Exact two-sided Wakimoto resolution for an admissible affine module

Stabilizing the polarization. Choose dominant coroot translations ${\gamma }_n$ in the integral translation lattice with every finite positive root pairing tending to infinity, and put $y_n=t_{-{\gamma }_n}$. They can be chosen with successive differences dominant; original lengths add. The rank-one quotient-localization inclusions give maps $M^{y_n}(\nu )\to M^{y_{n+1}}(\nu )$, fixing the top line. In ordered PBW coordinates a new root dualization changes only basis vectors involving that root. On a fixed depth beta, once every newly added inversion root has height greater than ht(beta), the map on that depth is an isomorphism. There are only finitely many positive roots below a given depth. Thus each weight block stabilizes algebraically, not just in dimension.

Call the resulting module W(nu). Its character is the Verma character. Its polarization dualizes exactly the positive roots with positive finite part, while the remaining negative-current and Cartan directions stay free. The canonical map M(nu)→W(nu) has at each depth the stable determinant above, with gamma ranging over positive real roots of positive finite part. The half-Jantzen argument therefore remains valid in the stable block.

If beta has negative finite part and $\mu =s_{\beta }\cdot \nu$ is a Bruhat cover, its singular vector consequently survives in W(nu). This is an actual proof of the nonzero edge test used after translating a semi-infinite cover into an ordinary cover. It supplies the missing nonvanishing; merely taking a limit of nonzero maps would not have sufficed.

Compatible polarization flips. The rank-one four-term sequence is needed, including its zero composites. For an ORIGINAL simple ${\alpha }_i$ and integer $⟨\nu +\widehat{\rho },{\alpha }_i^{\vee }⟩=m>0$, embed $M(s_i\cdot \nu )$ into M(nu) by $f_i^m$. Its quotient P is ${\mathfrak{s}\mathfrak{l}}_{2,i}$-integrable. The Ore derived sequence has $T_i$ $P=0$ and $L_1$ $T_i$ $P=P$, with the actual finite Tits identification. Applying it to this short exact sequence gives 0→M(s_i dot nu)→M(nu)→M^si(nu)→M^si(s_i dot nu)→0.

This proves exactness and the zero composite, rather than guessing the middle map. Conjugate by exact nonintegral-wall transports to obtain the analogous sequence for an integral simple root. This uses the genuine shifted action: it does not replace the exponent by nu(beta-coroot)+1 on an unshifted root sl2.

At an integral Bruhat cover $u>v$, the simple L(u dot lambda) has multiplicity1 in M(v dot lambda). Indeed the ordinary Jantzen sum and strong linkage allow only the reflection giving that cover to contribute this simple; its contribution is the highest line of M(u dot lambda). The actual embedding supplies one occurrence, so it is exactly1. This controls all Hom and nonzero-composite tests used in a flip.

Write $y^{\mathrm{\prime }}=y{s}_{\alpha }$ with integral length increasing by one and $\beta =y\alpha$. The canonical maps ${\varphi }_w:M^y(w\cdot \lambda )\to M^{y^{\mathrm{\prime }}}(w\cdot \lambda )$ are the one-root Ore middle maps, transported as in step 5.1 and normalized on the top line. On unchanged-degree summands $w^{-1}\beta$ is positive; the kernel of the middle map has strictly lower highest weight, so its top-line value is nonzero. The transition chain map is zero on changed-degree summands. To check a persistent cover with both endpoints unchanged, use the restricted adjunction of [F2, Proof 12.1] to remove the common $y$ twist, and the exact nonintegral-wall equivalence of [F1, Proof 28.1–34.1] to make $\alpha$ an original simple root ${\alpha }_i$. Denote the resulting ordinary labels by $a>b$, a Bruhat cover, with $s_{i}a>a$ and $s_{i}b>b$.

