Finite generation of C2-cofinite generalized joint modules

The coefficient form of the module Jacobi identity is the iterate identity $(a_{(m)}b{)}_{(n)}={\sum }_{t\ge 0}(-1{)}^t\left(\genfrac{}{}{0px}{}{m}{t}\right)(a_{(m-t)}{b}_{(n+t)}-(-1{)}^m{b}_{(n+m-t)}{a}_{(t)})$. Here $m,n\in \mathbb{Z}$, $\left(\genfrac{}{}{0px}{}{m}{t}\right)=m(m-1)\cdots (m-t+1)\mathrm{/}t!$, and the sum is finite on each vector by truncation. It is obtained by multiplying the Jacobi identity by $z_0^m{z}_2^n$, taking the three formal residues, and using $(z_1-z_2{)}^m={\sum }_{t\ge 0}(-1{)}^t\left(\genfrac{}{}{0px}{}{m}{t}\right)z_1^{m-t}{z}_2^t$ in the first expansion and its reversed expansion in the second. This formula holds on $V$ itself by [F4]. On any module, substitute $b=\mathbf{1},m=-2$ and use $a_{(-2)}\mathbf{1}=Ta$ from creation and ${\mathbf{1}}_{(j)}={\delta }_{j,-1}I$: the surviving terms give $(Ta{)}_{(n)}=-n{a}_{(n-1)}$, including $n=0$. Thus $Y_M(Ta,z)={\mathrm{\partial }}_z{Y}_M(a,z)$. The commutator coefficient of Jacobi then gives $[L_M(0),a_{(n)}^M]=(r-n-1)a_{(n)}^M$ for $a\in V_r$: explicitly it is $[{\omega }_{(1)}^M,Y_M(a,z)]=z{Y}_M({\omega }_{(0)}a,z)+Y_M({\omega }_{(1)}a,z)=z{\mathrm{\partial }}_z{Y}_M(a,z)+r{Y}_M(a,z)$. The $i$th component mode therefore shifts joint generalized weights by $(r-n-1)e_i$. Finally, [F4] gives the state-product weight $r+s-n-1$; decomposing $a,b$ into their finitely many homogeneous components shows $C_2(V)$ is a homogeneous subspace.

Define $\mathrm{\Omega }(M)=\{v:a_{(n)}v=0\text{ for all homogeneous }a\in V_r\text{ and }n\ge r\}$ for any module $M$. Define $a\circ b={\sum }_{j=0}^r\left(\genfrac{}{}{0px}{}{r}{j}\right)a_{(j-2)}b$, $a\ast b={\sum }_{j=0}^r\left(\genfrac{}{}{0px}{}{r}{j}\right)a_{(j-1)}b$, and $O(V)=\mathrm{span}\{a\circ b:a\text{ homogeneous},b\in V\}$, extending the products linearly. We claim $o(O(V))\mathrm{\Omega }(M)=0$ and $o(a\ast b)v=o(a)o(b)v$ on $\mathrm{\Omega }(M)$. For homogeneous $a\in V_r,b\in V_s$, the state-product weight formula in [F4] identifies the zero-mode indices, so apply step 1.1 to each summand of $o(a\circ b)$ with $m=j-2,n=r+s-j$. Every first-product term kills $v$ because its rightmost $b$-index is $r+s-j+t\ge s$. The second-product terms have the form $b_{(r+s-2-t)}{a}_{(t)}v$; those with $t\ge r$ vanish, and for $0\le t<r$ their coefficient is $-(-1{)}^t{\sum }_{j=0}^r(-1{)}^j\left(\genfrac{}{}{0px}{}{r}{j}\right)\left(\genfrac{}{}{0px}{}{j-2}{t}\right)=0$. The last equality is the vanishing of the $r$th finite difference of a polynomial in $j$ of degree $t<r$ (it follows by applying $r$ successive differences, each of which lowers degree). For $o(a\ast b)$ use $m=j-1,n=r+s-j-1$. The first-product terms vanish except $j=r,t=0$, which is $a_{(r-1)}{b}_{(s-1)}v$; the second-product coefficients vanish by the identical finite-difference argument with $\left(\genfrac{}{}{0px}{}{j-1}{t}\right)$. The case $r=0$ has no $0\le t<r$ terms, so it is included. Linearity proves both claims for arbitrary states.

Any component-stable subspace $U\subseteq W$ decomposes into its intersections with the joint generalized-weight spaces. Indeed, a vector has only finitely many joint components. For each coordinate, polynomials in $L_i(0)$ project onto each of its finitely many eigenvalues on the finite-dimensional cyclic space of that vector: the polynomials exist by the finite Chinese remainder identity for the pairwise coprime powers $(x-{\lambda }_i{)}^{n_i}$. Products of those coordinate projections isolate every joint component, and preserve $U$. Therefore $W\mathrm{/}U$ inherits the direct-sum joint generalized-weight decomposition, coordinatewise lower bounds, finite-dimensional joint spaces, and weight-shift rule of step 1.1.

