Affine Zhu quotient and G-integrable admissible descent

Restricted current representations give actual vertex modules. Let E be any unital U(g)-module and induce $M_k(E)$ from g[t]+CK with positive current modes zero and $K=k$. PBW gives $U(g[t^{-1}]t^{-1})$ tensor E, graded by total negative loop degree. Its current fields $J_x(z)$ are lower truncated and satisfy [J_x(z1),J_y(z2)] = J_[x,y](z2) z1^-1 delta(z2/z1) +k(x,y) partial_z2 z1^-1 delta(z2/z1).

Thus they are pairwise local. This Lie action is not simply assumed to be a VOA module. Close the fields under derivatives and the nth products (A_n B)(z)=Res_w ((w-z)^n A(w)B(z)-(-z+w)^n B(z)A(w)),

with the two specified binomial expansions; negative n give normal-ordered derivative products. These are fields by lower truncation. To check locality of such a product with C, multiply by powers clearing the three pairwise poles of A,B,C, commute C past both factors, and take the displayed residue; a power larger than the sum of the three pole orders plus |n|+1 kills the resulting commutator. This proves the needed closure, including negative products.

For local A,B, multiply their two product expansions by a common clearing power of w-z. Taylor expansion at $w=z+x$ and coefficient extraction then gives the three-delta Jacobi identity with iterate ${\sum }_n(A_{n}B)(z)x^{-n-1}$. Every coefficient has finitely many rightmost modes on a fixed vector. This proves the action identity, rather than invoking it as a missing axiom. On the field space itself, identity field is the vacuum and the derivative is translation; the same residue identity gives its state-field structure. The affine current vacuum relations hold for the identity field, so the induced-vacuum universal property gives $V^k(g)\to$ this local-field vertex algebra. Current PBW induction shows it preserves all nth products. Evaluation of its fields on $M_k(E)$ is therefore an actual weak $V^k$-module.

Its negative-loop grading is admissible: current $x_n$ has degree -n, and derivatives/nth products give degree wt(a)-n-1 for every state mode. Grade zero is E, with x0 acting as the original x. This argument allows infinite-dimensional E, including the left regular U(g)-module. It supplies the actual modules needed for the Zhu injection below.

Sugawara and the Zhu algebra. Noncriticality gives the already proved Sugawara modes. Their commutators with current modes make L0 the loop-degree operator on the vacuum, since it kills1 and $[L0,x_n]=-n$ $x_n$. Similarly L(-1) is the vertex translation, since both kill1 and have the same commutator on every current PBW word. Thus $V^k$ is the nonnegative CFT-type VOA with currents of weight1 used by the basic Zhu theorem.

The classes x=[x(-1)1] generate $A(V^k)$. For $n\ge 2$, the circle of $T^{n-2}x$ with v is $(n-1)!(x_{-n}v+x_{-n+1}v)$, so higher negative current modes reduce to $x_{-1}$. The star formula $[x_{-1}v]$=[x][v]-[x0 v] then reduces every PBW state by current length. Their star commutator is the finite Lie bracket: xy-y*x equals [x,y]-2 1+2[x,y]-1 1, whose class is [x,y]_-1 1 since [Ta]=-[L0 a]. Consequently x↦[x(-1)1] gives a surjective algebra map $U(g)\to A(V^k)$.

On the actual module $M_k(E)$ just constructed, zero modes on grade zero are the given g action. The basic Zhu action theorem therefore makes the composite $U(g)\to A(V^k)$ act on E exactly as its original action. Taking E=the left regular U(g)-module shows that the surjection is injective. Hence $A(V^k)=U(g)$.

If Nk is the supplied simple vacuum ideal, the quotient map sends $O(V^k)$ onto $O(V^k\mathrm{/}Nk)$, by the circle definition and lifting homogeneous states. Thus A(V^k/Nk)=U(g)/Ik,

where Ik is the image of Nk. The complete $n=0$ induction/top correspondence applies to this quotient, not merely to $V^k$: an Ik-annihilated simple g-module E has a simple admissible quotient module with that actual top. Conversely every simple highest quotient module has such a top. Its current highest module is the corresponding affine L(lambda), because induced negative currents give its unique simple highest quotient. No invalid implication from vanishing on the top of a prechosen module to vanishing of all ideal modes is used.

The singular generator and admissible finite tops. Let lambda0=k Lambda0 and let the positive integral simple roots be the finite original simple roots alpha1,...,alphar together with beta0. In the coprime branch beta0=-theta+q delta and m=<lambda0+rhohat,beta0-coroot>=p-hvee+1.

In the coprincipal branch ${\beta }_0=-{\theta }_s+(q\mathrm{/}r)\delta$ and $m=p-h+1$. Both are positive integers. The arithmetic/root-arrangement identification is the one already proved in the shared arithmetic work; it is not being weakened to an unshifted integrability claim.

