Exact principal minus reduction and vanishing of minimal-series extensions

Separate the two categories before making any finite-length assertion. Write $O^{\mathrm{\Lambda }}$ for the FULL affine category of F3 supported in $\mathrm{\Lambda }-Q^{+}$, at the fixed level k, and write ${\dot{O}}^{\mathrm{\Lambda }}$ for its full subcategory whose old-degree slices are objects of the ordinary finitely generated BGG category O of the finite-dimensional g. The latter finite g category, not the weakened affine convention, has finite length and is closed under subquotients, extensions and restricted duality by F5. Consequently ${\dot{O}}^{\mathrm{\Lambda }}$ is Serre and is closed under full-Cartan restricted Chevalley duality. This dual is the direct sum of the finite full-weight duals, fixes weights, and is exact and involutive weight by weight; its degree-d slice is the usual finite-g restricted dual of $M_d$. The affine current action on it is restricted because sufficiently positive loop modes move a fixed full weight outside $\mathrm{\Lambda }-Q^{+}$. For $\mu \le \mathrm{\Lambda }$, form the positive-Borel module $T_{\mu }$ from $U(b_{\mathrm{a}\mathrm{f}\mathrm{f}}){\otimes }_{U(h_{\mathrm{a}\mathrm{f}\mathrm{f}})}{C}_{\mu }$ by quotienting its weight spaces not lying below Lambda. Those spaces form a Borel submodule, since adding a positive root cannot repair a simple-root coefficient already exceeding the bound. The surviving positive-root PBW words have bounded total height, so $T_{\mu }$ is finite dimensional. It is generated by its mu vector, and ${\mathrm{Hom}}_b(T_{\mu },M)=M_{\mu }$ for every M in $O^{\mathrm{\Lambda }}$. PBW induction is exact and gives $P_{\mu }=U(g_{\mathrm{a}\mathrm{f}\mathrm{f}}){\otimes }_{U(b_{\mathrm{a}\mathrm{f}\mathrm{f}})}{T}_{\mu }$, with $\mathrm{Hom}(P_{\mu },M)=M_{\mu }$. This proves projectivity directly, without an Artinian hypothesis. A finite descending-weight flag of $T_{\mu }$ has one-dimensional Borel character quotients, so $P_{\mu }$ has a finite affine Verma flag. Each full weight piece is finite and its support is below Lambda. A finitely generated M in $O^{\mathrm{\Lambda }}$ is a quotient of a FINITE sum of these $P_{\mu }$, by choosing finitely many homogeneous generators. At each fixed old energy, an affine Verma is a finite-dimensional adjoint-loop PBW factor tensored with its finite-g Verma top. Such a tensor is finitely generated over U(g): move a generator across the finite tensor factor using $x(f\otimes m)=xf\otimes m+f\otimes xm$, inducting on word length. It has finite-cone support and finite weight spaces, so lies in ordinary finite-g O. Finite flags and quotients show that every finitely generated affine M lies in ${\dot{O}}^{\mathrm{\Lambda }}$. The restricted dual $P_{\mu }^{\ast }$ is injective in $O^{\mathrm{\Lambda }}$, since $\mathrm{Hom}(-,P_{\mu }^{\ast })=\mathrm{Hom}(P_{\mu },(-{)}^{\ast })$ is exact, and has a finite dual-Verma flag. Only these finite flags will be used.

