Affine semiregular tilting and the admissible BGG resolution

Principal grading and semiregular bimodule. Give the full untwisted affine algebra the principal grading $\mathrm{deg}(e_i)=1$, $\mathrm{deg}(f_i)=-1$ and $\mathrm{deg}(h)=0$. Then g0 is the finite-dimensional extended Cartan. The algebra is generated by these degrees. Every negative graded piece is finite dimensional, and $N=U(nminus)$ has $N0=C$ and finite-dimensional $N_{-r}$. Put $B=U(h+nplus)$. Both ordered PBW products N tensor B and B tensor N identify with U. Use the graded dual $N\mathrm{\#}=direct$ $su{m}_r$ $(N_{-r}{)}^{\ast }$ and finite-degree internal Hom, never unrestricted completed Hom.

Set $\gamma =2\widehat{\rho }$ on h and zero on nplus. This is a character of B's Lie algebra. Define the algebraic vector space $S=N\mathrm{\#}$ tensor B with its right U-module structure N# $tenso{r}_N$ U. Give it its left U-action from S = internal-Hom_B(U, C_gamma tensor B), (X G)(u)=G(uX).

The equality is literal, with no completion: a homogeneous B-linear map of degree d sends $N_{-r}$ into $B_{d-r}$, so it vanishes for $r>d$; there are finitely many remaining r and each domain is finite dimensional. This also handles the infinite-dimensional $B0=U(h)$.

The two actions commute. The following is the needed normal-order verification, adapted locally from the fully read proof of the published finite-type parabolic semiregular lemma; that lemma's public $g{l}_n$ Statement is not being applied to affine g. For X in g1, PBW uniquely writes aX = Xa + Σ_j H_j C_j^X(a) + R_X(a), a in N,

with a fixed Cartan basis $H_j$ and degree-one maps $C_j^X,R_X$ on N. For Y in $g_{-1}$, right action on f tensor1 is f $L_Y$ tensor1, where $L_Y$ is left multiplication on N. The coinduced left formulas are H(f tensor1)=-(f ad H) tensor1+f tensor(H+gamma(H)), X(f tensor1)=f tensor X +Σ_j(f C_j^X) tensor(H_j+gamma(H_j))+(f R_X) tensor1.

Comparing the PBW expansions of $(Ya)X=Y(aX)$ gives C_j^X L_Y=L_Y C_j^X+c_j id, R_X L_Y=L_Y R_X+Σ_j L_[Y,H_j] C_j^X,

where $[Y,X]={\mathrm{\Sigma }}_j$ $c_j$ $H_j$. Substitution cancels all nonconstant terms in the commutator of the left X action and right Y action. The remaining scalar is gamma([X,Y])-Tr_g0(ad X ad Y).

For $X=e_i,Y=f_j$ this is zero: the trace is 2 ${\delta }_i,j$ because on h the composite is $H\mapsto {\alpha }_j(H)$ ${\delta }_i,j$ $h_i$; $\gamma ([e_i,f_j])=2$ ${\delta }_i,j$. By bilinearity these identities cover $g1,g_{-1}$. Cartan crossing is [ad $H,L_Y]=L_{[}H,Y]$. Inducting on a B word extends commutation from f tensor1 to all f tensor b, and generation by g0,g±1 extends it to U. This calculation uses only finite-dimensional g0, not finite-dimensional g. In particular $\gamma (K)=2h^{\vee }$, since $K={\mathrm{\Sigma }}_i$ $a_i-coroot$ $h_i$.

The map U $tenso{r}_N$ N#→S induced by left multiplication is also an isomorphism. With PBW domain B tensor N#, filter by the NONNEGATIVE PRINCIPAL DEGREE OF THE B WORD and then its PBW length. For X in g1 the leading term is f tensor X; every correction has B principal degree0 rather than1, even though its dual coordinate degree can increase. For H in g0 the extra term has no B factor, hence lower B length. Multiplying a right B word preserves these strict decreases, since the left and right actions commute. Thus the leading map is b tensor f↦f tensor b. Induction on these two well-founded indices proves surjectivity; the largest nonzero pair in a finite relation proves injectivity. All expressions are finite. This correct filtration is essential: it would be wrong to claim that the dual degree decreases (already the sl2 term t f tensor H increases that degree). Thus S has both required PBW models.

