0.0.1 Introduction↩︎

The study of logarithmic conformal field theory (LCFT) has become increasingly important in recent years. However, LCFT is much more complicated than rational cases and has not been well studied. In 2010, Feigin–Tipunin [1] proposed a geometric construction of LCFTs, known as the Feigin–Tipunin (FT) construction, defined as the \(0\)-th sheaf cohomology \(H^0(G\times_BV)\) of \(G\)-equivariant VOA bundle \(G\times_BV\) over the flag variety \(G/B\). They further claimed that multiplet W-algebras¹ can be constructed and studied using their method.

The first author [3], [4] provided rigorous proofs of several claims made in [1].

In [5], the authors investigated the \(V^{(p)}\)-algebra (\(=\) doublet affine \(\mathfrak{sl}_2\)) [6], [7] by combining the method in [3], [4] with the inverse quantum Hamiltonian reduction [8]. On the other hand, a close examination of [3], [4] suggests that, although formulated in the framework of VOAs, the essential arguments ultimately reduce to properties of an underlying Lie algebraic setting. As detailed below, extracting such a Lie algebraic setting and argument not only provides a clear perspective for the study of LCFTs, but also sheds light [9], [10] on the expected correspondence between LCFTs and \(3\)-manifolds via the \(\hat{Z}\)-invariants [11]–[14].

In Part 1, we introduce such a purely Lie algebraic setting, tentatively named shift system, and prove that the main results of [3] can be understood and derived from the setting (Theorem 1). In other words, if a VOA-module \(V\) (e.g. irreducible lattice VOA-module) fits into a shift system, the corresponding module over the multiplet W-algebra satisfies Theorem 1. In particular, when it admits the FT construction \(H^0(G\times_BV)\), it can be studied using methods from geometric representation theory.

As demonstrated in Section 2.0.1.4, the verification of the axiom of shift system is basically reduced to a relatively easy computation (\(=\) Serre relation of long screening operators) on \(V\) and the rank \(1\) case (\(=\) “Felder complex" of short screening operators). Hence, the shift system provides a useful framework for reducing the study of higher rank LCFTs to much easier cases: the rank \(1\) cases and free field algebras. Moreover, with some information about the corresponding”W-algebra" \(H^0(G\times_BV)^G\) (e.g. Kazhdan–Lusztig decomposition), the main results of [4] are proved (Section 2.0.1.2). Using these results, in Part 2, we will give the shift system and the FT constructions corresponding to the (multiplet) principal W-(super)algebras \(\mathbf{W}^k(\mathfrak{g})\) for any simple Lie algebra \(\mathfrak{g}\) and Lie superalgebra \(\mathfrak{g}=\mathfrak{osp}(1|2n)\), respectively, and prove almost all the main results of [3], [4] for these new cases (Theorem 2). At first glance, these results themselves may appear to be just a straightforward extension of [3], [4]. However, as mentioned briefly at the end of the last paragraph, the value of this paper lies in a change of perspective, which greatly increases the transparency and flexibility of the discussion in LCFTs, by focusing not on individual examples of LCFT and their complicated VOA-structures but on a simple Lie algebraic pattern behind them in common. In that sense, the application to the two examples in Part 2 is a demonstration of this new perspective which will itself be further developed and applied to a much broader class of expected LCFTs corresponding to \(3\)-manifolds in the future (see the last of introduction below).

Let us describe the overview of this paper. For a finite dimensional simple Lie algebra \(\mathfrak{g}\) of rank \(r\), we consider a triple \((\Lambda,\uparrow,\{V_\lambda\}_{\lambda\in\Lambda})\), named shift system (see Definition 1 for the full axioms). Here \(\Lambda\) has the action \(\ast\colon W\times\Lambda\rightarrow\Lambda\) of the Weyl group \(W\) of \(\mathfrak{g}\),

\(\uparrow\colon W\times\Lambda\rightarrow\mathfrak{h}^\ast\) is a map (shift map) satisfying a few simple axioms, and \(V_\lambda\) is a graded weight \(B\)-module with a parameter \(\lambda\in\Lambda\) and \(B\)-module homomorphisms called short screening operators \(Q_{i,\lambda}\colon V_\lambda\rightarrow V_{\sigma_i\ast\lambda}(\sigma_i\uparrow\lambda)\) satisfying the “Felder complex" [15] for each direction \(i\in I:=\{1,\dots,r\}\), where \((\sigma_i\uparrow\lambda)\) means the shift of Cartan weight. Recall that in [1], [3], [4], \(V_\lambda\) is an irreducible module \(V_{\sqrt{p}(Q+\lambda)}\) over the lattice VOA \(V_{\sqrt{p}Q}\) with a conformal grading \(V_\lambda=\bigoplus_{\Delta}V_{\lambda,\Delta}\) and a parameter \(\lambda\in\Lambda \simeq (\sqrt{p}Q)^\ast/\sqrt{p}Q \simeq \tfrac{1}{p}Q^\ast/Q\), and the long/short screening operators define \(B\)- and \(W\)-actions on \(V_\lambda\) and \(\Lambda\), respectively. The shift of Cartan weights by the short screening operators satisfies the axioms of the shift map above. The situation in [5] is a little more complicated, but essentially the same. Therefore, the shift system can be regarded as an abstraction of the relationship between such VOA-modules and screening operators acting on them, forgetting the VOA-structure, and the shift map \(\uparrow\colon W\times\Lambda\rightarrow P\) records how the Cartan weight is shifted when short screening operators are applied along a Weyl group element. The first main theorem of Part 1 asserts that all the main results of [3] hold even under such abstraction.

Note that the weak/strong conditions in Theorem 1 are special cases of Definition 1[length32and32carry-over]. That is, these conditions are constraints on the shifts of Cartan weights.

Since the constructions and homomorphisms in Theorem 1 are natural, they are compatible with additional algebraic (e.g, VOA-module) structures. In particular, if \(V_0\) is a VOA and \(V_\lambda\) are \(V_0\)-modules, then \(H^0(G\times_BV_0)\) has the induced VOA-structure and \(H^n(G\times_BV_\lambda)\) are \(H^0(G\times_BV_0)\)-modules (see [4]). It allows us to develop a geometric representation theory of multiplet W-(super)algebras, traditionally defined in the form of the right-hand side of Theorem 1[main32theorem32140141]. Except for rank 1 cases, these VOAs are difficult to study algebraically due to their complicated VOA structures. However, the geometric construction allows us to avoid the bottleneck and obtain a variety of results qualitatively, as in [3]–[5].

In Part 2, we demonstrate the usefulness of the shift system and Theorem 1 by applying them to specific VOSAs. Let us consider the rescaled root lattice \(\sqrt{p}Q\) for some \(p\in\mathbb{Z}_{\geq 1}\). We consider the case where \(\sqrt{p}Q\) is positive-definite integral, namely, \(p\in r^\vee\mathbb{Z}_{\geq 1}\) for the lacing number \(r^\vee\) or \(p\) is odd and \(\mathfrak{g}=B_r\). In the first case, \(\sqrt{p}Q\) is even and we consider \(V_\lambda=V_{\sqrt{p}(Q+\lambda)}\). On the other hand, in the second case, \(\sqrt{p}Q\) is odd and we need a modification \(V_\lambda=V_{\sqrt{p}(Q+\lambda)}\otimes F\) by the free fermion \(F\). In Section 2.0.1.4, from minimal natural assumptions 32 or 33 , shift systems \((\Lambda,\uparrow,\{V_\lambda\}_{\lambda\in\Lambda})\) are uniquely constructed (Theorem 5). As noted in the second paragraph above, only relatively easy computations on \(V_\lambda\) and some basic results in rank \(1\) cases ([2] and [16], [17]) are used here. Therefore, Theorem 1 is applied to our cases (for more detail, see Section 2.0.2).

Let us explain the two consequences of Theorem 1[main32theorem32140241], namely, the Weyl-type character formula and the simplicity theorem (see Section 1.0.0.5 and 2.0.1.2). The natural \(G\)-action on \(H^0(G\times_BV_\lambda)\) gives the decomposition \[\begin{align} \label{the32decomposition} H^0(G\times_BV_\lambda) \simeq \bigoplus_{\beta\in P_+} L_\beta\otimes\mathcal{W}_{-\beta+\lambda}, \end{align}\tag{1}\] where \(L_\beta\) is the irreducible \(\mathfrak{g}\)-module with highest weight \(\beta\), and \(\mathcal{W}_{-\beta+\lambda}\) is the multiplicity of a weight vector of \(L_\beta\). In Part 2, we consider additional VOA structures, where \(\mathcal{W}_0\simeq H^0(G\times_BV_0)^G\) is a subalgebra of the multiplet W-(super)algebra \(H^0(G\times_BV_0)\) and \(\mathcal{W}_{-\beta+\lambda}\) is a \(\mathcal{W}_0\)-modules. If \(\lambda\) satisfies ?? , then by combining \(H^{n>0}(G\times_BV_\lambda)\simeq 0\) with the Atiyah–Bott localization formula [18], we obtain the Weyl-type character formula \[\begin{align} \label{character32formula32in32introduction} \operatorname{ch}_q H^0(G\times_BV_\lambda) &= \sum_{\beta\in P_+}\operatorname{dim}L_{\beta} \operatorname{ch}_q\mathcal{W}_{-\beta+\lambda}\\ &= \sum_{\beta\in P_+}\operatorname{dim}L_{\beta} \sum_{\sigma\in W}(-1)^{l(\sigma)}\operatorname{ch}_q V_{\sigma\ast\lambda}^{h=\beta-\sigma\uparrow\lambda}, \end{align}\tag{2}\] where \(V_\mu^{h=\gamma}\) is the subspace of \(V_\mu\) at the Cartan weight \(h=\gamma\). On the other hand, by combining the case \(n=0\) with Serre duality, we have \(H^0(G\times_BV_\lambda)\simeq H^0(G\times_BV_{\lambda'})^\ast\) for some \(\lambda'\in\Lambda\). In the VOA side, it leads to the self-duality and simplicity of the vacuum case \(H^0(G\times_BV_0)\). By the quantum Galois theory [19], [20], \(\mathcal{W}_{-\beta}\) are also simple as \(\mathcal{W}_0\)-modules. In addition, if a Kazhdan–Lusztig-type character formula for the simple quotient of \(\mathcal{W}_{-\beta+\lambda}\) is known, by comparing it with 2 , we can extend the simplicity to the whole \(\lambda\in\Lambda\) satisfying ?? . For more detail, see Section 2.0.1.2.