The degree-one term of the ordinary BGG resolution gives L(lambda0)=M(lambda0)/sum_(beta integral simple) M(s_beta dot lambda0),

with actual embedded Verma submodules. On the other hand the universal affine vacuum module is V^k(g)=M(lambda0)/sum_(i=1)^r M(s_i dot lambda0).

The latter is simply parabolic induction from the one-dimensional trivial finite-g module: the relations $f_i$ $v=0$ for all finite simple i make the horizontal highest module trivial; PBW then leaves exactly the negative-loop induced vacuum module. Therefore the maximal ideal Nk of $V^k$ is generated as an affine module by the image s of the beta0 singular vector.

That image is nonzero. The beta0 label and each finite simple label are distinct length-one integral Weyl elements. Strong linkage says no $M(s_i\cdot {\lambda }_0)$, i finite, contains a simple subquotient $L(s_{{\beta }_0}\cdot {\lambda }_0)$, because distinct length-one elements are incomparable in Bruhat order. If the beta0 highest vector belonged to their sum, its generated copy of $M(s_{{\beta }_0}\cdot {\lambda }_0)$ would give precisely that forbidden simple subquotient. Thus s survives.

Its finite-g weight is eta=m theta in the coprime branch, and eta=m ${\theta }_s$ in the coprincipal branch. The homogeneous grade containing s is finite dimensional and g-integrable: it is a sum of finite tensor powers of the adjoint representation in the current PBW model. Since eta is dominant and s is finite-g highest, its horizontal orbit is the irreducible finite module $E_{\eta }$. The full ideal Nk consists of negative-current descendants of that finite orbit; positive affine modes annihilate s and all PBW words can be reordered into negative modes and horizontal words.

Now use the locally proved affine Zhu identification $A(V^k)=U(g)$. The image Ik of Nk in U(g) is the two-sided ideal generated by the image of $E_{\eta }$. Here is the needed ideal-generation calculation. For a weight-one current x, the circle of $T^{n-2}x$ with v is (T^(n-2)x) circle v=(n-1)! (x_-n v+x_(-n+1)v), n>=2.

Indeed derivative modes annihilate the terms $j=2,\mathrm{.}\mathrm{.}\mathrm{.},n-1$ in the circle sum, while $j=0,1$ have equal coefficient (n-1)!. Thus $[x_{-n}v]=-[x_{-n+1}v]$. Also the star identity gives $[x_{-1}v]$=[x]*[v]-[x0 v]. Repeatedly reduce every negative-current descendant by these two formulas. Horizontal zero modes act by adjoint commutator in U(g). This proves that the whole image Ik is generated by the horizontal orbit of [s], not just that this orbit is contained in Ik.

Let $E_{\lambda }$ be a finite-dimensional simple g-module of dominant integral highest weight lambdabar satisfying the admissible alcove inequality. In the coprime branch lambdabar(theta-coroot)<=p-hvee=m-1;

in the coprincipal branch the corresponding inequality is $\bar{\lambda }({\theta }_s^{\vee })\le m-1$. These follow directly by evaluating regular dominance on beta0; all finite-root pairings are integral, so the integral root set is the vacuum one.

Write alpha for theta or ${\theta }_s$ according to the branch. Its coroot is dominant: alpha is the dominant root of its length orbit, so ${\alpha }_i({\alpha }^{\vee })\ge 0$ for every finite simple ${\alpha }_i$. Hence every weight of $E_{\lambda }$ has alpha-coroot value at most lambdabar(alpha-coroot). Reflection $s_{\alpha }$ makes the weight spectrum symmetric under sign for this coroot. Therefore every weight of $\mathrm{End}(E_{\lambda })$ has alpha-coroot value at most 2 $\bar{\lambda }({\alpha }^{\vee })\le 2m-2$. But $E_{m\alpha }$ contains a vector of alpha-coroot weight2m. There is consequently no nonzero g-map E_(m alpha)→End(E_lambda).

The zero-mode action of the singular-state orbit is such a g-map. It must vanish. The ideal-generation calculation implies that the whole Ik annihilates $E_{\lambda }$.

Finally apply the actual $n=0$ Zhu induction/top correspondence to the quotient $V^k\mathrm{/}Nk$: $E_{\lambda }$ is an A(L(k Lambda0))-module, so its simple admissible induced quotient exists as an actual L(k Lambda0)-module. Viewed as a $V^k$-module it is the simple affine highest module L(lambdabar+k Lambda0), because the current top is $E_{\lambda }$ and the induced negative-current module has that unique simple highest quotient. Thus every G-integrable admissible highest module in the vacuum alcove descends. The proof does not assume that killing the top alone kills every ideal mode on a prechosen module; it uses induction for the QUOTIENT and identifies its unique simple highest quotient.

The same argument also proves that Nk is the maximal vacuum ideal and supplies its single finite-g singular-generator type; both conclusions came from the proved BGG H0, not from a guessed singular-vector formula.