By homogeneity from step 1.1, choose a finite homogeneous spanning set for $V\mathrm{/}{C}_2(V)$, and let $B\ge 0$ bound its weights. If $n>B$ then any state in $V_n$ is a finite sum of $a_{(-2)}b$ with homogeneous $a,b$ and $\mathrm{w}\mathrm{t}(a)+\mathrm{w}\mathrm{t}(b)+1=n$. Modulo $O(V)$, each such product equals $-{\sum }_{j=1}^r\left(\genfrac{}{}{0px}{}{r}{j}\right)a_{(j-2)}b$, whose summands have weights $n-j<n$ by [F4]. Induction on $n$ shows that $V\mathrm{/}O(V)$ is spanned by the image of $V_{\le B}$, a finite-dimensional space. Neither the ideal property of $O(V)$ nor associativity of the Zhu product is needed for this conclusion.

Every nonzero quotient $Q=W\mathrm{/}U$ has a nonzero joint generalized-weight vector in its simultaneous top ${\mathrm{\Omega }}_k(Q)={\bigcap }_{i=1}^k{\mathrm{\Omega }}_i(Q)$. To see this, select any occurring weight ${\lambda }^0$ and consider the occurring weights in its single coset ${\lambda }^0+{\mathbb{Z}}^k$. Write these as ${\lambda }^0+n$ with $n\in {\mathbb{Z}}^k$. Each $n_i$ is bounded below by the corresponding coordinatewise real-part bound, so the integer sum ${\sum }_i{n}_i$ has a minimum. Choose an occurring joint weight $\lambda$ attaining this minimum and any nonzero vector $q\in Q_{[\lambda ]}$. A component mode $a_{(n)}^{(i)}$ with $n\ge r$ would lower that sum by the positive integer $n+1-r$, so its target weight space is absent. Thus it kills $q$, which proves $q\in {\mathrm{\Omega }}_k(Q)$.

Put $p_0=\mathbf{1}$ and recursively $p_{n+1}=\omega \ast p_n$, using the linear extension of $\ast$ if $p_n$ is inhomogeneous. Finite dimensionality in step 3.1 supplies a relation ${\sum }_{n=0}^d{c}_n{p}_n\in O(V)$ with not all $c_n$ zero. Set $P(x)={\sum }_{n=0}^d{c}_n{x}^n$, a nonzero polynomial, and let $E$ be its finite set of complex roots. Step 2.1 and induction give $o(p_n)v=L(0{)}^{n}v$ for $v\in \mathrm{\Omega }(M)$: at each induction stage the star identity is applied to the original vector $v\in \mathrm{\Omega }(M)$, so no invariance of $\mathrm{\Omega }(M)$ is being assumed. Consequently $P(L(0))v=0$ on $\mathrm{\Omega }(M)$ for every module $M$. In particular, [F4] makes $V$ a valid regular module, and creation puts its vacuum in $\mathrm{\Omega }(V)$; applying the polynomial identity to that vacuum gives $P(0)=0$, so $E$ is nonempty. The relation and $P$ depend only on $V$.

Apply step 4.1 to the $i$th module action on the vector $q$ from step 3.2. Then $P(L_i(0))q=0$. Since $q$ lies in the joint generalized-weight space of $\lambda$, this forces $P({\lambda }_i)=0$: if $P({\lambda }_i)\mathrm{\ne }0$, the operator $P(L_i(0))$ is invertible on the finite-dimensional cyclic space of $q$, because it is a nonzero scalar plus a nilpotent operator. Hence every nonzero quotient of $W$ has a nonzero joint generalized-weight space at some $\lambda \in E^k$. This conclusion uses generalized eigenvectors and does not assume their nilpotent parts vanish.

The space $G(W)$ is finite dimensional because $E^k$ is finite and each of its summands is finite dimensional. Let $U$ be the submodule generated by it. The quotient $Q=W\mathrm{/}U$ has zero joint generalized-weight spaces at every $\lambda \in E^k$, by step 2.2. If $Q$ were nonzero, step 5.1 would contradict that fact. Thus $U=W$, and a finite basis of $G(W)$ is a finite generating set.

By the mode-shift rule, generation in step 6.1 puts every weight of $W$ in $E^k+{\mathbb{Z}}^k$. In each of the finitely many such cosets, coordinatewise lower bounds and coordinatewise upper bounds leave only finitely many lattice points. A bound on the sum of real parts also leaves finitely many points: the coordinatewise lower bounds convert it to an upper bound on each coordinate separately. The corresponding spaces are therefore finite sums of finite-dimensional joint generalized-weight spaces.

For each of the finitely many occurring cosets in step 7.1, choose a representative $\alpha$. For each coordinate let $d_i$ be the minimum integer $n_i$ among its occurring weights $\alpha +n$, and set $\beta =\alpha +(d_1,\dots ,d_k)$. These minima exist by the coordinatewise lower bounds; $\beta$ itself need not occur. The occurring weights of this coset lie in $\beta +{\mathbb{N}}^k$. Sum $W_{[\beta +n]}$ over the distinct cosets to define $W(n)$. These are finite sums of finite-dimensional spaces, and the weight-shift rule gives exactly the stated mode-degree rule. This is an admissible multigrading; its semisimple degree operators are auxiliary and do not replace the possibly nonsemisimple conformal operators $L_i(0)$. This proves the remaining assertion.