Exact free/cofree equivalence and tilting. A homogeneous N-endomorphism of N# transposes to a homogeneous right N-endomorphism of N. Such an endomorphism is multiplication by its value at1. Finite-dimensional graded pieces and finite-degree internal Hom therefore give $internal-En{d}_N(N\mathrm{\#})=N^{\mathrm{o}\mathrm{p}}$. This would be false for unrestricted Hom, which admits completed power series.

On graded U-modules free of finite graded rank over N, $T=S$ $tenso{r}_U$ - takes a finite sum of shifts of N to the corresponding finite sum of shifts of N#. Conversely $H=internal-Ho{m}_U(S,-)$, using the left PBW model and the endomorphism identity, takes finite cofree objects to finite free objects. Tensor-Hom adjunction supplies the unit and counit. On the shifted rank-one N summands these are respectively the endomorphism identity and the evaluation map, so both are isomorphisms. A short exact sequence of finite free objects splits over N; the cofree version splits because N# is graded injective, its Hom being the dual of the appropriate graded component. Hence these are exact equivalences for the indicated exact categories.

Take principal restricted dual with antipode action (Xf)(v)=-f(Xv). It interchanges finite cofree and finite free N-modules. Therefore t(M)=(S tensor_U M)#

is an exact contravariant equivalence on finite-Verma-flag modules, with inverse H composed with restricted dual. For $M(\lambda )=U$ $tenso{r}_B$ $C_{\lambda }$, tensor cancellation gives T $M(\lambda )=N\mathrm{\#}$ tensor $C_{\lambda }$. On the augmentation vector its Cartan action is lambda+gamma. Dualizing yields a highest generator of weight -lambda-gamma, with a free negative action. Consequently t M(lambda) = M(-lambda-2rhohat).

Finite Verma flags are finite free N-modules because each extension splits over N; reversing their exact sequences yields the required finite Verma flags on t(M). Conversely its inverse reverses these flags and takes each standard back to a standard. Principal grading origins may be chosen separately on the finitely many Cartan-weight cosets. All actual U maps preserve the grading-element eigenvalues, so this descends from graded to the usual finite-flag category; no arbitrary completed-Hom equivalence is asserted.

Hom uniqueness in the admissible orbit. Let lambda be regular dominant with full integral root system of affine type, and $t\mathrm{\ne }0$. Put lambda*=-lambda-2rhohat. It is regular antidominant relative to its integral positive roots, and has level parameter -t. The determinant proved in the adjacent note makes M(lambda*) simple: no real positive-root pairing is a positive integer and the critical factor is nonzero.

For every w in the integral Weyl group, M(w dot lambda*) contains M(lambda*). Choose a sequence of left descents of w. If $s_{\beta }$ decreases length, $w^{-1}$ beta is negative, so $<w(\lambda \ast +\widehat{\rho }),\beta -coroot>$ is positive integral. The positive-real-root embedding already proved gives $M(s_{\beta }w<mrow><mi\; mathvariant="normal">d</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">t</mi></mrow>{\lambda }^{\ast })\to M(w<mrow><mi\; mathvariant="normal">d</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">t</mi></mrow>{\lambda }^{\ast })$. Iteration reaches lambda*. Every nonzero map is injective.

If two maps between any two orbit Vermas were independent, restrict them to the common simple Verma submodule of their source. The simple-source growth argument gives proportional restrictions. Their scalar difference vanishes on a nonzero submodule, hence is zero by PBW injectivity. Thus Hom between any two negative-orbit Vermas has dimension at most1. Applying the fully constructed exact tilting equivalence proves the same bound for the original positive-orbit Vermas. This is not an assumption of Fiebig block equivalence.