The kernel of $c_i:M(b\cdot \lambda )\to M^{s_i}(b\cdot \lambda )$ is $M(s_{i}b\cdot \lambda )$ by step 4.1. This kernel can contain the weight $a\cdot \lambda$; the required exclusion concerns the simple subquotient $L(a\cdot \lambda )$. Strong linkage [F3] would require $a\ge s_{i}b$ in Bruhat order. But $\mathrm{\ell }(a)=\mathrm{\ell }(s_{i}b)$ and $a\mathrm{\ne }{s}_{i}b$, so that is impossible. Thus the singular line of the embedded $M(a\cdot \lambda )$ survives $c_i$. For the other comparison composite, put $N=M(s_{i}b\cdot \lambda )\mathrm{/}M(s_{i}a\cdot \lambda )$. The Ore long exact sequence makes the kernel of $M^{s_i}(a\cdot \lambda )\to M^{s_i}(b\cdot \lambda )$ the image of $L_1{T}_{i}N$. By [F2, Proof 24.1], this is the Tits twist of the $f_i$-torsion submodule of $N$. That submodule is a $\widehat{\mathfrak{g}}$-submodule, since $\mathrm{ad}{f}_i$ is locally nilpotent on each enveloping word; it is integrable for the original ${\mathfrak{s}\mathfrak{l}}_2^i$, since $e_i$ is locally nilpotent on $N$. Its finite Tits operator identifies the twist with the same submodule. Consequently $L_1{T}_{i}N$ has no $L(a\cdot \lambda )$ subquotient, as $N$ is a quotient of $M(s_{i}b\cdot \lambda )$. The nonzero highest vector $c_i(v_a)$ therefore survives the second map. Both comparison composites are nonzero. The cover multiplicity one of step 6.1 bounds their common Hom space by one, using the Verma character and the restricted adjunction; hence they span it.

Here is the actual coefficient normalization. For a new cover w1→w2 classify the signs of $w_1^{-1}\beta ,w_2^{-1}\beta$.

The length-two nonvanishing needed for a diamond wholly in the unchanged-degree region has the same simple-subquotient proof. In the reduced ordinary picture let $a>b$ have length difference two and let all four vertices of its Bruhat diamond satisfy $s_{i}x>x$. If $s_{i}b\le a$, then $s_{i}b$ is one of its two middle vertices, contrary to $s_i(s_{i}b)<s_{i}b$. Thus strong linkage again excludes $L(a\cdot \lambda )$ from $M(s_{i}b\cdot \lambda )$, and the composite of the ordinary inclusion with $c_i$ is nonzero. This assertion is restricted to those diamonds; no general exclusion of their endpoint weights, or general nonvanishing for every length-two path, is used. Diamonds with a vertical side are handled by the four-term zero composite and the subsequent side-square normalization.

• If both are positive, it was an old cover with unchanged degrees; set its scalar so that the nonzero comparison square commutes. - If $w1=s_{\beta }$ w2, it reverses the old vertical cover; take the old vertical scalar with the rank-one Ore normalization. - If both are negative, translate both vertices by $s_{\beta }$ to the positive case. The new edge and the two already assigned vertical edges complete a square. Set its scalar to make the two path products opposite. - In the remaining nonvertical mixed case (negative at w1, positive at w2), lifting gives the unique extra middle vertex completing the square from $s_{\beta }$ w1 to w2. The other three coefficients were assigned in the preceding cases; define the last one by their negative product/quotient.

No coefficient being divided by is zero, by the cover and length-two tests above. Every new diamond is consistent with these assignments. If it has no vertical side, its beta-translated diamond lies in the positive region; multiplying that relation by its four comparison side relations proves the new relation, since auxiliary coefficients occur once on each side. If it has a vertical side, either it is a defining side square or it is the product of three such squares; the same cancellation gives the negative two-path relation. These exhaust the lifting possibilities, exactly as in the finite-ideal sign construction. Thus all new differentials square to zero and all unchanged-degree comparison squares commute.

Define the transition chain map by phi_w on summands whose two twisted degrees agree, and by zero on the others. The preceding commuting squares prove the chain-map equation on persistent covers. In the changed-degree cases the four-term zero composite proves that equation. On the degree-zero lambda highest line the map is1. Each twisted complex has only the simple L(lambda) in homology, so the transition is a quasi-isomorphism.

The module comparisons are the actual semiregular comparisons. We specify the maps, rather than equating abstract resolutions by their characters. For an original length-additive factorization $z=yu$, write the bimodule factorization of [F2, Proof 10.1] as $S_z=S_y{\otimes }_U{}^y{S}_u$, using fixed compatible Tits lifts. If $f_1,\dots ,f_r$ are the ordered root coordinates of the new factor, let $1_u^{\ast }=f_1^{-1}\otimes \cdots \otimes f_r^{-1}$ in its quotient-localization model; under its polynomial-dual model this is the augmentation functional. Set $j_{z,y}(s)=s\otimes 1_u^{\ast }$ and define ${\nu }_{z,y}^{\mu }:M^y(\mu )\to M^z(\mu )$ by $s\otimes v_{y^{-1}\cdot \mu }\mapsto j_{z,y}(s)\otimes v_{z^{-1}\cdot \mu }$, with the indicated Tits relabelings. These are algebraic maps. Balancing follows by moving a generator across the new root coordinates using the two PBW bimodule models: a positive-Borel generator annihilates the new top vector, and a Cartan generator gives the shift $\widehat{\rho }-z^{-1}\widehat{\rho }=(\widehat{\rho }-u^{-1}\widehat{\rho })+u^{-1}(\widehat{\rho }-y^{-1}\widehat{\rho })$ in the factorization convention. All reordering sums terminate by [F2, Proof 1.1–10.1]. Thus the formula descends to the induced modules.