Let us go back to our special cases above. In our cases, \(\mathcal{W}_0\) is the principal W-(super)algebra \(\mathbf{W}^k(\mathfrak{g})\) or \(\mathbf{W}^k(\mathfrak{osp}(1|2r))\), and thus we have the Kazhdan–Lusztig type character formula derived from [21], [22] and the exactness of \(+\)-reduction \(H^0_{\mathrm{DS},+}(\cdot)\) in [23] (however, in the second case we assume the latter). By applying the discussion in the last paragraph, we can extend the main results of [24] to our cases (Theorem 7 and 8).

The decomposition 1 with Theorem 2 is regarded as the Schur-Weyl type duality for the multiplet principal W-algebras. In the non-simply laced case, in light of [20], the tensor category of \(\mathbf{W}^k(\mathfrak{g})\)-modules is expected to exhibit a symmetry governed by the Langlands dual Lie algebra \({}^L\mathfrak{g}\).

Finally, let us discuss two directions to which Theorem 1 points. Our geometric representation theory of FT construction has similar aspects to the modular representation theory. Indeed, in our special cases, the strong condition in Lemma 12[lemm:32strong32condition32for32non-super32case] also appears in the BWB theorem for \({}^LG\) [25]. Ultimately, such a similarity should come from the log Kazhdan–Lusztig correspondence (e.g. [1], [26]–[30]): an expected categorical equivalence between a multiplet \(W\)-algebra and a small quantum group. For example, in our case, \(H^0(G\times_BV_{\sqrt{p}Q})\) is expected to correspond to the small quantum group \(u_{\zeta}({}^L\mathfrak{g})\) at \(\zeta=\frac{\pi i}{p}\).³ Although the case \(\mathfrak{g}=\mathfrak{sl}_2\) was already treated in [29], it seems necessary to develop both our geometric methods and W-algebraic approaches for higher rank generalizations (see Remark 10).

On the other hand, under the correspondence of the BWB-type theorems above, Theorem 1[main32theorem32140141] clearly does not hold in the modular representation theory [25]. In this regard, the first author conceived the following idea a few years ago: Let \(\vec{\lambda}=(\lambda_1,\dots,\lambda_N)\in\Lambda_1\times\cdots\times\Lambda_N\) be a parameter sequence and \(\mathbb{V}_{\vec{\lambda}}\) be a \(G\)-module.

We consider the situation such that for each \(0\leq n\leq N-1\), an appropriate quotient \[\begin{align} \underbrace{\tilde{H}^0(G\times_B\tilde{H}^0(G\times_B\cdots\tilde{H}^0(G\times_B}_{\text{n-times}}\tilde{\mathbb{V}}_{\vec{\lambda}})\cdots)) \end{align}\] of \[\begin{align} \underbrace{{H}^0(G\times_B\tilde{H}^0(G\times_B\cdots\tilde{H}^0(G\times_B}_{\text{n-times}}\tilde{\mathbb{V}}_{\vec{\lambda}})\cdots)) \end{align}\] is a shift system with respect to \(\lambda_{n+1}\in\Lambda_{n+1}\) (note that when \(n=0\), the quotient \(\tilde{\mathbb{V}}_{\vec{\lambda}}\) of \(\mathbb{V}_{\vec{\lambda}}\) is a shift system with respect to \(\lambda_1\in\Lambda_{1}\). In the case of the triplet Virasoro algebra at level \(p\in\mathbb{Z}_{\geq 2}\), \(\mathbb{V}_\lambda\) is a projective cover of its irreducible module and \(\tilde{\mathbb{V}}_{\lambda}\) is an irreducible lattice VOA-module, respectively). The reason we must consider the quotient \(\tilde{H}^0(G\times_B\cdots)\) is that the functor \(H^0(G\times_B-)\) does not change a \(G\)-module; in other words, to apply the \(n\)-th FT construction, we must break the \(G\)-module structure of the \((n-1)\)-th FT construction to make a shift system (\(B\)-module structure) again. Then we finally obtain the nested FT constructions \[\begin{align} \underbrace{H^0(G\times_B\tilde{H}^0(G\times_B\cdots\tilde{H}^0(G\times_B}_{\text{N-times}}\tilde{\mathbb{V}}_{\vec{\lambda}})\cdots)). \end{align}\] Because a shift system appears at each stage of nesting, Theorem 1 can be applied repeatedly. In particular, for \(\mathfrak{g}=\mathfrak{sl}_2\), the \(q\)-series obtained by iteratively applying 2 along the nested FT construction surprisingly almost coincides with the \(\hat{Z}\)-invariant [14] of the corresponding negative definite Seifert \(3\)-manifold, after restricting to the \(h=0\) sector (conjecturally singlet-type VOA). This suggests that such nested FT constructions yield rich examples of LCFT whose representation theories are to some extent reducible and controllable by the geometric representation theory of FT construction, providing a blueprint for the “dictionary" between LCFTs and negative definite plumbed \(3\)-manifolds expected in [11]–[13], etc.

In [9], [10], for \(\mathfrak{g}=\mathfrak{sl}_2\) and any negative definite plumbed 3-manifold \(Y\), the first author realized the nested FT constructions on an abstract abelian category \(\mathcal{C}^Y\). Here, \(\mathcal{C}^Y\) is intended to be a module category of the conjectural “Virasoro VOA”, whose existence is predicted based on the “dictionary” mentioned above. Indeed, the \(\hat{Z}\)-invariants of \(Y\) are reconstructed in the Grothendieck group \([\mathcal{C}^Y]\) of \(\mathcal{C}^Y\) up to formal substitution of certain lattice theta functions \(\Theta^Y_{\vec{\lambda}}(q)\) into the representatives \([V_{\vec{\lambda}}]\in[\mathcal{C}^Y]\) of the corresponding indecomposable objects \(V_{\vec{\lambda}}\in\mathcal{C}^Y\), which are intended to be the irreducible modules of the conjectural “lattice VOA” whose characters are \(\Theta^Y_{\vec{\lambda}}(q)\). The Lie algebraic framework developed here — formulating the FT construction without presupposing VOA structure — makes it possible to study such conjectural VOAs before their full VOA-theoretic description is available.

Acknowledgments: S.S. thanks Naoki Genra and Hiroshi Yamauchi for useful comments on naming. H.L. thanks Antun Milas for motivating questions, and to Drazen Adamovic and Naoki Genra for useful discussions. The authors thank Tomoyuki Arakawa for useful feedback. The authors also thank anonymous referee for useful suggestions. The majority of this work was done when the authors were postdocs in YMSC, Tsinghua University.

1 Shift system↩︎

In Part 1, we describe the discussion of [1], [3], [4] in an axiomatic (in other words, purely Lie Algebraic) manner. We also generalize the discussion in these papers to include non-simply laced cases.

1.0.0.1 Preliminaries on Lie Algebras

We present the basic notations and facts used throughout this paper. Let \(\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}\) be a finite dimensional simple Lie algebra of rank \(r\) and lacing number \(r^\vee\), and its triangular decomposition. Let \(\Delta=\Delta_+\sqcup\Delta_{-}\) be the root system of \(\mathfrak{g}\) (resp. positive roots and negative roots). For the Weyl group \(W\) of \(\mathfrak{g}\), we consider the canonically normalized \(W\)-invariant bilinear form \((\cdot,\cdot)=(\cdot,\cdot)_Q\), namely, for a long root \(\alpha\in\Delta^l\), the length \(|\alpha|^2\) is always \(2\) (denote \(\Delta^s\) the short roots). For a root \(\alpha\in\Delta\), the coroot \(\alpha^\vee\) is defined by \(\tfrac{2}{|\alpha|^2}\alpha\). For \(1\leq i\leq r\), \(\alpha_i\), \(\alpha_i^\vee\), \(\alpha_i^\ast\), \(\varpi_i\) and \(\sigma_i\in W\) are the simple root, simple coroot, fundamental coweight (dual of \(\alpha_i\)), fundamental weight (dual of \(\alpha_i^\vee\)), and the simple reflection \(\sigma_i(\mu)=\mu-(\mu,\alpha_i^\vee)\alpha_i\) corresponding to \(\alpha_i\), respectively. In this paper, the labeling of the Dynkin diagrams of \(\mathfrak{g}\) is given by \[\begin{align} & \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/uaortlch.png}\tag{3}\end{figure} \;\; \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/czpxbwhk.png}\tag{4}\end{figure} \;\; \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/rywelhqs.png}\tag{5}\end{figure} \;\; \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/uqeslghr.png}\tag{6}\end{figure} \;\; \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/atixpsgl.png}\tag{7}\end{figure} \\ &\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/mptkrgnu.png}\tag{8}\end{figure} \;\; \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/kiwaombg.png}\tag{9}\end{figure} \;\; \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/bqlokdvy.png}\tag{10}\end{figure} \;\; \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fpmqxjyg.png}\tag{11}\end{figure} \end{align}\] respectively. We sometimes identify the set of simple roots \(\Pi=\{\alpha_1,\dots,\alpha_r\}\) with the set \(I=\{1,\dots,r\}\). The fundamental weight \(\varpi_i\) is the dual vector to the coroot. Denote \(Q=\sum_{i\in I}\mathbb{Z}\alpha_i\), \(Q^\vee=\sum_{i\in I}\mathbb{Z}\alpha_i^\vee\), \(Q^\ast=\check{P}=\sum_{i\in I}\mathbb{Z}\alpha_i^\ast\) and \(P=\sum_{i\in I}\mathbb{Z}\varpi_i\) the root lattice, coroot lattice, coweight lattice and weight lattice, respectively. The set of dominant integral weights \(P_+\) and dominant integral coweights \(\check{P}_+=Q^\ast_+\) are given by \(P_+=\sum_{i\in I}\mathbb{Z}_{\geq 0}\varpi_i\), \(\check{P}=\sum_{i\in I}\mathbb{Z}_{\geq 0}\alpha_i^\ast\), respectively. We use the letter \(P_{\mathrm{min}}\subseteq P_+\) for the family of minuscule weights. Let \(G\) be the corresponding simply-connected simple algebraic group with \(B\) the Borel subgroup. Given a subset \(J\subset I\), we denote by \(P_J\) the corresponding parabolic subgroup⁴ and \(j\in J\), by \(SL_2^j\) the corresponding subgroup isomorphic to \(SL_2\). In the case \(J=\{j\}\), we denote \(P_J\) as \(P_j\). Denote \(\rho=\sum_{i\in I}\varpi_i\) and \(\rho^\vee=\sum_{i\in I}\alpha_i^\ast\) the Weyl vector and Weyl covector, \(\theta\) and \(\theta_s\) the highest (long) root and highest short root, \(h\) and \(h^\vee\) the Coxeter and dual Coxeter number, respectively. For \(\sigma\in W\), \(l(\sigma)\) and \(w_0\) denote the length of \(\sigma\) and the longest element in \(W\), respectively. For a minimal expression \(\sigma=\sigma_{i_n}\cdots\sigma_{i_1}\) of \(\sigma\in W\), we sometimes use \(\sigma_{i_0}=\operatorname{id}\) for convenience. For \(\beta\in P_+\), denote \(L_\beta\) the finite-dimensional irreducible \(\mathfrak{g}\)-module with the highest weight \(\beta\). We sometimes use the letters \(y_\beta\) and \(y'_{\beta}\) for a highest-weight vector and lowest-weight vector of \(L_\beta\), respectively. The Langlands dual \({}^L\mathfrak{g}\) is defined by replacing the root system by the coroot system. Namely, \[\begin{align} &(Q_{{}^L\mathfrak{g}},(\cdot,\cdot)_{{}^LQ}) = (Q^\vee,\tfrac{1}{r^\vee}(\cdot,\cdot)_Q),\\ &\alpha_{i,{}^L\mathfrak{g}}=\alpha_i^\vee,\;\; \alpha_{i,{}^L\mathfrak{g}}^\vee=r^\vee\alpha_{i},\;\; \alpha_{i,{}^L\mathfrak{g}}^\ast=r^\vee\varpi_i,\;\; \varpi_{i,{}^L\mathfrak{g}}=\alpha_i^\ast \end{align}\] (in particular, \(\rho_{{}^L\mathfrak{g}}=\rho^\vee\) and \(\rho_{{}^L\mathfrak{g}}^\vee=r^\vee\rho\)). If we want to emphasize that a certain object \(X\) (e.g., \(\mathfrak{g}\), \(Q\),...) is derived from \({}^L\mathfrak{g}\) rather than \(\mathfrak{g}\), we often use symbols such as \({}^LX\), \(X_{{}^L\mathfrak{g}}\), \(\check{X}\), etc. By abuse of notation, for \(\theta=\sum_{i\in I}a_i\alpha_i\), denote \({}^L\theta=\sum_{i\in I}a_i\alpha_{r+1-i}^\vee\in\mathfrak{h}^\ast\) (i.e., we regard \({}^L\theta_{\mathfrak{g}}=\theta_{{}^L\mathfrak{g}}\) as an element in \(\mathfrak{h}^\ast\)). For more detailed data in each case, see e.g. [31] (but note that the labeling of Dynkin diagrams is different).