Put $D_0=\mathrm{\Lambda }(D)$ and $d=D_0-N$ with N a nonnegative integer; degree pieces outside these old-degree congruence classes are zero. Within this coset, an old degree is at least another exactly when their difference is a nonnegative integer; no order on arbitrary complex numbers is used. All ghost creators have nonnegative old loop-energy cost, so $C_{-}(M)$ has no old degree above D0. In its degree-d component a nonzero-loop ghost monomial of energy cost s has $0\le s\le N$ and absolute nonzero-loop ghost charge at most s. For fixed s its nonzero-loop ghost space $G_s$ is finite dimensional. Decompose the ghost space as its nonzero-loop part tensor Lambda(n-), where the zero-mode wedge is in nonpositive cohomological degrees. Filter by nonzero-loop ghost charge. In the actual BRST formulas the filtration-preserving part is precisely character-twisted Chevalley homology for the finite zero-mode n- acting diagonally on the coefficient $M_{d+s}\otimes G_s$; the other terms increase nonzero-loop charge. This follows by separating terms with all zero modes, terms with a single zero mode, and terms with none: their filtration increases are respectively 0, 0 or 2, and 1. At fixed total old degree there are only finitely many filtration values, so this is an ordinary bounded spectral sequence, not an unbounded convergence assertion. The zero-mode n- action on $G_s$ is nilpotent: it preserves its energy and lowers finite-root height in its finite set of Cartan weights. A weight-height flag, refined within a weight, has trivial one-dimensional n- quotients. If M lies in ${\dot{O}}^{\mathrm{\Lambda }}$, the induced flag on $M_{d+s}\otimes G_s$ has quotients $M_{d+s}$. F2 and its homology long sequences therefore kill every positive zero-mode homology group, and give finite-dimensional zeroth homology. After this first page the zero-mode homological degree is zero. Thus a surviving total cohomological degree i must equal a nonzero-loop ghost charge of energy cost at most N. We obtain $H_{-}^i(M{)}_d=0$ for $\mathrm{\mid }i\mathrm{\mid }>N$, and finite-dimensional $H_{-}^i(M{)}_d$, for M in ${\dot{O}}^{\mathrm{\Lambda }}$. These bounds use the SAME Lambda, independently of finite generators and of the number of later kernel/cokernel steps. The support assertion for complexes and tails uses no dot O hypothesis; no finite-dimensionality assertion is made here for arbitrary full affine O.

First reduce a full affine-O object to finitely many root-lattice support cosets. Within each coset, the finitely many given upper cones have a common upper bound Lambda: choose a base weight in the coset and take coefficientwise maxima of their simple-root coordinates. The full Cartan, including D, makes those coordinates unambiguous. The complexes split over these finitely many submodules, so fix one $O^{\mathrm{\Lambda }}$. Every M in $O^{\mathrm{\Lambda }}$ is the filtered union of its finitely generated submodules, each in ${\dot{O}}^{\mathrm{\Lambda }}$ by step 1.1. The actual BRST complex is an algebraic tensor complex, with finite support on each vector. Hence it commutes with this union, as does its cohomology: a cycle has finitely many module coefficients in some finitely generated submodule, and a chosen primitive witnessing a boundary has finitely many coefficients in a possibly larger one. Thus concentration for ${\dot{O}}^{\mathrm{\Lambda }}$ implies concentration for ALL $O^{\mathrm{\Lambda }}$, and the uniform bound of step 2.1 also passes to the union. This is not a finite-generation restriction on the Statement. In particular the union may have infinite-dimensional reduced old-degree pieces. For dimension shifting inside ${\dot{O}}^{\mathrm{\Lambda }}$ fix $d=D_0-N$. The N+1 finite-g modules $M_{D_0},\mathrm{.}\mathrm{.}\mathrm{.},M_d$ are each finitely generated. Select finite g generating sets for all of them; the affine submodule M' generated by their union is finitely generated and has exactly the same slices in this range, since g has old degree zero. Therefore M/M' has no old degree at least d. Its BRST complex has zero degree-d component by the support observation in step 2.1, so the actual long exact sequence gives $H_{-}^i(M{)}_d=H_{-}^i(M^{\mathrm{\prime }}{)}_d$ for every i. Apply this same construction to the full-Cartan restricted dual $M^{\ast }$ in ${\dot{O}}^{\mathrm{\Lambda }}$. A finitely generated N subset $M^{\ast }$ then has $(M^{\ast }\mathrm{/}N{)}_t=0$ for $t\ge d$. Exact weightwise duality yields $0\to (M^{\ast }\mathrm{/}N{)}^{\ast }\to M\to N^{\ast }\to 0$; its kernel has the same vanishing range. Hence $H_{-}^i(M{)}_d=H_{-}^i(N^{\ast }{)}_d$ for every i. All replacement modules remain in the SAME $O^{\mathrm{\Lambda }}$ and ${\dot{O}}^{\mathrm{\Lambda }}$.