Building compatible maps. For a Bruhat cover $u>v$ in the integral Coxeter group, $u=s_{\beta }$ v with beta positive and $v^{-1}$ beta positive. Since lambda is regular dominant, $<v(\lambda +\widehat{\rho }),\beta -coroot>$ is positive integral. The real-root embedding gives M(u dot lambda)→M(v dot lambda). Composing along a descending Bruhat path gives an embedding $j_w:M(w<mrow><mi\; mathvariant="normal">d</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">t</mi></mrow>\lambda )\to M(\lambda )$. Hom uniqueness implies every cover map can be normalized so that $j_v$ $i_{u,v}=j_u$. Therefore all composites are literal inclusions of the same submodule and are path-independent.

Here is a sign construction for the infinite integral Coxeter group. The required subword/lifting and rank-two diamond properties follow from strong exchange; these are combinatorial Coxeter statements and do not require a longest element. The actual generic strong-exchange public interface is thm-cg-root-inversion-formulas-and-strong-exchange, whose Statement was read. The finite-ideal construction below adapts the fully read proof of lem-compatible-signs-exist-on-the-bruhat-graph; its finite-Weyl public Statement is not silently enlarged.

For a principal Bruhat ideal I(v), induct on length(v). Pick a right descent s and put $J=I(vs)$. An element x outside J must descend under s, and xs lies in J, by lifting. Assign sign1 to its vertical edge x→xs. For any other cover x→y with x outside J, y descends too; otherwise lifting would force $ys=x$, the excluded vertical edge. Thus xs→ys is an edge inside J. Assign epsilon(x,y)=-epsilon(y,ys) epsilon(xs,ys).

This makes the side diamond (x,y,xs,ys) negative. Every diamond is negative: if its top is in J use induction. Otherwise, if neither middle is the vertical mate of its top, translating all four vertices by s gives a diamond in J; multiplying that diamond with the four side diamonds cancels all auxiliary edges, leaving product $(-1{)}^5=-1$. If one middle is the vertical mate, either the whole diamond is already a side diamond, or it is the product of three side/inside diamonds (the extra vertices are the s-mates of its other middle and its bottom). Their auxiliary edges again occur twice and cancel, leaving $(-1{)}^3=-1$. These cases exhaust the two-middle diamond. This constructs signs on every finite I(v).

Two signings on I(v) differ by a vertex gauge. Their ratio r has diamond product1. In the same induction choose the gauge c on J, and put $c(x)=r(x,xs)c(xs)$ outside J. The side diamond gives $r(x,y)=c(x)\mathrm{/}c(y)$ on every remaining lateral edge, so the gauges agree everywhere. Fix $c(e)=1$.

Every finite collection of Weyl elements has a common Bruhat upper bound. Concatenate their reduced words and take the successive Demazure maxima u→max(u,us). Lifting shows by induction that this maximum dominates every subword, hence each chosen element. Enumerate the countable group by finite words, and recursively choose upper bounds $v_n$ so $I(v_n)$ exhaust the group and increase with n. Construct signs on each new ideal, gauge their restriction to match the old signs, and extend that finite vertex gauge arbitrarily outside the old ideal. This yields a global compatible signing. Choices can be made lexicographically from finite sign assignments. Every diamond lies in one finite ideal, so its product is -1.

Use these signs on the normalized inclusions. Each coefficient of d squared is the sum of the two diamond composites; their maps coincide and their signs are opposite, so it is zero. This is an actual complex, with finitely many terms in each homological degree. The following argument proves its exactness independently of its alternating character.

Rank-one coset filtration and homology integrability. Fix an integral simple root beta. Integrability below ALWAYS means integrability after the explicitly indicated nonintegral-wall transport; it never means ordinary integrability under the unshifted root sl2 of beta or under $g[t^q]$. For example at sl2 $k=-1\mathrm{/}2$, $\beta =2\delta -\alpha =s0(\alpha 1)$ has lambda(beta-coroot)=-1. The nonintegral $T_{s0}$ sends lambda to s0 dot lambda with original alpha1-highest weight1, so the transported pairing is2. It is this original sl2 action that is used.