For a weight \(B\)-module \(M\) and \(\beta\in\mathfrak{h}^\ast\), denote \(M^{h=\beta}\) the weight space with Cartan weight \(\beta\). For \(\mu\in\mathfrak{h}^\ast\), let \(\mathbb{C}_\mu\) be the one-dimensional \(B\)-module such that \(\mathfrak{n}_{-}\)-action is trivial. For a \(B\)-module \(V\) and \(\mu\in\mathfrak{h}^\ast\), we write \(V(\mu)\) for the \(B\)-module \(V\otimes_\mathbb{C}\mathbb{C}_\mu\). We use the same notation for a vector bundle and its sheaf of sections. For a sheaf \(\mathcal{F}\) over a topological space \(X\) and an open subset \(U\) of \(X\), \(\mathcal{F}(U)\) denotes the space of sections of \(\mathcal{F}(U)\) on \(U\). For an algebraic variety \(X\), let \(\mathcal{O}_X\) be the structure sheaf of \(X\). For \(\mu\in\mathfrak{h}^\ast\), we write \(\mathcal{O}(\mu)\) for the line bundle (or invertible sheaf) \(G\times_B\mathbb{C}_\mu\) over the flag variety \(G/B\). In particular, \(\mathcal{O}(0)\) is \(\mathcal{O}_{G/B}\). For an \(\mathcal{O}_{G/B}\)-module \(\mathcal{F}\), we use the letter \(\mathcal{F}(\mu)\) for \(\mathcal{F}\otimes_{\mathcal{O}_{G/B}}\mathcal{O}(\mu)\). In particular, for a \(B\)-module \(V\) and a homogeneous vector bundle \(G\times_BV\), we have \((G\times_BV)(\mu)=G\times_BV(\mu)\). For \(n\in\mathbb{Z}\), the \(n\)-th sheaf cohomology of \(\mathcal{F}\) is denoted by \(H^n(\mathcal{F})\).

For later convenience, we will list several facts.

The isomorphism of varieties \(P_j/B\simeq SL_2^j/B^j(\simeq \mathbf{P}^1)\) actually extends to an isomorphism of equivariant bundles \(P_j\times_B M\simeq SL_2^j\times_{B^j}M\) over \(\mathbf{P}^1\) and thus induces an isomorphism of the global sections \[\begin{align} H^0(P_j\times_B M)\simeq H^0(SL_2^j\times_{B^j}M) \overset{\text{Lemma \ref{Sug321143211532lemma324465}}}{\simeq} M. \end{align}\] Since the left-hand side implies that it is naturally a \(P_j\)-module, we obtain the assertion.

It suffices to show \(\mathrm{ad}_{e_i}^{1-c_{ij}}(e_j)=0\) for \(i\neq j\) on \(\bigcap_{j\in J}M_j\). Since \([e_i,f_j]=0\), we have \[\begin{align} [\mathrm{ad}_{e_i}^{1-c_{ij}}(e_j),f_i]=\sum_{m=0}^{-c_{ij}}\mathrm{ad}_{e_i}^{-c_{ij}-m}\mathrm{ad}_{h_i}\mathrm{ad}_{e_i}^m(e_j)=\sum_{m=0}^{-c_{ij}}h_i(m \alpha_i-\alpha_j)\mathrm{ad}_{e_i}^{-c_{ij}}(e_j)=0 \end{align}\] Then, it suffices to show \(\mathrm{ad}_{e_i}^{1-c_{ij}}(e_j)y^i_\alpha=0\) for any highest weight vector \(y^i_\alpha\) of \(\bigcap_{j\in J}M_j\) with respect to the \(SL_2^{i}\)-action (where \(\alpha\in P\) such that \((\alpha,\alpha_i^\vee)\geq 0\)). Since \(h_i((1-c_{ij})\alpha_i+\alpha_j+\alpha)=2-c_{ij}+h_i(\alpha)\), it suffices to show \(f_i^{2-c_{ij}+h_i(\alpha)}\mathrm{ad}_{e_i}^{1-c_{ij}}(e_j)y^i_\alpha=0\). Note that \(f_i^{2-c_{ij}+h_i(\alpha)}e_i^{1-c_{ij}}=Xf_i^{h_i(\alpha)+1}\) for some \(X\in U(\mathfrak{sl}_2^i)\) by weight consideration. Then \[\begin{align} f_i^{2-c_{ij}+h_i(\alpha)}\mathrm{ad}_{e_i}^{1-c_{ij}}(e_j)y^i_\alpha =f_i^{2-c_{ij}+h_i(\alpha)}e_i^{1-c_{ij}}e_jy^i_\alpha =Xf_i^{s+1}e_jy^i_\alpha =X e_j f_i^{s+1}y^i_\alpha=0. \end{align}\] This completes the proof.

1.0.0.2 Shift system

In this subsection, we introduce a new concept, named shift system,

which is the main tool in the present paper, and gives its basic examples and properties. The name “shift system” was chosen because, as seen in Theorem 1, the shift \(\sigma\uparrow\lambda\) plays a central role in our representation theory.

As mentioned in Section 0.0.1, the motivation for introducing the shift system lies in the study of logarithmic CFTs/VOAs, particularly multiplet W-algebras (Definition 7). In the case of \(\mathfrak{g}=\mathfrak{sl}_2\), the short exact sequence in Lemma 15 formed by the irreducible modules over the multiplet W-algebra (i.e., triplet Virasoro algebra) and the lattice VOA is called the Felder complex [15]. The shift system can be viewed as a purely combinatorial and Lie algebraic formulation of this Felder complex, as well as a generalization to arbitrary \(\mathfrak{g}\).

The following lemma is shown in exactly the same manner as [3], so the proof is omitted.

1.0.0.3 Feigin–Tipunin conjecture/construction

Let \((\Lambda,\uparrow,\{V_\lambda\}_{\lambda\in\Lambda})\) be a shift system. In this subsection, we will give a brief necessary and sufficient condition for the Feigin-Tipunin conjecture [1], [3] \[\begin{align} H^0(G\times_BV_\lambda)\simeq\bigcap_{i\in I}W_{i,\lambda} \subseteq V_\lambda \end{align}\] to hold. For \((i,j,\lambda)\in I\times I\times \Lambda\), we consider the following condition (see [3]): \[\begin{align} \label{Sug-98} \text{\lambda\in\Lambda^{\sigma_j} or ({\sigma_j}\uparrow{\lambda},\alpha_i^\vee)=-\delta_{ij}}. \end{align}\tag{13}\] For a subset \(J\subseteq I\), we also consider the following condition: \[\begin{align} \label{Sug-99} \text{(i,j,\lambda) satisfies \eqref{Sug-98} for any (i,j)\in J\times J}. \end{align}\tag{14}\] When \(J=I\), the condition 14 is stated as follows (this is the weak condition ?? in Theorem 1[main32theorem32140141]): \[\begin{align} \label{Sug-100} \text{(i,j,\lambda) satisfies \eqref{Sug-98} for any (i,j)\in I\times I.} \end{align}\tag{15}\]

If \(\lambda\in\Lambda^{\sigma_j}\) or \(i=j\), then we have \(W_{i,\lambda}\cap W_{j,\lambda}=W_{i,\lambda}\) and the assertion is clear. Let us assume that \(i\not=j\) and \(({\sigma_j}\uparrow{\lambda},\alpha_i^\vee)=0\). It suffices to show that if \(f_i^nA_\beta\in W_{i,\lambda}\cap W_{j,\lambda}\) for some \(A_\beta\in(W_{i,\lambda})_{\beta}^{e_i}\), \((\beta,\alpha_i^\vee)>0\), and \(0<n\leq(\beta,\alpha_i^\vee)\), then \(A_\beta\) (and thus \(f_i^{n-1}A_\beta\)) is also in \(W_{i,\lambda}\cap W_{j,\lambda}\). Since \(f_i^{(\beta,\alpha_i^\vee)}A_\beta\in W_{j,\lambda}\), we have \[\begin{align} f_{i}^{(\beta,\alpha_i^\vee)}Q_{j,\lambda}A_{\beta} =Q_{j,\lambda}f_{i}^{(\beta,\alpha_i^\vee)}A_{\beta}=0, \end{align}\] By the assumption \(({\sigma_j}\uparrow{\lambda},\alpha_i^\vee)=0\) and Remark 2[Sug3234614-34616], we have \(Q_{j,\lambda}A_{\beta}=0\), that is, \(A_\beta\in W_{j,\lambda}\). The last assertion follows from Lemma 2 and Lemma 3.