Work first with M in ${\dot{O}}^{\mathrm{\Lambda }}$ and $d=D_0-N$. For $i>0$ replace M at degree d by the finitely generated M' of step 3.1, and take a surjection $P\to M^{\mathrm{\prime }}$ from a finite sum of the projectives in step 1.1. Its kernel K is in ${\dot{O}}^{\mathrm{\Lambda }}$, because that subcategory is Serre. The finite Verma flag and F1 give $H_{-}^j(P)=0$ for $j\mathrm{\ne }0$. The long sequence identifies $H_{-}^i(M{)}_d$ with $H_{-}^{i+1}(K{)}_d$. Repeat the SAME fixed-degree replacement for K and its successive kernels. At stage r the new module is still supported below the original Lambda, so step 2.1 kills the shifted group as soon as $i+r>N$. No kernel is presumed finitely generated or of finite affine length. For $i<0$, step 3.1 replaces M at degree d by $N^{\ast }$ with N finitely generated. Dualizing a finite projective surjection $P\to N$ gives an embedding $N^{\ast }\to I=P^{\ast }$. Its cokernel K is in ${\dot{O}}^{\mathrm{\Lambda }}$. F1 and the finite dual-Verma flag make I acyclic outside degree zero. The long sequence now gives $H_{-}^i(M{)}_d=H_{-}^{i-1}(K{)}_d$. Iterating with the same Lambda kills the group once $\mathrm{\mid }i-r\mathrm{\mid }>N$. Therefore ${\dot{O}}^{\mathrm{\Lambda }}$ has no nonzero-degree cohomology. The filtered-union argument of step 3.1 extends this to every object of FULL $O^{\mathrm{\Lambda }}$ and hence to full affine O. Finally the cohomology long sequence makes $H_{-}^0$ exact as an actual W-module functor. No target ordinary-module or finite-grade assertion is inferred for an arbitrary full-O input.

Let L(lambda) be simple. Exactness makes its reduction a quotient of the W Verma from F1, generated by the image of its top. At the highest old degree the complex is precisely the finite n− homology complex of its finite simple highest top. F2 says that top is zero unless lambdabar is anti-dominant in its explicit no-positive-integer sense. Zero top forces zero reduction because the preceding Verma quotient is generated by its top. Otherwise the top is one-dimensional. Dualizing the affine Verma-to-simple surjection embeds L(lambda) into its affine dual Verma. Exact reduction and F1 embed its nonzero image into the corresponding W dual Verma. That dual Verma has simple socle meeting its one-dimensional top: the annihilator of a submodule in its primal Verma, which is generated by its top, proves every nonzero submodule meets that line. Its one-dimensional top already generates the simple socle. Thus a quotient of the W Verma generated by that line and embedded in the dual Verma must equal that simple socle. This proves the asserted simple-image formula.

The composition passage is applied to an AFFINE VERMA and its subquotients, not to an arbitrary full-O object. Such objects lie in ${\dot{O}}^{\mathrm{\Lambda }}$ by steps 1.1 and 3.1. Fix $d=D_0-N$ and put ${\mathrm{\ell }}_d(Q)={\sum }_{j=0}^N{\mathrm{length}}_g(Q_{D_0-j})$, using ordinary finite-g composition length. This is a finite nonnegative integer, additive on short exact sequences; it is zero exactly when Q has no old degrees at least d. If ${\mathrm{\ell }}_d(Q)>0$, choose a maximal full weight mu of Q among its weights of old degree at least d. Such a maximal weight exists because every interval above a weight in $\mathrm{\Lambda }-Q^{+}$ is finite. Every positive affine root raises old degree by a nonnegative integer, so a nonzero vector v at mu is killed by all positive root spaces. Its cyclic highest module $V=U(g_{\mathrm{a}\mathrm{f}\mathrm{f}})v$ is a Verma quotient, with a one-dimensional highest line, and therefore has a unique simple quotient L(mu): the sum of all proper submodules misses that line. Write J for its kernel. Both J and Q/V are still in ${\dot{O}}^{\mathrm{\Lambda }}$, and ${\mathrm{\ell }}_d(J)+{\mathrm{\ell }}_d(Q\mathrm{/}V)={\mathrm{\ell }}_d(Q)-{\mathrm{\ell }}_d(L(\mu ))<{\mathrm{\ell }}_d(Q)$, because L(mu) has a nonzero slice at $\mu (D)\ge d$. Strong induction on ${\mathrm{\ell }}_d(Q)$, using $0\to J\to V\to L(\mu )\to 0$ and $0\to V\to Q\to Q\mathrm{/}V\to 0$, gives a FINITE filtration with leaves either affine simple highest modules or tails with no old degrees at least d. There are at most ${\mathrm{\ell }}_d(Q)$ simple leaves visible in that range. This is a finite local composition construction, not an affine Artinian hypothesis. Apply this to $Q=M_{\mathrm{a}\mathrm{f}\mathrm{f}}(\lambda )$. Exactness in step 4.1 transports the finite filtration through the actual $H_{-}^0$; by step 5.1 every affine-simple leaf has zero or simple W output. Each tail has no reduced old degree at least d by step 2.1. To detect any chosen local simple subquotient S of the W Verma, choose d no greater than its highest old degree. The mode-degree identification in F1 makes the old grading on this W Verma its relative conformal grading; since it is cyclic with a one-dimensional top, D+(L0 minus its top scalar) is constant. Consequently its W submodules and subquotients respect that grading. S cannot occur in the reduction of a tail below d. Refining the finite transported filtration against a subquotient realizing S therefore makes S a factor of one nonzero simple $H_{-}^0(L(\mu ))$, so S is that simple module. The same finite-filtration argument gives additivity of local multiplicities, allowing different affine labels to have the same W top. Thus every local simple factor needed in the following cross-factor argument comes from an actual affine Verma simple factor. No finite g composition length, finite affine generation or finite affine length is asserted for a general object of the original full affine O.