This formula also specifies every current map. In the common transported bimodule model, for $X=a\otimes t^m$, $K$ or the grading Cartan, left multiplication gives ${\nu }_{z,y}^{\mu }(X\xi )=X{\nu }_{z,y}^{\mu }(\xi )$. In polynomial coordinates this is the identity ${\mathrm{Ad}}_{n_1{n}_2}X={\mathrm{Ad}}_{n_1}{\mathrm{Ad}}_{n_2}X$ in the explicit action $X(f\otimes v)={\sum }_{j}f{p}_j\otimes X_{j}v$ of [F2, Proof 3.1–7.1]. The Cartan correction is exactly the shift in step 12.1. Consequently this equality holds for every finite word of current modes, including words with positive modes; it is stronger than equality of top-line scalars. Associativity of the balanced tensor product and $1_{uv}^{\ast }=1_u^{\ast }\otimes 1_v^{\ast }$ in ordered coordinates give ${\nu }_{z_3,z_2}^{\mu }{\nu }_{z_2,z_1}^{\mu }={\nu }_{z_3,z_1}^{\mu }$.

The locally finite limit is exact. Take dominant translations ${\gamma }_n$ in the integral translation lattice, tending strictly into every finite positive-root direction, with successive differences dominant, and $y_n=t_{-{\gamma }_n}$. Their original and integral lengths are additive along the increments. Refine the increments into integral simple flips and apply the compatible construction above.

Here are the hypotheses needed to apply the restricted adjunction, including at infinite PBW rank. At a finite twist $y$, put $A=U({\mathfrak{n}}_y)$ and $D=A^{\mathrm{\#}}$. Each Verma is $A\otimes V$ by the complementary PBW words. Both support sums in [F2, Proof 12.1] are finite: the first is the affine partition count, and after the $y$ relabeling each creator in $D\otimes V$ has negative-root weight, with the original imaginary multiplicities, so the second is the same count with a fixed normal-order shift. Finite sums, finite Verma flags, and the finite additional polarizations in a factorization satisfy the same test. Thus the unit $m\mapsto (s\mapsto s\otimes m)$ and counit $s\otimes \phi \mapsto \phi (s)$ of that supplier are actual inverse $U$ maps for every finite-twist Hom comparison used here. In particular a comparison from $M^y(\mu )$ to a further polarization of the same top weight is determined by its adjoint map from $M(y^{-1}\cdot \mu )$; the target's top-weight space is one-dimensional. Its nonzero top-normalized map is therefore precisely ${\nu }_{z,y}^{\mu }$. The nonzero unchanged-degree maps of steps 7.1–11.1 have this normalization. Their composite along an integral refinement of an original length-additive increment is consequently ${\nu }_{z,y}^{\mu }$, by the unit/counit identities, not an arbitrary comparison quasi-isomorphism.

The HOMOLOGICAL degree of the w summand is d_n(w)=ell_int(t_gamma_n w)-ell_int(t_gamma_n).

This identifies the actual finite complexes to be limited. Admissibility gives both regular dominance and full affine rank by [F5], and $k+h^{\vee }>0$ is assumed, so every hypothesis of the ordinary BGG seed [F1] holds. At $y\in W_{\lambda }$, apply $T_y$ to that seed and shift by ${\mathrm{\ell }}_{\mathrm{i}\mathrm{n}\mathrm{t}}(y)$. Every individual Verma is $T_y$-acyclic and each degree contains finitely many labels. [F2, Proof 24.1–31.1] computes the derived image of its sole homology as $L(\lambda )$ in precisely that shift, by the finite gallery of nonintegral exact transports and integral original-simple integrable steps. Hence the resulting complex $B^y$ has $B_p^y={⨁}_{{\mathrm{\ell }}_{\mathrm{i}\mathrm{n}\mathrm{t}}(y^{-1}w)-{\mathrm{\ell }}_{\mathrm{i}\mathrm{n}\mathrm{t}}(y)=p}{M}^y(w\cdot \lambda )$ and sole homology $L(\lambda )$ in degree zero. Use the scalars of steps 8.2–10.1. The maps of step 11.1 then form chain maps between these very complexes; on a persistent summand their module map is the semiregular $\nu$ just identified, and on a changing-degree summand it is zero. This supplies finite complexes, current-linear comparison maps and their compatible normalizations before any limit is taken.