Let us assume that \((J,\lambda)\) does not satisfy 14 . Then there exists a pair \((j,i)\in J\times J\) such that \(i\not=j\), \(\sigma_i\uparrow\lambda\not=-\alpha_i\) and \((\sigma_i\uparrow\lambda,\alpha_j^\vee)>0\). For \(\beta\in P\) such that \((\beta,\alpha_k^\vee)\geq 0\) for \(k\in J\), we take nonzero vectors \[\begin{align} x\in\bigcap_{k\in J} (W_{k,\sigma\ast\lambda}^{h=\beta})^{e_k},\;\; y=f_i^{(\beta,\alpha_i^\vee)}x,\;\; z=f_j^{\sigma_i(\beta),\alpha_j}y. \end{align}\] Clearly, we have \(f_iy=0\). Furthermore, since \(y\in (W_{j,\sigma_i\ast\lambda}^{h=\sigma_i(\beta)})^{e_j}\), we have \(f_jz=0\). Denote \(w\in V_\lambda^{h=\beta+\sigma_i\uparrow\lambda}\) a preimage of \(x\), namely, \(Q_{i,\lambda}w=x\). Then \[\begin{align} \label{assumptionX} 0 \not= f_i^{(\beta,\alpha_i^\vee)}w \in\bigcap_{i\not=k\in J}(W_{k,\lambda}^{h=\sigma(\beta)+\sigma_i\uparrow\lambda})^{e_k}. \end{align}\tag{16}\] In fact, using the Serre relation \(\mathrm{ad}(f_k)^{1-(\alpha_k^\vee,\alpha_i)}f_i=0\) repeatedly, we have \[\begin{align} f_k^{(\sigma_i(\beta)+\sigma_i\uparrow\lambda,\alpha_k^\vee)+1}f_i^{(\beta,\alpha_i^\vee)}w = f_k^{(\beta+\sigma_i\uparrow\lambda,\alpha_k^\vee)+1}(f_k^{-(\alpha_k^\vee,\alpha_i)(\beta,\alpha_i^\vee)}f_i^{(\beta,\alpha_i^\vee)})w=0. \end{align}\] Let us define the nonzero vector \(z'\) by \[\begin{align} z'=f_j^{(\sigma_i(\beta)+\sigma_i\uparrow\lambda,\alpha_j^\vee)}f_i^{(\beta,\alpha_i^\vee)}w. \end{align}\] Then by the assumption \((\sigma_i\uparrow\lambda,\alpha_j^\vee)>0\), we have \[\begin{align} Q_{i,\lambda}z' =f_j^{(\sigma_i(\beta)+\sigma_i\uparrow\lambda,\alpha_j^\vee)}Q_{i,\lambda}f_i^{(\beta,\alpha_i^\vee)}w =f_j^{(\sigma_i(\beta)+\sigma_i\uparrow\lambda,\alpha_j^\vee)}y =f_j^{(\sigma_i\uparrow\lambda,\alpha_j^\vee)}z=0. \end{align}\] By 16 , we have \(z'\in \bigcap_{k\in J}W_{k,\lambda}\). On the other hand, if \(\bigcap_{k\in J}W_{k,\lambda}\) is closed under \(e_j\), then we have \[\begin{align} f_i^{(\beta,\alpha_i^\vee)}w\in \bigcap_{k\in J}W_{k,\lambda}, \end{align}\] and thus the image \(y\) of \(Q_{i,\lambda}\) is zero. It contradicts the fact \(y\not=0\) above, and thus \(\bigcap_{k\in J}W_{k,\lambda}\) is not closed under \(e_j\), and hence not a \(P_J\)-submodule of \(V_\lambda\).

Let us prove Theorem 1[main32theorem32140141] (see also [3]). By Theorem 3 and Remark 2[maximality32of32W95i], the first half of Theorem 1[main32theorem32140141] is proved. By Lemma 5 and Lemma 6, \(\bigcap_{i\in I}W_{i,\lambda}\) has the \(G\)-module structure (and thus, in \(H^0(G\times_BV_\lambda)\)) if and only if \(\lambda\in\Lambda\) satisfies the weak condition ?? . This completes the proof.

1.0.0.4 Borel–Weil–Bott-type theorem

Let \((\Lambda,\uparrow,\{V\}_{\lambda\in\Lambda})\) be a shift system. For \(\mu\in P\) and \(\lambda\not\in\Lambda^{\sigma_i}\), we have \[\begin{align} \label{the32short32exact32sequence} 0\rightarrow W_{i,\lambda}(\mu) \rightarrow V_{\lambda}(\mu) \rightarrow W_{i,\sigma_i\ast\lambda}(\mu+\sigma_i\uparrow\lambda) \rightarrow 0. \end{align}\tag{17}\]

Let us apply the cohomology functor \(H^\bullet(P_i\times_B\text{-})\) to 17 with \[\begin{align} (\lambda,\mu)\mapsto(\sigma\ast\lambda, \sigma\uparrow\lambda),\quad (\lambda,\mu)\mapsto (\sigma_i\sigma\ast\lambda, (\sigma_i\sigma)\uparrow\lambda) \end{align}\] respectively. Then the assertions follow from Lemma 1 if we have \[\begin{align} H^1(P_i\times_B\mathbb{C}_{\sigma\uparrow\lambda})= H^0(P_i\times_B\mathbb{C}_{\sigma_i\sigma\uparrow\lambda+\sigma_i\uparrow(\sigma_i\sigma\ast\lambda)})=0. \end{align}\] By Lemma 1 again, it suffices to show that \[\begin{align} (\sigma\uparrow\lambda,\alpha_i^\vee)\geq 0,\; (\sigma_i\sigma\uparrow\lambda+\sigma_i\uparrow(\sigma_i\sigma\ast\lambda),\alpha_i^\vee)<0. \end{align}\] They immediately follow from the assumption and axioms in Definition 1.

First we consider the case \(\sigma\ast\lambda\not\in\Lambda^{\sigma_i}\). By the axiom [-132property32of32carry-over] and the assumption \((\sigma\uparrow\lambda,\alpha_i^\vee)=0\), we have \((\sigma\uparrow\lambda+\sigma_i\uparrow(\sigma\ast \lambda),\alpha_i^\vee)=-1\). Therefore, by Lemma 1, we have \[\begin{align} \label{Sug-132} H^n(P_i\times_B \mathbb{C}_{\sigma\uparrow\lambda})\simeq \delta_{n,0}\mathbb{C}_{\sigma\uparrow\lambda},\quad H^n(P_i\times_B \mathbb{C}_{\sigma\uparrow\lambda+\sigma_i\uparrow(\sigma\ast\lambda)})=0 \end{align}\tag{18}\] and by applying 18 to the first short exact sequence in Lemma 7, we have \[\begin{align} \label{Sug-133-1} H^n(P_i\times_B V_{\sigma\ast\lambda}(\sigma\uparrow\lambda)) \simeq \delta_{n,0}W_{i,\sigma\ast\lambda}(\sigma\uparrow\lambda). \end{align}\tag{19}\] By the assumption \(\sigma\ast\lambda\not\in\Lambda^{\sigma_i}\) (and thus, \(\sigma_i\sigma\ast\lambda\not\in\Lambda^{\sigma_i}\)) and axiom [-132property32of32carry-over], we have \[\begin{align} \label{pre-Sug-131} \sigma_i\sigma\uparrow\lambda=\sigma_i\circ(\sigma\uparrow\lambda+\sigma_i\uparrow(\sigma\ast\lambda)), ~ \sigma\uparrow\lambda=\sigma_i\circ(\sigma_i\sigma\uparrow\lambda+\sigma_i\uparrow(\sigma_i\sigma\ast\lambda)). \end{align}\tag{20}\] By applying Lemma 1 to 20 , we obtain that \[\begin{align} \label{pre-Sug-133-2} H^n(P_i\times_B \mathbb{C}_{\sigma_i\sigma\uparrow\lambda})=0, ~H^n(P_i\times_B \mathbb{C}_{\sigma_i\sigma\uparrow\lambda+\sigma_i\uparrow(\sigma_i\sigma\ast\lambda)})\simeq \delta_{n,1} \mathbb{C}_{\sigma\uparrow\lambda}, \end{align}\tag{21}\] where the first one follows from the assumption \((\sigma\uparrow\lambda,\alpha_i^\vee)=0\) and [-132property32of32carry-over], and the second one follows from \((\sigma\uparrow\lambda,\alpha_i^\vee)=0\) and Lemma 1. Hence, by combining the second short exact sequence with 21 , we obtain \[\begin{align} \label{Sug-133-2} H^n(P_i\times_B V_{\sigma_i\sigma\ast \lambda}(\sigma_i\sigma\uparrow\lambda))\simeq \delta_{n,1}W_{i,\sigma\ast\lambda}(\sigma\uparrow\lambda). \end{align}\tag{22}\] By combining 19 and 22 , we obtain \[\begin{align} \label{Sug-133} H^a(P_i\times_B V_{\sigma\ast \lambda}(\sigma\uparrow\lambda))\simeq \delta_{a,0}W_{i,\sigma\ast\lambda}(\sigma\uparrow\lambda) \simeq H^b(P_i\times_B V_{\sigma_i\sigma\ast \lambda}(\sigma_i\sigma\uparrow\lambda)) \end{align}\tag{23}\] for \((a,b)=(0,1)\), \((1,0)\). Now, we have the Leray spectral sequences \[\begin{align} &E_2^{a,b}=H^a(G \times_{P_i}H^b(P_i\times_B V_{\sigma\ast \lambda}(\sigma\uparrow\lambda)))\Rightarrow H^{a+b}(G \times_B V_{\sigma\ast \lambda}(\sigma\uparrow\lambda)),\\ &E_2^{a,b}=H^a(G \times_{P_i}H^b(P_i\times_B V_{\sigma_i\sigma\ast \lambda}(\sigma_i\sigma\uparrow\lambda)))\Rightarrow H^{a+b}(G \times_B V_{\sigma_i\sigma\ast \lambda}(\sigma_i\sigma\uparrow\lambda)), \end{align}\] which differ by the 1-shift for \(b\) by 23 . Thus, we obtain the assertion.