Identify the real level-one affine weight space modulo $\mathbb{R}\delta$ with the finite real weight space $E$ using the full affine conventions in [F1, Facts F3]. For $\beta =\alpha +n\delta$, its coroot is ${\alpha }^{\vee }+2nK\mathrm{/}(\alpha ,\alpha )$, so the finite part of $s_{\beta }$ is $x\mapsto x-2((x,\alpha )+n)\alpha \mathrm{/}(\alpha ,\alpha )$. Its wall is therefore $(x,\alpha )=-n$, exactly the Euclidean wall reflection of [F4]. The original simple walls are $(x,{\alpha }_i)=0$ and $(x,\theta )=1$; hence their fundamental alcove and reflection action coincide with those of [F4]. This verifies the comparison rather than attributing a loop-realization identification to that supplier. The positive integral chamber is the intersection of the half-spaces $(x,\alpha )+n>0$ for inherited positive integral roots, so it is convex and contains the original fundamental alcove. At a relative interior point of its $\beta$ facet, avoid the other original walls; local finiteness permits this because no distinct reduced positive affine roots have the same wall. On the chamber side there is then an original alcove adjacent to that facet. A generic segment inside the chamber from the original fundamental alcove to this alcove crosses only nonintegral walls and gives a finite gallery by [F4]. If $g$ takes the fundamental alcove to the chosen alcove, put $w=g^{-1}$. Then $w\beta ={\alpha }_i$ for its corresponding original simple wall, with positive orientation. Every positive integral root is positive on both alcoves, so $w$ sends it to an original positive root. Thus the inversions used in this transport avoid the integral root set.

At every crossed nonintegral simple wall, use the exact functor $T_i(M)={\sigma }_i^{\ast }(M_{fi}\mathrm{/}M)$. For a weight module with $e_i$ locally nilpotent and no integral $h_i$ eigenvalues, $f_i$ is injective: $f_i$ torsion and $e_i$-local nilpotence would generate a finite-dimensional sl2 module and thus an integral $h_i$ eigenvalue. Ore localization is exact, and the quotient exactness follows from the natural short exact sequence $0\to M\to M_{fi}\to M_{fi}\mathrm{/}M\to 0$. For a Verma M(nu), $T_i$ $M(\nu )=M(s_i<mrow><mi\; mathvariant="normal">d</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">t</mi></mrow>\nu )$. Indeed its nonzero highest vector is $f_i^{-1}$ $v_{\nu }$; use parabolic PBW order with the negative roots other than $-{\alpha }_i$ first. The new negative i-generator acts on $f_i^{-r}$ with coefficients $r(\nu (h_i)+r+1)$, all nonzero because $\nu (h_i)$ is nonintegral. Triangular induction in the parabolic PBW filtration shows that the new negative words on that vector give exactly the Verma PBW basis. This proves both generation and absence of additional relations.

Local rank-one lemma and the precise category. Write $e=e_i$, $f=f_i$, $h=h_i$, $N=\mathbb{C}[f]$ and $D=\mathbb{C}[f,f^{-1}]\mathrm{/}\mathbb{C}[f]$, with $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$. Let $\mathcal{C}$ be all subquotients of finite direct sums of the orbit Vermas in the current nonintegral block; let ${\mathcal{C}}^{\mathrm{\prime }}$ be the analogous full subquotient category generated by their reflected Vermas. Every $h$-eigenvalue on these modules is nonintegral: weights differ from the block weight by the root lattice, whose pairing with $h_i$ is integral. In particular $f$ is injective and $e$ locally nilpotent on every object of either category. For the injectivity assertion, if $fv=0$ and $e^{r+1}v=0$ with $e^{r}v\mathrm{\ne }0$, then $f{e}^{r+1}v=-(r+1)(a+r)e^{r}v$ for $hv=av$, so $a=-r$ is integral, a contradiction.