It stabilizes for each w by the finite affine-root inversion formula. Define its COHOMOLOGICAL semi-infinite degree p(w)=-lim $d_n(w)$. The minus sign is essential: for the rank-one affine extra reflection the eventual homological degree is1 and the cohomological degree is-1. At every flip d(w) either stays fixed or decreases by2. Once its translation endpoint values have stabilized, no intermediate degree change can occur. The tail module maps on that summand are therefore precisely the finite-polarization inclusions, whose limit is W(w dot lambda).

Every vector in a term has finitely many labels w. A changed-degree comparison kills that old summand; a stable-degree tail yields its W module. Hence the limit term is the actual direct sum over labels of the prescribed semi-infinite degree. At a fixed weight only finitely many labels occur: $w\cdot \lambda \le \lambda$, these weights are distinct, and the interval of root-lattice depths above a fixed lower weight is finite. Ghost and PBW constructions consequently remain locally finite, even when a semi-infinite degree has infinitely many labels globally.

Take the directed system of the actual complexes $B^{y_n}$ in step 15.1 with cohomological degree $-p$. Its chain maps are current-linear by step 13.1. At every full weight the eventual term maps are the isomorphisms of steps 1.1 and 15.2; they identify both the vectors and every fixed current-mode action once the source and target weights have stabilized. A cycle in the colimit has a finite representative; its differential is zero at some common later stage, so it has a cycle representative there. If its class maps to zero, a finite bounding vector and its equation likewise occur at one later stage. These two observations prove directly. The finite-stage homologies are $L(\lambda )$ in degree zero, and their transition maps are the identity under the highest-line identifications of step 11.1. Therefore the limit has $H^0=L(\lambda )$ and $H^p=0$ for $p\mathrm{\ne }0$. The resulting homology identification commutes with every current mode, since every comparison and differential does.

Stable edges are nonzero and unique. A stable semi-infinite cover is an ordinary Bruhat cover after sufficiently large positive translation. Write its source highest weight as $\mu =s_{\eta }\cdot \nu$ in that translated picture, with eta positive and of negative finite part. The finite-polarization determinant note proves that the singular vector of M(mu) has nonzero image in W(nu): eta is excluded from the positive-finite-part half determinant, and the half-Jantzen kernel has no simple cover factor L(mu).

Restricted adjunction on the stabilized modules. For a fixed dominant translation $\gamma$, undo only the finite ${\mathfrak{n}}_{t_{-\gamma }}$ polarization. The ordered PBW model of $W(\mu )$ is cofree in its transported finite dual coordinates. Its multiplicity space consists of words in the remaining creators. Both support sums required in [F2, Proof 12.1] are finite: undoing these finitely many coordinates and relabeling by $t_{\gamma }$ leaves precisely the negative affine-root creator multiset and its fixed top shift. Hence that supplier's actual counit is an isomorphism on this object; no assertion for arbitrary cofree modules is used. Its inverse Hom model, by moving those finite coordinates back across the ordered products, is $G_{t_{-\gamma }}W(\mu )=W(t_{\gamma }\cdot \mu )$. On maps it is $G(f)(\phi )=f\circ \phi$, with inverse supplied by evaluation. Thus it is faithful on these Wakimoto Hom spaces. Under this identification the adjoint of $r_y^{\mu }:M^y(\mu )\to W(\mu )$, the structure map of the limit, is the canonical top-normalized map $M(y^{-1}\cdot \mu )\to W(y^{-1}\cdot \mu )$: on the augmentation it is $d\mapsto j_y(d)\otimes 1$, and the counit sends $s\otimes \phi$ to $\phi (s)$. This is equality of actual $U$ maps. For a stable edge apply this adjunction to $r_y^{w^{\mathrm{\prime }}\cdot \lambda }{d}_y=d{r}_y^{w\cdot \lambda }$. Its top evaluation is the ordinary translated Verma cover embedding followed by that canonical map. Step 18.1 makes this composite nonzero. Therefore $d$ is nonzero.