Second, we consider the case where \(\sigma\ast\lambda\in\Lambda^{\sigma_i}\). By the axiom [-132property32of32carry-over], we have \(\sigma_i\sigma\uparrow\lambda=\sigma_i\circ(\sigma\uparrow\lambda)\). By combining Lemma 1 with the assumption that \(V_{\sigma\ast\lambda}\) (and thus, \(V_{\sigma_i\sigma\ast\lambda}\)) is a \(P_i\)-module, we have \[\begin{align} \label{Sug-136} \begin{aligned} H^a(P_i\times_B V_{\sigma\ast \lambda}(\sigma\uparrow\lambda)) &\simeq H^a(P_i\times_B \mathbb{C}_{\sigma\uparrow\lambda})\otimes V_{\sigma\ast \lambda}\\ &\simeq H^b(P_i\times_B \mathbb{C}_{\sigma_i\sigma\uparrow\lambda})\otimes V_{\sigma_i\sigma\ast \lambda}\\ &\simeq H^b(P_i\times_B V_{\sigma_i\sigma\ast \lambda}(\sigma_i\sigma\uparrow\lambda)) \end{aligned} \end{align}\tag{24}\] for \((a,b)=(0,1)\), \((1,0)\). By applying the assumption \((\sigma\uparrow\lambda,\alpha_i^\vee)=0\) and Lemma 1 to 24 , we have \[\begin{align} \label{Sug-138} H^1(P_i\times_B V_{\sigma\ast \lambda}(\sigma\uparrow\lambda)) \simeq H^0(P_i\times_B V_{\sigma_i\sigma\ast \lambda}(\sigma_i\sigma\uparrow\lambda))\simeq 0. \end{align}\tag{25}\] By combining the Leray spectral sequences above with 24 and 25 , we obtain the assertion.

1.0.0.5 Weyl-type character formula

For a graded vector space \(V=\bigoplus_{\Delta}V_\Delta\), \(\dim_\mathbb{C}V_\Delta<\infty\), denote \[\begin{align} \operatorname{ch}_q V=\sum_{\Delta}\dim_\mathbb{C}V_\Delta q^\Delta \end{align}\] the character (where \(q\) and \(\Delta\) can take multiple variables, i.e. \(q=(q_i)_{1\leq i\leq n}\), \(\Delta=(\Delta_i)_{1\leq i\leq n}\), \(q^\Delta=\Pi_{i=1}^nq^{\Delta_i}\)). In the same manner as [3], by combining the cohomology vanishing⁵ \(H^{n>0}(G\times_BV_\lambda)=0\) and the Atiyah–Bott localization formula [18], we obtain the Weyl-type character formula of the FT construction \(H^0(G\times_BV_\lambda)\). An application of the case \(n=0\) in Theorem 1[main32theorem32140241] will be discussed in Section 2.0.1.2.

2 Application to vertex operator superalgebras↩︎

In Part 2, we consider several free-field algebras and check the axioms of shift system (Definition 1). It enables us to use the results in Part 1 for the study of the corresponding new multiplet W-(super)algebras.

2.0.1 Shift systems and free field algebras↩︎

2.0.1.1 Preliminary from VOSA

First, let us introduce the vertex operator superalgebra and some basic notation. Let \(V\) be a superspace, i.e., a \(\mathbb{Z}_{2}\)-graded vector space \(V=V_{\overline{0}} \oplus V_{\overline{1}}\), where \(\{ {\overline{0}}, \overline{1} \}=\mathbb{Z}_2\). For \(a\in V\), we say that the element \(a\) has parity \(p(a)\in \mathbb{Z}_2\) if \(a\in V_{p(a)}\). A field \(a(z)\) is a formal series of the form \[\begin{align} a(z)=\displaystyle\sum_{n\in \mathbb{Z}}a_{(n)}z^{-n-1},\; \; a_{(n)}\in \operatorname{End}(V) \end{align}\] such that for any \(v\in V\), one has \(a_{(n)}v =0\) for some \(n\gg 0\).

Note that from the locality axiom, one can derive the following commutator formula \[\begin{align} a_{(n)}b(z)=b(z)a_{n}+\sum_{i\geq 0}\binom{n}{i}z^{n-i}(a_{(i)}b)(z). \end{align}\] We call \(a_{(0)}\) a zero-mode. By the commutator formula, zero-mode is a derivation: \[\begin{align} \label{eqref:32zero32mode32is32derivation} [a_{(0)},b_{(n)}]=(a_{(0)}b)_{(n)}. \end{align}\tag{26}\]

If we allow the exponent of \(z\) in field expansion to be a rational number, then one can define the generalized V(O)SA similarly (see [33] for more details). Let us introduce the following two operations that create another VOSA module from a given one.

2.0.1.2 Simplicity theorem

This subsection will not be used until Section 2.0.2 and may be skipped on a first reading. In this subsection, we formalize the discussion in [4] to enhance the perspective and flexibility of the discussion.⁷ Throughout this subsection, \((\Lambda,\uparrow,\{V_\lambda\}_{\lambda\in\Lambda})\) is a shift system.

By Theorem 1[main32theorem32140141] and the Serre duality, we have \[\begin{align} H^0(G\times_BV_\lambda) \simeq H^{l(w_0)}(G\times_BV_{w_0\ast\lambda}(w_0\uparrow\lambda)) &\simeq H^{l(w_0)}(G\times_BV_{\lambda'}^\ast(-2\rho)))\\ &\simeq H^0(G\times_BV_{\lambda'})^\ast \end{align}\] as \(G\)-modules. For the naturality discussion, see [4].

In Section 2.0.3, we prove the simplicity theorem for the multiplet W-(super)algebras and W-(super)algebras by comparing the Weyl-type character formula ?? with the “Kazhdan–Lusztig-type" character formula described by the affine Weyl group (see Lemma 18 and 20). The following lemma summarizes that argument.

In the remainder of this subsection, we consider the setting of Example 1 under the following conditions:

1. \(W_{Q}:=H^0(G\times_BV_0)\) is a generalized cyclic VOSA and \(W_{Q+\lambda}:=H^0(G\times_BV_\lambda)\) is a graded \(W_Q\)-module for each \(\lambda\in\Lambda\). Furthermore, the natural \(G\)-action is in \(\operatorname{Aut}(W_Q)\).

2. \(W_{P}:=\bigoplus_{\substack{\mu\in\Lambda\\\mu_\bullet=0}}H^0(G\times_BV_\mu)\) is a generalized cyclic VOSA and \(W_{P+\lambda_\bullet}:=\bigoplus_{\substack{\mu\in\Lambda\\\mu_\bullet=\lambda_\bullet}}H^0(G\times_BV_\mu)\) is a graded \(W_P\)-module for each \(\lambda\in\Lambda\). Furthermore, the natural \(G\)-action is in \(\operatorname{Aut}(W_P)\) and \(W_P\) is an abelian intertwining algebra with abelian group \(P/Q\) (see [20]).

If [assumption:32global32VOA32structure32for32P] holds, then so does [assumption:32global32VOA32structure32for32Q] (but the converse does not in general). If [assumption:32global32VOA32structure32for32Q] (resp. [assumption:32global32VOA32structure32for32P]) holds, then \(\mathcal{W}_0\simeq W_Q^G\simeq W_P^G\) is a vertex operator super subalgebra of \(W_Q\) and \(\mathcal{W}_{-\beta}\) is a \(\mathcal{W}_0\)-module for any \(\beta\in P_+\cap Q\) (resp. for any \(\beta\in P_+\)) and \(\lambda\in\Lambda\). In the same manner as [4], we obtain the following.

In the same manner as [4] or [5], we have the following.

We now explain how these results are applied to the simplicity theorems for multiplet \(W\)-(super)algebras. We consider the case [assumption:32global32VOA32structure32for32Q] or [assumption:32global32VOA32structure32for32P]. Then, by Lemma 10, \(\mathcal{W}_{-\beta}\) is simple as \(\mathcal{W}_0\)-module. To extend the simplicity of \(\mathcal{W}_{-\beta}\) to \(\mathcal{W}_{-\beta+\lambda}\) such that \(\lambda\in\Lambda\) satisfies ?? , we use Lemma 9 to \(\lambda_0=0\) and \(\lambda_1=\lambda\). Here, \(\mathbb{M}(\mu)\) and \(\mathbb{L}(\mu)\) represent the Verma module and the irreducible module over \(\mathcal{W}_0\) with highest weight \(\mu\), respectively. As we show later, sometimes \(\mathbb{L}(-\beta+\lambda_i)\) (\(i=0,1\)) has a “Kazhdan–Lusztig-type" character formula in Lemma 9[formal32KL32decomposition]. Since \(\mathcal{W}_{-\beta+\lambda_0}\) is irreducible, by Lemma 9, so is \(\mathcal{W}_{-\beta+\lambda_1}\). In particular, \(\mathcal{W}_{-\beta+\lambda}\) are cyclic. Furthermore, if they are cyclic, by Lemma 11, \(W_{Q+\lambda_\bullet}\) or \(W_{Q+\lambda}\) is also simple as \(W_Q\)-module.