Finite rank in the exact direction being localized. Decompose by full Cartan-weight cosets modulo $\mathbb{Z}{\alpha }_i$, not by $h_i$ alone. Each such slice of a Verma is a finite-rank free $N$-module. Indeed put $f_i$ first in negative PBW order. The remaining positive roots all have a positive coefficient at some other simple root. At any fixed vector of those other coefficients there are finitely many such roots and finitely many PBW monomials: for an untwisted affine root this follows directly from its form $n\delta +\gamma$, since the other coefficients bound $n$; a root supported solely at $i$ is just ${\alpha }_i$. Their root spaces have finite multiplicity. Thus fixing the other coefficients leaves a finite set of PBW words, each with an arbitrary $f_i$ power. A finite sum of Vermas has the same property. There is no assertion that the whole module has finite rank over $N$.

Nonintegral strings, including subquotients. A weight ${\mathfrak{s}\mathfrak{l}}_2$-module with $e$ locally nilpotent, nonintegral weights and support bounded above in one ${\alpha }_i$-slice is the algebraic direct sum of its highest strings. Here is an explicit splitting. On a vector $v$ of weight $a$, put $P(v)={\sum }_{n\ge 0}(-1{)}^n{f}^n{e}^{n}v\mathrm{/}(n!(a+2)(a+3)\cdots (a+n+1))$, with empty product $1$. The sum is finite by local nilpotence and every denominator is nonzero. The identity $e{f}^n{e}^{n}v=f^n{e}^{n+1}v+n(a+n+1)f^{n-1}{e}^{n}v$ proves $eP(v)=0$ by adjacent cancellation. Repeatedly subtracting the highest-string contribution to $v$ and inducting on its $e$-nilpotence degree expresses $v$ as a finite sum $\sum f^r{v}_r$ with $e{v}_r=0$: equivalently start with $e^{R}v$ and use $e^R{f}^{R}w=R!{\prod }_{j=0}^{R-1}(b-j)w$ for a highest vector of weight $b$, whose factors are nonzero. Applying the largest occurring $e$ power proves independence. These strings are simple Verma modules, since $e{f}^{r}w=r(b-r+1)f^{r-1}w$ never vanishes for $r>0$. Their submodules split: in one weight coset two different highest weights $b,b-2r$ have distinct Casimir eigenvalues unless $b=r-1$ is integral; at a fixed highest weight the multiplicity space is an ordinary vector space. Thus every subquotient of a finite sum of strings is again a finite sum of strings. By step 29.1 every slice of every object of $\mathcal{C}$ and ${\mathcal{C}}^{\mathrm{\prime }}$ is consequently finite free over $\mathbb{C}[f]$; after the inverse Tits twist, every slice of an object of ${\mathcal{C}}^{\mathrm{\prime }}$ is finite cofree over $\mathbb{C}[f]$, namely a finite sum of shifts of $D$. Here cofree means this explicit restricted-dual module, with $f{f}^{-1}=0$ and $f{f}^{-r}=f^{-(r-1)}$ for $r>1$.