Suppose lambda is nondegenerate admissible and a different minimal-series simple is a factor of its reduced W Verma. By step5.1 some affine Verma factor L(mu) reduces to it. Thus mubar is finite anti-dominant and has the same finite central character as a nondegenerate admissible label lambda'. By HC it is in the finite dot-Weyl orbit of lambdabar'; choose the arbitrary derivation shift of lambda' to equal that of mu, which does not affect admissibility or its central character. Pure finite Weyl transformations preserve positivity of all affine roots with nonzero positive loop index; finite roots may change sign but all their shifted pairings at lambda' are nonintegral. Therefore mu is itself regular dominant admissible. Yet F3 puts mu in the integral affine Weyl orbit of lambda, where two regular dominant labels cannot differ. Thus $\mu =\lambda$, contradicting the supposed different W top. This proves exactly the cross-minimal-series multiplicity vanishing needed here; it does not assert a generic one-to-one correspondence for all affine composition labels. This uses the full actual affine-classification interface of F3 only for the admissible label and its quotient module, after the local affine-Verma composition passage in step 6.1. Exactness and that passage already hold on the full affine-O domain by the fixed-degree dimension shift and filtered unions above; no finite global affine composition length is inserted into this cross-factor argument.

Let $0\to S_j\to E\to S_i\to 0$ be an A-module extension in the stated generalized finite-grade category. If their lowest conformal weights satisfy ${\mathrm{\Delta }}_i<{\mathrm{\Delta }}_j$, the generalized ${\mathrm{\Delta }}_i$ space of E is the unchanged simple top of $S_i$, and every lowering mode kills it. F4 gives a map of its universal W Verma into E. Its image P surjects onto $S_i$. If P intersects the simple kernel nontrivially, it contains $S_j$ and hence $E=P$; then $S_j$ is a factor of that Verma, contrary to step6.1, since unequal lowest weights imply different characters. Otherwise P is a splitting. For ${\mathrm{\Delta }}_i>{\mathrm{\Delta }}_j$ apply the exact finite-grade contragredient. It permutes the same finite minimal-series list, preserves conformal weights, preserves A-modules because the kernel ideal is stable under the state Virasoro operators in the opposite-field formula, and reverses the extension. The preceding argument splits it and dualizing back splits E. The lowest weights are real rational here: finite simple-coroot eigenvalues obey the finite rank-one admissible polynomial, and the reduced Sugawara zero mode is a rational quadratic expression at rational noncritical k.

If ${\mathrm{\Delta }}_i={\mathrm{\Delta }}_j=\mathrm{\Delta }$, the generalized Delta spaces form an exact sequence of actual Zhu modules. Semisimplicity of the assumed actual A Zhu algebra splits it, including the central L0 action. Choose the lifted simple top. Lowering modes kill it and its induced Verma image has degree-Delta space exactly this chosen top, by the ordered negative-word PBW basis of F1. It cannot contain the independent kernel top, so intersects the simple kernel trivially and gives a splitting. This handles equal characters and self-extensions as well. Induction on finite composition length now proves semisimplicity of the specified finite-length category. Passage to arbitrary admissible modules is separate and is not claimed from finite length alone.