In [3], [4] and [5], the cases where \(\mathcal{W}_0\) is the principal W-algebra \(\mathcal{W}_k(\mathfrak{g})\) for simply-laced \(\mathfrak{g}\) and the affine VOA \(V_k(\mathfrak{sl}_2)\) were considered, respectively, and the corresponding simplicity theorems were proved in the same manner explained above. In Section 2.0.2 and Section 2.0.3 of the present paper, we consider the cases where \(\mathcal{W}_0\) is the principal W-(super)algebra \(\mathcal{W}_k(\mathfrak{g})\) for non-simply-laced \(\mathfrak{g}\) and \(\mathfrak{g}=\mathfrak{osp}(1|2r)\), respectively. In the former case, [assumption:32global32VOA32structure32for32P] holds and the simplicity theorem will be proved for any \(\lambda\in\Lambda\) satisfying ?? , but in the later case, it will be proved only for \(\lambda^\bullet=0\) case (i.e., assume only [assumption:32global32VOA32structure32for32Q]) because of some technical difficulty.

2.0.1.3 Free field algebras

Let us introduce two VOSAs that will serve as material for shift systems later.

2.0.1.3.1 Fermionic VOSA

The Clifford algebra is generated by \(\{\phi(n),n\in \frac{1}{2}+\mathbb{Z}\}\cup \{{1}\}\) satisfying the relations \[\begin{align} \{\phi(n),\phi(m)\}=\delta_{n,-m},\quad n,m\in \tfrac12+\mathbb{Z}. \end{align}\] Let \(F\) be the module of Clifford algebra generated by \(\boldsymbol{1}\) such that \(\phi(n)\boldsymbol{1}=0\) (\(n>0\)). Then there is a unique VOSA structure on \(F\), with the field and conformal vector \[\begin{align} \label{conformal32vector32of32free32fermion} Y(\phi(-\tfrac12)\boldsymbol{1},z)=\phi(z)=\sum_{n\in \frac{1}{2}+\mathbb{Z}}\phi(n)z^{-n-\frac{1}{2}},\;\; \omega^{(s)}=\tfrac12 \phi(-\tfrac32)\phi(-\tfrac12)\boldsymbol{1}, \end{align}\tag{28}\] which gives the central charge \(\tfrac12\). The conformal weight of \(\Delta_\phi\) of \(\phi(-\tfrac{1}{2})\mathbf{1}\) is \(\tfrac{1}{2}\). The character and supercharacter of \(F\), and the character of the \(\iota\)-twisted \(F\)-module \(\iota^*F\), are given by \[\operatorname{ch}_q F = \frac{\eta(q)^2}{\eta(q^2)\eta(q^{1/2})}, \qquad \operatorname{sch}_q F = \frac{\eta(q^{1/2})}{\eta(q)}, \qquad \operatorname{ch}_q \iota^*F = 2\frac{\eta(q^2)}{\eta(q)}.\] Here \(\iota\) denotes the parity automorphism. Moreover, \((\phi(-\frac{1}{2})v^*,v)=(v^*,\text{i} \phi(-\frac{1}{2})v)\) for the dual \(v^\ast\in F^\ast\) of \(v\in F\).

2.0.1.3.2 Lattice VOSAs

Let \(L\) be a positive definite integral lattice of rank \(r\). We extend the bilinear form on \(L\) to \(\mathfrak{h}_L=\mathbb{C}\otimes_{\mathbb{Z}} L\). We also consider the affinization \(\hat{\mathfrak{h}}_L=\mathbb{C}[t,t^{-1}]\otimes \mathfrak{h}_L\oplus \mathbb{C}c\) of \(\mathfrak{h}_L\) and its subalgebras \(\hat{\mathfrak{h}}_L^{\geq 0}=\mathbb{C}[t]\otimes \mathfrak{h}_L\), \(\hat{\mathfrak{h}}_L^-=t^{-1}\mathbb{C}[t^{-1}]\otimes \mathfrak{h}_L\). For \(\mu\in \mathfrak{h}\), the induced representation \(M(\mu)\) of \(\hat{\mathfrak{h}}_L\) is defined by \[\begin{align} M(\mu)=U(\hat{\mathfrak{h}}_L)\otimes_{U(\hat{\mathfrak{h}}_L^{\geq 0}\oplus \mathbb{C}c)}\mathbb{C}_{\mu}, \end{align}\] where \(h\in\hat{\mathfrak{h}}_L\) acts as \((\mu,h)\) on \(\mathbb{C}_{\mu}\). Then there exists a simple V(S)A structure (see [37]) on \[\begin{align} V_{L} = M(0)\otimes \mathbb{C}[L] \simeq \bigoplus_{\mu\in L}M(\mu) \end{align}\] with the vacuum vector \(\mathbf{1}=1\otimes 1\), such that \(Y(ht^{-1}\otimes 1,z)=h(z)=\sum_{n\in \mathbb{Z}}h_{(n)}z^{-n-1}\), \(h_{(n)}=h\otimes t^n\). The group algebra of \(\mathbb{C}[L]\) is generated by \(e^\mu\), \(\mu\in {L}\). The vertex operator of \(Y(e^\mu,z)\) is defined as \[\begin{align} Y(e^\mu,z)=e^{\mu}z^{\mu_{(0)}}\exp\left(-\sum_{j<0}\tfrac{z^{-j}}{j}\mu_{(j)}\right)\exp\left(-\sum_{j>0}\tfrac{z^{-j}}{j}\mu_{(j)}\right)\epsilon(\mu,\cdot), \end{align}\] where \(e^\mu(e^{\lambda})=e^{\mu+\lambda}\) for \(\lambda\in \mathbb{C}[{L}]\), and \(\epsilon(\cdot,\cdot)\) is the \(2\)-cocycle satisfying [37]. The parity of the element \(v\otimes e^{\mu}\) is \(|\mu|^2\mod{2}\). One can define a (shifted) conformal vector \[\begin{align} \label{shifted32conformall32vector} \omega_{\mathrm{sh}}=\omega_{\mathrm{st}}+\gamma_{(-2)}\mathbf{1},\;\; \omega_{\mathrm{st}} =\tfrac{1}{2} \sum_{i=1}^{r} v_{i (-1)}v_i^\ast, \;\; \gamma\in \mathfrak{h}_L \end{align}\tag{29}\] with central charge is \(r-12(\gamma,\gamma)\), where \(\{v^i\}_{i=1}^{\ell}\) and \(\{v_i^\ast\}_{i=1}^{r}\) are bases of \(L\) and \(L^\ast\), respectively, such that \((v_i,v_j^\ast)=\delta_{i,j}\). The dual lattice \(L^\ast\) of \(L\) is defined by \(L^\ast:=\{\beta\in \mathbb{Q}\otimes L\;|\; (\alpha,\beta)\in \mathbb{Z}\,\,\text{for all}\,\, \alpha\in L\}\). Then the lattice VO(S)A has finitely many irreducible modules parametrized by \(L^\ast/L\) [38]; these modules are \[\begin{align} V_{L+\lambda}:=M(0)\otimes e^{\lambda}\mathbb{C}[L],\;\;(\lambda\in L^\ast/L) \end{align}\] up to isomorphism. For \(\mu\in \lambda+L\), the conformal weight \(\Delta_\mu\) of \(e^\mu\) under the action of \((\omega_{\mathrm{sh}})_{(1)}=L_0\) is \[\begin{align} \label{conformal32weight32of32lattice32point32vector} \Delta_{\mu} =\tfrac{1}{2}|\mu-\gamma|^2+\tfrac{c-r}{24} =\tfrac{1}{2}|\mu|^2-(\mu,\gamma) . \end{align}\tag{30}\] According to [34], the contragredient dual \(V_{L+\lambda}^\ast\) of \(V_{L+\lambda}\) is \(V_{L+\lambda'}\), for some \(\lambda'\in L^\ast/L\).

2.0.1.3.3 Ramond sector

Let us consider the setup in the last two sections and Definition 5[def:32spectral32flow32twist]. For \[\begin{align} R=\tfrac{1}{2}\sum_{i\in I_{\mathrm{odd}}}v_i^\ast,\;\; I_{\mathrm{odd}} = \{1\leq i\leq r\;|\;\text{|v_i|^2 is odd}\}, \end{align}\] we can easily check that \(R\) is a simple current element and \(g_R=\operatorname{exp}(2\pi iR_{(0)})\) gives the parity automorphism \(\iota_{V_L}\) of \(V_L\). Let \(M\) be a \(1\otimes\iota_{F}\)-twisted \(V_L\otimes F\)-module. Note that \(\iota_{V_L}\otimes 1\in\operatorname{Aut}(V_L\otimes F)\). The Ramond sector of \(M\) is defined by the \(\iota_{V_L\otimes F}\)-twisted \(V_L\otimes F\)-module \(\mathcal{S}_{R}(M)\).

2.0.1.4 Construction of shift systems

Let us consider step-by-step if we can build a shift system (Definition 1) using the two VOSA(-modules), free fermion \(F\) and lattice VOSA-module \(V_{L+\lambda}\) with conformal vectors 28 and 29 , respectively, introduced above. To consider \(V_{L+\lambda}\) as a weight \(B\)-module, it is natural to consider the case \(L=xQ\) for some \(x\in\mathbb{C}^\times\) and the \(\mathfrak{h}\)-action \[\begin{align} \label{the32Cartan32action32for32our32cases} h_i=\lceil -\tfrac{1}{x}\alpha_{i (0)}^\vee\rceil. \end{align}\tag{31}\] In other words, we consider the Fock space \(M(-x\mu)\) as the Cartan weight space with weight \(\lceil \mu\rceil\), where \(\lceil \mu\rceil\in P\) is uniquely determined by \(\lceil(\mu,\alpha_i^\vee)\rceil=(\lceil \mu\rceil,\alpha_i^\vee)\) for \(i\in I\). We consider the case where \(L=xQ\) is positive integral (see Section 2.0.1.3.2), which is equivalent to the cases \(x=\sqrt{p}\) for some \(p\in\mathbb{Z}_{\geq 1}\) such that

1. \(p=r^\vee m\) (where \(r^\vee\) is the lacing number of \(\mathfrak{g}\) and \(m\in\mathbb{Z}_{\geq 1}\)), or

2. \(p=2m-1\), \(m\in\mathbb{Z}_{\geq 1}\) and \(\mathfrak{g}=B_r\).