The Hom bound uses finite sources, not cyclicity of a Wakimoto module. The universal property of the algebraic colimit gives . By the explicit adjunction in step 19.1 its $n$th space is ${\mathrm{Hom}}_U(M(t_{{\gamma }_n}w\cdot \lambda ),W(t_{{\gamma }_n}{w}^{\mathrm{\prime }}\cdot \lambda ))$. For large $n$ the two translated labels form an ordinary Bruhat cover. This Hom space is the singular-vector space at the source highest weight and has dimension at most one: each independent singular vector supplies an independent highest simple factor, while equality of the target's character with that of its Verma and the cover-multiplicity calculation of step 6.1 give multiplicity one. This argument concerns only the finite-source Verma Hom after adjunction; it does not assert that $W(w\cdot \lambda )$ is generated by its top line. The character multiplicity is taken in the finite interval of weights between that cover weight and the target top, so no finite-length assertion about the entire Wakimoto module is needed. The nonzero differential of step 19.1 restricts nontrivially at every sufficiently late stage by its same translated singular-vector calculation. Hence restriction between these one-dimensional tail Hom spaces is nonzero, and the inverse limit is one-dimensional, spanned by $d$.

The actual vacuum singular line. In rank one write $\lambda =k{\mathrm{\Lambda }}_0$, let ${\beta }_0$ be the extra positive integral simple root (its finite part is negative), and put $\sigma =s_{{\beta }_0}\cdot \lambda$. Let $j_0:M(\sigma )\to M(\lambda )$ be the actual ordinary BGG component, including its chosen nonzero scalar, and $v_{\sigma }$ its top vector. The label $s_{{\beta }_0}$ has ordinary homological degree one and eventual homological degree one. Degrees can only decrease by two along the chosen polarization path, so it and the identity label keep their degrees at every intermediate flip. Consequently the chain-map equations of steps 11.1, 12.1 and 13.1 give the literal square $d{r}_e^{\sigma }=r_e^{\lambda }{j}_0$, not merely a proportionality inferred from Hom dimensions. Normalize its top vectors by the augmentations of step 12.1. In particular $d(v_{\sigma }^W)=r_e^{\lambda }(j_0{v}_{\sigma })$. The half-Jantzen cover test applies because ${\beta }_0$ has negative finite part and is excluded from every $I(y_n)$ and its stable half. Thus this vector is nonzero.

To identify it with the actual universal-vacuum singular state, let $q:M(k{\mathrm{\Lambda }}_0)\to V^k$ be the quotient imposing the finite-simple highest-vector relations $f_i{v}_{\lambda }=0$ (and hence the trivial finite-dimensional top representation). The canonical map $r_e^{\lambda }$ kills those vectors: for each finite simple root ${\alpha }_i$, its one-root polarization map has kernel the BGG image of $M(s_i\cdot \lambda )$ by step 4.1, and it factors into a sufficiently large translation comparison because ${\alpha }_i\in I(y_n)$. Therefore $r_e^{\lambda }$ factors through an actual current-linear map $r_V:V^k\to W(k{\mathrm{\Lambda }}_0)$. Set $v_0=q(j_0{v}_{\sigma })$. It is a singular vector in $V^k$ and is nonzero because $r_V(v_0)=d(v_{\sigma }^W)\mathrm{\ne }0$. The degree-zero exactness of the ordinary BGG seed says that the kernel of $M(\lambda )\to L(\lambda )$ is generated by its integral-simple degree-one images. After quotient by the finite-simple images, its remaining generator is precisely $v_0$. Thus $U(\widehat{\mathfrak{g}})v_0$ is the actual maximal ideal of $V^k$, not a formal Verma-character contribution. The displayed square embeds this particular singular line into $W(k{\mathrm{\Lambda }}_0)$ and identifies its image with the first Wakimoto boundary. It requires no unproved assertion that the whole map $r_V$ is injective.

Finally, for every finite current-mode word $P$, the same square reads $d(P{v}_{\sigma }^W)=r_V(P{v}_0)$. Hence the induced map from the actual ideal $U(\widehat{\mathfrak{g}})v_0$ into the vacuum Wakimoto term lands in the actual image of $d$. This is the current-equivariant singular-line and descendant interface supplied to a parabolic consumer; it identifies a vector and all its mode descendants, rather than only an abstract one-dimensional Hom space.

The remaining application is the actual parabolic BRST induction/restriction construction, including its central-level anomaly and functorial nonzero edge. It is treated separately; the present exactness theorem does not silently supply that interface.