The inverse and its adjunction maps. Set $S=U_f\mathrm{/}U$. Ore reordering gives both models $S=D{\otimes }_{N}U$ as a right $U$-module and $S=U{\otimes }_{N}D$ as a left $U$-module: powers of $f$ move across each enveloping word by the finite commutator expansion because $\mathrm{ad}f$ is locally nilpotent. Their maps send $f^{-r}\otimes u$ and $u\otimes f^{-r}$ to the indicated classes in $U_f\mathrm{/}U$; the inverse maps follow from the two PBW orders, so these are isomorphisms, not completed tensor products. Choose the Tits lift $\sigma$ with $\sigma (e)=-f$, $\sigma (f)=-e$ and $\sigma (h)=-h$; pullback means $u\cdot x=\sigma (u)x$. Put $T(M)={\sigma }^{\ast }(S{\otimes }_{U}M)$ and $G(X)={\mathrm{Hom}}_U^{\mathrm{w}\mathrm{t}}(S,({\sigma }^{\ast }{)}^{-1}X)$. Precisely, the superscript selects the direct sum over $\nu \in {\widehat{\mathfrak{h}}}^{\ast }$ of left $U$-linear maps $\phi$ satisfying $\phi (sH)=\nu (H)\phi (s)$ for every $H\in \widehat{\mathfrak{h}}$ and $s\in S$. Equivalently, $\phi (f^{-r})$ has weight $\nu +r{\alpha }_i$ in the inverse-twisted target. It excludes infinite sums of distinct weight maps. Give $G$ its left $U$-action $(u\phi )(s)=\phi (su)$. Tensor-Hom adjunction has unit ${\eta }_M(m)(s)=s\otimes m$ (with the twists canceled) and counit ${ϵ}_X(s\otimes \phi )=\phi (s)$, followed by ${\sigma }^{\ast }$. These are actual $U$-maps by the commuting bimodule actions. The left PBW model identifies $G(X)$ with ${\mathrm{Hom}}_N^{\mathrm{w}\mathrm{t}}(D,({\sigma }^{\ast }{)}^{-1}X)$. On one string the identities ${\mathrm{End}}_N^{\mathrm{w}\mathrm{t}}(D)=N$ and $D{\otimes }_{N}N=D$ identify the unit with $N\to N$ and the counit with evaluation $D{\otimes }_{N}N\to D$. To check the endomorphism identity explicitly, $f$-linearity forces the value on $f^{-r}$ to be the truncated translate of that on later basis vectors; a homogeneous map is a scalar multiple of a nonnegative power of $f$, and negative powers cannot commute with $f$ at $f^{-1}$. Finite sums of homogeneous maps give exactly $N$, whereas unrestricted Hom would give $\mathbb{C}\mathrm{⟦}f\mathrm{⟧}$. The finite number of strings in each full-Cartan slice in step 30.1 makes these rank-one checks valid slice by slice, with no product over infinitely many strings. Therefore both ${\eta }_M$ and ${ϵ}_X$ are isomorphisms on the categories under consideration.

Exactness, closure and actual equivalence. Localization and the injection in step 28.1 make $T$ exact. On the inverse-twisted slices of ${\mathcal{C}}^{\mathrm{\prime }}$, all short exact sequences split as $N$-modules by the string decomposition, so the restricted Hom defining $G$ is exact as well. The construction in step 27.1 identifies $T$ of each generating Verma with its reflected Verma. Hence exactness puts $T(\mathcal{C})$ inside ${\mathcal{C}}^{\mathrm{\prime }}$. Conversely apply $G$ to a submodule and quotient of a finite sum of reflected Vermas: its exactness and $GTM\cong M$ identify the result with a subquotient of the corresponding original finite Verma sum. Thus $G({\mathcal{C}}^{\mathrm{\prime }})\subset \mathcal{C}$. This also proves that the restricted Hom has the required category-$\mathcal{O}$ support and local finiteness; those conditions were not silently assumed for arbitrary unrestricted Hom. The unit and counit of step 31.1 now prove mutually inverse exact equivalences $T:\mathcal{C}\simeq {\mathcal{C}}^{\mathrm{\prime }}:G$. These full subquotient categories suffice even if a kernel in the affine complex is not known a priori to be finitely generated.

Simple images and highest labels. For any simple highest subquotient $L(\mu )$ of an orbit Verma sum, $TL(\mu )$ is nonzero by the unit, and is simple as an actual $U$-module: any of its $U$-submodules belongs to the full subquotient category ${\mathcal{C}}^{\mathrm{\prime }}$, and the exact inverse would produce a proper nonzero submodule of $L(\mu )$. Apply $T$ to $M(\mu )\twoheadrightarrow L(\mu )$. Its source is $M(s_i\cdot \mu )$ by step 27.1, so its nonzero simple quotient is precisely $L(s_i\cdot \mu )$. Thus every simple subquotient, rather than just a formal character or a highest line in a localization, has the asserted image.