Note that in the case [non-super32case], \(L=\sqrt{p}Q\) is even, but in the case [super32case], \(\sqrt{p}Q\) is odd (because \(|\sqrt{p}\alpha_r|^2\) is odd). Also, since \(\sqrt{p}Q\) is an integral, \(\sqrt{p}Q\subseteq (\sqrt{p}Q)^\ast\) and thus we can consider \((\sqrt{p}Q)^\ast/\sqrt{p}Q\). For the moment, we will not fix the representative \(\Lambda\) of \((\sqrt{p}Q)^\ast/\sqrt{p}Q\simeq \tfrac{1}{p}Q^\ast/Q\), and \(\lambda\) will simply represent an element of \(\tfrac{1}{p}Q^\ast\) and we use the letter \(V_{\sqrt{p}(Q+\lambda)}\) for the corresponding irreducible \(V_{\sqrt{p}Q}\)-module. The representative \(\Lambda\) will be determined later from considerations on the choice of conformal weight vector.

Since the \(\mathfrak{h}\)-action is given by 31 , it is natural to consider whether the zero-modes \[\begin{align} e^{\sqrt{p}\alpha_i}_{(0)}\colon V_{\sqrt{p}(Q+\lambda)}^{h=\lceil\mu\rceil}=M(-\sqrt{p}\mu) \rightarrow M(-\sqrt{p}(\mu-\alpha_i))=V_{\sqrt{p}(Q+\lambda)}^{h=\lceil\mu-\alpha_i\rceil=\lceil\mu\rceil-\alpha_i} \end{align}\] define a \(B\)-action of a shift system. For a \(B\)-action to be compatible with the structure of VOSA, it should be even, but in the case [super32case], \(e^{\sqrt{p}\alpha_r}_{(0)}\) is odd as we checked above. To avoid this problem, in the case of [super32case], let us consider \(V_{\sqrt{p}(Q+\lambda)}\otimes F\) and \((e^{\sqrt{p}\alpha_r}\otimes\phi)_{(0)}\) instead of \(V_{\sqrt{p}Q}\) and \(e^{\sqrt{p}\alpha_r}_{(0)}\), respectively. Then since \(\phi\in F\) is odd, \((e^{\sqrt{p}\alpha_r}\otimes\phi)_{(0)}\) is even. From now on we consider the following two cases: For case [non-super32case] and [super32case], \[\begin{align} &V_\lambda=V_{\sqrt{p}(Q+\lambda)},\;\; \omega=\omega_{\mathrm{sh}},\;\; h_i=\lceil-\tfrac{1}{\sqrt{p}}\alpha_{i(0)}^\vee\rceil,\;\; f_i=e^{\sqrt{p}\alpha_i}_{(0)}\;\;(1\leq i\leq r), \tag{32} \\ &V_\lambda=V_{\sqrt{p}(Q+\lambda)}\otimes F,\;\; \omega=\omega_{\mathrm{sh}}+\omega^{(s)},\;\; h_i=\lceil-\tfrac{1}{\sqrt{p}}\alpha_{i(0)}^\vee\rceil,\\ &f_i= \begin{cases} e^{\sqrt{p}\alpha_i}_{(0)}&(1\leq i\leq r-1),\\ (e^{\sqrt{p}\alpha_r}\otimes \phi)_{(0)}&(i=r), \end{cases} \tag{33} \end{align}\] respectively. Recall that in Definition 1[conformal32grading32of32FT32data], we need a grading of \(V_\lambda\) which decomposes \(V_\lambda\) into finite-dimensional weight \(B\)-modules. Here we shall employ the conformal grading by \(\omega\) above. Then, for the operators \(\{f_i,h_i\}_{i\in I}\) to define the \(B\)-action in the shift system, in addition to the Serre relation and integrability, the conformal weights must be preserved⁸. In each case [non-super32case], [super32case], let us determine such a conformal vector \(\omega\) and describe the shift system naturally derived from the choice of \(\omega\). In particular, we will give the pair \((\Lambda,\uparrow)\) as special cases of Example 1, and thus Definition 1[FT32data321],[FT32data322] is satisfied.

2.0.1.4.1 The case [non-super32case]

By 30 , the operators \(f_i\) (\(i\in I\)) preserve the conformal weight iff \[\begin{align} \label{conformal32vector32for32the32case32of32non-super} \omega =&\tfrac{1}{2}\sum_{i\in I}\alpha_{i(-1)}\alpha_i^\ast+ \sqrt{p}(\rho-\tfrac{1}{p}\rho^\vee)_{(-2)}\mathbf{1} \end{align}\tag{34}\] and thus we consider this conformal vector. On the other hand, under the choice 34 , another solution of \(\Delta_{e^{x\alpha_i}_{(0)}}=0\) is \(x=-\tfrac{2}{\sqrt{p}(\alpha_i,\alpha_i)}\), namely the case \(x\alpha_i=-\tfrac{1}{\sqrt{p}}\alpha_i^\vee\). By [39], if \((p\lambda+\rho^\vee,\alpha_i)\equiv s\;(\operatorname{mod}\;p)\) for \(1\leq s\leq p-1\), then we have the non-zero (conformal weight-preserving) screening operators \[\begin{align} \label{short32screening32operator32in32non-super32case} Q_{i,\lambda}= \int_{[\Gamma]} e^{-\tfrac{1}{\sqrt{p}}\alpha_i^\vee}(z_1)\cdots e^{-\tfrac{1}{\sqrt{p}}\alpha_i^\vee}(z_s)dz_1...d_{z_{s}} \colon V_{\lambda}\rightarrow V_{\lambda-\tfrac{s}{p}\alpha_i^\vee}, \end{align}\tag{35}\] where \([\Gamma]\) is the cycle defined in [27]. To use 35 as short screening operators of a shift system, we have to regard \(\lambda-\tfrac{s}{p}\alpha_i^\vee\) as \(\sigma_i\ast\lambda\) for some \(W\)-action \(\ast\). If we choose \(x=\tfrac{1}{p}\rho^\vee\) in Example 1, then the representative \(\Lambda\) of \(\tfrac{1}{p}Q^\ast/Q\) is given by \[\begin{align} \label{the32representatives32in32non-super32case} \Lambda=\{ -\lambda^\bullet+\lambda_\bullet\;|\; \lambda^\bullet\in P_{\mathrm{min}},\; \lambda_{\bullet}=\sum_{i\in I}\tfrac{\lambda_i-1}{p}\alpha_i^\ast,\; 1\leq\lambda_i\leq r_i^\vee m \},\; r_i^\vee= \begin{cases} r^\vee&(\text{\alpha_i is long}),\\ 1&(\text{\alpha_i is short}), \end{cases} \end{align}\tag{36}\] where \(P_{\mathrm{min}}\) is the set of minuscule weights, and has the \(W\)-action \(\ast\) and the shift map \(\uparrow\) induced from \[\begin{align} \label{W-action32in32non-super32case} \sigma\ast\mu=\sigma(\mu+\tfrac{1}{p}\rho^\vee)-\tfrac{1}{p}\rho^\vee. \end{align}\tag{37}\] Then under the \(W\)-action, the short screening operator 35 is described as \(Q_{i,\lambda}\colon V_{\lambda}\rightarrow V_{\sigma_i\ast\lambda}(\sigma_i\uparrow\lambda)\).

2.0.1.4.2 The case [super32case]

By 30 and \(\Delta_\phi=\tfrac{1}{2}\), the operators \(f_i\) (\(i\in I\)) preserve the conformal weight iff \[\begin{align} \label{conformal32vector32for32the32case32of32super} \omega =\omega^{(s)}+\tfrac{1}{2}\sum_{i\in I}\alpha_{i(-1)}\alpha_i^\ast+ \sqrt{p}(1-\tfrac{1}{p})\rho_{(-2)}\mathbf{1} \end{align}\tag{38}\] and thus we consider this conformal vector (note that the \(\omega_{\mathrm{sh}}\)-part is different from 34 ). On the other hand, under the choice 38 , another solution of \(\Delta_{e^{x\alpha_i}_{(0)}}=0\) (\(i\not=r\)) and \(\Delta_{(e^{x\alpha_r}\otimes\phi)_{(0)}}=0\) is \(x=-\tfrac{1}{\sqrt{p}}\). In the same manner as above, if \((p\lambda+\rho,\alpha_i^\vee)\equiv s\;(\operatorname{mod}\;p)\) for \(1\leq s\leq p-1\), then we have the non-zero (conformal weight-preserving) screening operators (see also [40]) \[\begin{align} \label{short32screening32operator32in32super32case} Q_{i,\lambda}= \begin{cases} \int_{[\Gamma]} e^{-\frac{1}{\sqrt{p}}\alpha_i}(z_1)\cdots e^{-\frac{1}{\sqrt{p}}\alpha_i}(z_s)dz_1...d_{z_{s}}&(i\not=r),\\ \int_{[\Gamma]} (e^{-\frac{1}{\sqrt{p}}\alpha_r}\otimes\phi)(z_1)\cdots (e^{-\frac{1}{\sqrt{p}}\alpha_r}\otimes\phi)(z_{s})dz&(i=r), \end{cases} \colon V_{\lambda}\rightarrow V_{\lambda-\frac{s}{p}\alpha_i}. \end{align}\tag{39}\] If we choose \(x=\tfrac{1}{p}\rho\) in Example 1, then the representative \(\Lambda\) of \(\tfrac{1}{p}Q^\ast/Q\) is given by \[\begin{align} \label{the32representatives32in32super32case} \Lambda=\{ -\lambda^\bullet+\lambda_\bullet\;|\; \lambda^\bullet\in P_{\mathrm{min}},\; \lambda_{\bullet}=\sum_{i\in I}\tfrac{\lambda_i-1}{p}\varpi_i,\; 1\leq\lambda_i\leq p, \;\; \lambda_r+(\lambda^\bullet,\alpha_r^\vee)\equiv 1\;(\operatorname{mod}\;2) \}, \end{align}\tag{40}\] and has the \(W\)-action \(\ast\) and the shift map \(\uparrow\) induced from \[\begin{align} \label{W-action32in32super32case} \sigma\ast\mu=\sigma(\mu+\tfrac{1}{p}\rho)-\tfrac{1}{p}\rho. \end{align}\tag{41}\] Then, under the \(W\)-action, the short screening operator 39 is described as \(Q_{i,\lambda}\colon V_{\lambda}\rightarrow V_{\sigma_i\ast\lambda}(\sigma_i\uparrow\lambda)\).

