ of the induced module $=0$ show Wh of the cokernel is zero. Step7.1 makes that cokernel zero. Thus evaluation is an isomorphism. Together with the natural $\mathrm{Wh}(Y\otimes E)=E$ of step 5.1 this proves the asserted inverse equivalences. The induction functor is exact by step 4.1, and its inverse Wh is therefore exact as well. The asserted cohomology formula is step 5.1, with no condition on E beyond being a Z-module.