Irreducible extremal cyclic Virasoro modules in boundary Fock spaces

The oscillator monomials ${\prod }_m{x}_m^{k_m}{e}^{t\alpha }$, where $k_m\in {\mathbb{Z}}_{\ge 0}$ is finitely supported, have degree $N=\sum m{k}_m$ and are indexed by partitions of $N$, so $\dim ({\mathcal{F}}_t{)}_N=P(N)$. Substitution in the current vacuum-weight formula [F2] gives $h(\beta -n)=a_n$ and $h(\delta -n)=b_n$. The charges are integral for the screening hypotheses: $2p(\beta -n)=i-2pn$ and $2p(\delta -n)=2p-2-i-2pn$ are integers. Thus each composition used in [F4] is defined under [F2].

Fix $n\ge 0$, let $t=\beta -n$, so $t^{\ast }=\delta +n$, and set $w=Q^{2n+1}{e}^{(\delta -(n+1))\alpha }$. By [F4] this vector is nonzero in ${\mathcal{F}}_{\delta +n}$; by [F2] it is singular. Screening commutation preserves its weight, which is $b_{n+1}$, while the target vacuum in ${\mathcal{F}}_{\delta +n}$ has weight $a_n$. Its relative oscillator level is $g=b_{n+1}-a_n=(i+1)(2n+1)>0.$

The highest-vector property defines a Verma map $A:M(c,a_n)\to {\mathcal{F}}_t$ sending its top to $e^{t\alpha }$; its image is the stated cyclic module. For a level-$g$ PBW word $X$, the adjoint identity in [F3] gives $⟨AX{v}_{a_n},w⟩=⟨e^{t\alpha },X^{\ast }w⟩=0$, because $X^{\ast }$ is a nonempty product of positive modes and each positive mode kills the singular vector $w$. Nondegeneracy of the pairing makes $u\mapsto ⟨u,w⟩$ a nonzero functional on $({\mathcal{F}}_t{)}_g$, so the level-$g$ image is proper. Its domain and target both have dimension $P(g)$ by [F1] and [F6]; hence $\ker A$ has a nonzero vector at level $g$.

The kernel of $A$ is a proper graded submodule because its highest vector maps to a nonzero vacuum. By [F5], it lies in the maximal proper submodule $J$ of $M(c,a_n)$, whose first nonzero level is $g$ and whose grade-$g$ space is one-dimensional. Step 3.1 then implies $\ker A$ contains that first singular vector; since this vector generates $J$, while $\ker A\subseteq J$, equality follows. Thus the actual cyclic image is $L(c,a_n)$, with the first character in the Statement by [F5].

For $n\ge 1$, put $t=\delta -n$, so $t^{\ast }=\beta +n$. The vector $w=Q^{2n}{e}^{(\beta -n)\alpha }\in {\mathcal{F}}_{\beta +n}$ is nonzero by [F4], singular by [F2], and has weight $a_n$; the target vacuum weight is $b_n$, so its positive relative level is $g=a_n-b_n=2nd$. The same adjoint calculation as in step 3.1 makes the level-$g$ Verma-map image proper. Equal level dimensions [F1, F6] give a nonzero kernel vector there. By [F5], the kernel is contained in the first maximal submodule, meets its one-dimensional first grade, and therefore equals it. This proves the second cyclic irreducibility and character formula.