At each gallery wall, all weights in the relevant integral orbit have nonintegral $h_i$ eigenvalues, since the wall is outside that integral root system and root-lattice shifts change $h_i$ by integers. Apply the local equivalence in steps 28.1–33.1 to the finite sums of orbit Vermas in each complex degree. Its exactness transports the actual kernels, images, homology and their simple subquotients, and step 33.1 gives the reflected simple highest labels. Iterating this construction through the finite gallery therefore gives exactly the simple-subquotient transport and integral-root labels needed below, with a restricted-Hom inverse at every step. No external equivalence theorem is used.

Pair its Weyl labels in left cosets of $1,s_i$. Filter by the length of the minimal coset representative. The Bruhat subword property shows this filtration is preserved by the differential. A cover with equal minimal-coset lengths is precisely the internal pair: write its upper word as $s_i$ a with a minimal; deleting a letter other than its initial $s_i$ leaves a reduced word still starting with $s_i$, so its lower element cannot be the minimal representative of a different equal-length coset. On its associated graded, each coset contributes the two-term rank-one embedding M(s_i dot nu)→M(nu), <nu+rhohat,alpha_i-coroot> positive integral.

The cokernel is locally $sl{2}_i$-integrable: $f_i$ has the stated finite relation on its highest vector, $ad(f_i)$ is locally nilpotent on every enveloping word, and $e_i$ is locally nilpotent by highest support. The kernel is zero. All pages of the filtered-complex spectral sequence therefore consist of $sl{2}_i$-integrable modules, because submodules, quotients and finite extensions preserve local nilpotence of both generators. In homological degree j only minimal-coset lengths j or j-1 occur. Thus the filtration in that degree is finite, and convergence gives actual $sl{2}_i$-integrability of the transformed homology.

Now suppose some original homology has a simple highest subquotient L(mu). Linkage for the orbit Verma terms forces mu in the integral dot orbit of lambda. Successively applying the exact nonintegral-wall functors, and at each step taking the highest subquotient above, produces L(w dot mu) as a subquotient of the transformed homology. Its $sl{2}_i$-integrability forces $<\mu +\widehat{\rho },\beta -coroot>$ to be positive integral. Do this for every integral simple beta. Thus mu is regular dominant for the integral Coxeter system. The orbit has a unique such weight: a nontrivial Weyl element has a left descent, whose inverse sends a simple root negative, contradicting simultaneous positive pairings. Hence $\mu =\lambda$.

For $j>0$ the complex term consists of M(v dot lambda) with $length(v)=j$; all their weights are strictly below lambda. No such homology can contain L(lambda), so it is zero. In degree zero the homology is generated by the original highest line. Any proper nonzero submodule would have a highest simple subquotient below lambda, contradicted by the same argument. The degree-zero homology is therefore L(lambda).

This proves exactness of the constructed ordinary BGG seed. The finite-coset filtration, not equality of alternating characters, proves the vanishing.

Opposite polarization. Reverse the principal grading and replace $2\widehat{\rho }$ by $-2\widehat{\rho }$ in the same construction. Every finite normal-order and PBW calculation remains valid. Write $N_{+}=U({\widehat{\mathfrak{n}}}_{+})$ and $B_{-}=U(\widehat{\mathfrak{h}}+{\widehat{\mathfrak{n}}}_{-})$, so its right PBW model is $N_{+}^{\mathrm{\#}}\otimes B_{-}$. For $H\in \widehat{\mathfrak{h}}$ the reversed version of step 5.1 is $H(f\otimes b)=-(f\circ \mathrm{ad}H)\otimes b+f\otimes (H-2\widehat{\rho }(H))b$. The adjoint term generally does not vanish. For $H=K$ it vanishes because $K$ is central, giving precisely left $K$ equal to right $K-2h^{\vee }$. This supplies the actual opposite bimodule used in relative BRST induction.