The following lemma is proved in the same manner as [3], so we will omit the proof.

2.0.1.4.3 Verification of Definition 1[FT32data323]

Let us prove the remaining conditions of the shift system. Since the proofs are similar, we will treat both cases [non-super32case] and [super32case] simultaneously.

From now on, we consider the \(B\)-action on \(V_\lambda\) in Lemma 13.

2.0.1.4.4 Ramond sector of case [super32case]

Let us recall the notation in Section 2.0.1.3.3. In the case [super32case], the simple current element \(R\) is given by \(R=\tfrac{\pi i}{\sqrt{2m-1}}\varpi_r\). Under the spectral flow twist \(\mathcal{S}_R\), the conformal grading is deformed as \[\begin{align} \label{ABCdefn} \begin{aligned} L_0 &=\operatorname{Res}_z z^2Y(\Delta(\tfrac{ \varpi_r}{\sqrt{2m-1}},z)\omega_{\mathrm{sh}}+e^{\Delta_x}\omega^{(s)},z)\\ &=\operatorname{Res}_z z^2Y(\omega_{\mathrm{sh}}+\omega^{(s)},z)+\tfrac{1}{16}+\operatorname{Res}_z z^2Y((\tfrac{(\varpi_r)_{(1)}}{ \sqrt{2m-1}}+\tfrac{1}{2}(\tfrac{(\varpi_{r})_{(1)}(\varpi_{r})_{(1)}-\sqrt{2m-1}(\varpi_{r})_{(2)}}{2m-1}))\omega_{\mathrm{sh}},z)\\ &=:\operatorname{Res}_z z^2Y(\omega_{\mathrm{sh}}+\omega^{(s)},z)+\tfrac{1}{16}+A(\alpha_r)_{(0)}+B(\alpha_{r-1})_{(0)}+C, \end{aligned} \end{align}\tag{45}\] where \(\Delta_x\) is defined in [17]. Here \(A,B,C\in\mathbb{C}\) are determined solely by the spectral-flow twist fixed above (case (2), \(B_r\), \(p=2m-1\)); in particular they do not depend on the lattice weight \(\mu\) or on \(\lambda\in\Lambda\). In the Ramond sector \(\mathcal{S}_RV_{\lambda}\), we have \[\begin{align} \label{twistedweight} \Delta_{\mu} = \tfrac{1}{2}|\mu|^2-\sqrt{2m-1}(1-\tfrac{1}{2m-1})(\mu,\rho)+A(\alpha_r,\mu)+B(\alpha_{r-1},\mu)+C+\tfrac{1}{16} \end{align}\tag{46}\]

Since a spectral flow twist is invertible, under the appropriate changes, the Ramond sector \(\{\mathcal{S}_RV_\lambda\}_{\lambda\in\Lambda}\) of case [super32case] also defines a shift system. In conclusion, we obtain the following theorem.

Finally, the following is proved in the same manner as [4] so we omit the proof.

2.0.2 Main results I↩︎

In this section, by applying Theorem 1 to the shift systems in Theorem 5, we show that the main results of [3] and part of [4] also hold for our cases. After stating the definitions and properties common to the two cases of Theorem 5, we will examine each case in Section 2.0.2.1 and 2.0.2.2, respectively.

On the other hand, in the same manner as [4], the \(0\)-th sheaf cohomology \[\begin{align} H^0(G\times_BV_0) \end{align}\] of the sheaf associated with the \(G\)-equivariant vector bundle \(G\times_BV_0\) over the flag variety \(G/B\) has the VO(S)A structure induced from \(V_0\), and the \(n\)-th sheaf cohomology \(H^n(G\times_BV_\lambda(\mu))\) is an \(H^0(G\times_BV_0)\)-module (for more detail, see [4] and the discussion just before [4]). Denote \[\begin{align} \label{classicaldecomposition} H^0(G\times_BV_\lambda) \simeq \bigoplus_{\alpha\in P_+\cap Q} L_{\alpha+\lambda^\bullet} \otimes \mathcal{W}_{-\alpha+\lambda} \end{align}\tag{47}\] the decomposition of \(H^0(G\times_BV_\lambda)\) as \(G\)-module, where \(L_\beta\) is the irreducible \(\mathfrak{g}\)-module with highest weight \(\beta\in P_+\) and \(\mathcal{W}_{-\alpha+\lambda}\) is the multiplicity of a weight vector of \(L_{\alpha+\lambda^\bullet}\). In our case, \[\begin{align} H^0(G\times_BV_0)^G =V_0^B =\bigcap_{i\in I}\ker f_i|_{V_0^{h=0}} =\mathcal{W}_0 \end{align}\] is a vertex operator (super) subalgebra of \(H^0(G\times_BV_0)\) and each \(\mathcal{W}_{-\alpha+\lambda}\) is a \(\mathcal{W}_0\)-module.

Let us use the same notation and setup in Theorem 5. The proofs are basically the same as in [3], [4], so we will only outline them. By Theorem 1 and Theorem 5 we obtain the following.

2.0.2.1 The non-super case [non-super32case]

By applying Corollary 1 and Lemma 10, we obtain the following:

2.0.2.2 The super case [super32case]

In the same manner as above, we obtain the following:

2.0.3 Main results II↩︎

In Section 2.0.3, first we show that \(\mathcal{W}_0\) is isomorphic to the corresponding principal W-algebra \(\mathbf{W}^k(\mathfrak{g})\). It enables us to use another type of character formula, i.e., Kazhdan–Lusztig type character formula (see [21], [23]) and the discussion in Section 2.0.1.2 (see the last two paragraphs). Using these results, we will attempt to extend the simplicity theorems (Corollary 3 and Corollary 5) to \(\lambda\in\Lambda\) satisfying condition ?? (or, equivalently, the conditions in Lemma 12). In the non-super case [non-super32case], the simplicity theorem will be proved in the same manner as [4]. On the other hand, the super case [super32case] has not been studied as much as the case [non-super32case], and we will prove the simplicity theorem under a particular assumption (exactness of \(+\)-reduction). Since the flow of the discussion is the same as that of [4], details are sometimes omitted.

2.0.3.1 Preliminary from W-algebra

Let us recall the notation in [31] (see also [43]) and [23], [24], [44]. For an affine Lie algebra \(\hat{\mathfrak{g}}\), the Cartan subalgebra \(\hat{\mathfrak{h}}\) and its dual \(\hat{\mathfrak{h}}^\ast\) are decomposed as \(\hat{\mathfrak{h}}=\mathfrak{h}\oplus(\mathbb{C}K+\mathbb{C}d)\) and \(\hat{\mathfrak{h}}^\ast=\mathfrak{h}^\ast\oplus(\mathbb{C}\delta+\mathbb{C}\Lambda_0)\), respectively. For \(\hat{\mu}\in\hat{\mathfrak{h}}^\ast\), we have \[\begin{align} \hat{\mu}=\mu+\langle \hat{\mu},K\rangle\Lambda_0+(\mu,\Lambda_0)\delta, \;\; (\mu\in\mathfrak{h}^\ast). \end{align}\] By [31], the affine Weyl group \(\hat{W}\) is \(W\ltimes Q^\vee\) (for \(\hat{\mathfrak{g}}=X^{(1)}_r\)) and \(W\ltimes Q_{B_r}\) (for \(\hat{\mathfrak{g}}=A^{(2)}_{2r}\)), respectively. The \(\hat{W}\)-action on \(\hat{\mathfrak{h}}^\ast\) is given by \[\begin{align} \sigma t_\beta(\hat{\mu}) = \sigma(\mu+\langle\hat{\mu},K\rangle\beta) + \langle\hat{\mu},K\rangle\Lambda_0 + ((\hat{\mu},\Lambda_0-\beta)-\tfrac{1}{2}|\beta|^2\langle\hat{\mu},K\rangle)\delta. \end{align}\] For \(\hat{\rho}:=\rho+h^\vee\Lambda_0\), the circle \(\hat{W}\)-action on \(\hat{\mathfrak{h}}^\ast\) is given by \(w\circ\hat{\mu}:=w(\hat{\mu}+\hat{\rho})-\hat{\rho}\).

For \(k\in\mathbb{C}\), denote \(\hat{\mathfrak{h}}^\ast_k\) the set of \(\hat{\mu}\in\hat{\mathfrak{h}}^\ast\) such that \((\hat{\mu},K)=k\). Let us define \(V^k(\mathfrak{g})=U(\hat{\mathfrak{g}})\otimes_{\mathfrak{g}[t]\oplus\mathbb{C}K}\mathbb{C}_k\) the universal affine vertex algebra at level \(k\), where \(\mathfrak{g}[t]\) (resp. \(K\)) act on \(\mathbb{C}_k\) trivially (resp. \(k\operatorname{id}\)). More generally, we can define the Weyl module \(\hat{V}_{\beta,k}=U(\hat{\mathfrak{g}})\otimes_{\mathfrak{g}[t]\oplus\mathbb{C}K}L_{\beta}\) over \(V^k(\mathfrak{g})\) in the same manner. Clearly, we have \(\hat{V}_{0,k}=V^k(\mathfrak{g})\). For \(\hat{\mu}\in\hat{\mathfrak{h}}^\ast_k\), define \(\hat{M}(\hat{\mu})=U(\hat{\mathfrak{g}})\otimes_{U(\hat{\mathfrak{b}})}\mathbb{C}_{\hat{\mu}}\) the Verma module and its simple quotient \(\hat{L}(\hat{\mu})\), respectively, where \(U(\hat{\mathfrak{n}})\) (resp. \(\mathfrak{h}\), \(K\)) act on \(\mathbb{C}_{\hat{\mu}}\) trivially (resp. by the character of \(\mu\), \(k\operatorname{id}\)).

By applying the Drinfeld–Sokolov reduction \(H^0_{\mathrm{DS}}(-)\) to \(V^k(\mathfrak{g})\), the (universal) principal W-algebra \(\mathbf{W}^k(\mathfrak{g})\) at level \(k\) and its unique simple quotient \(\mathbf{W}_k(\mathfrak{g})\) are obtained. Note that we have the Feigin–Frenkel duality \[\begin{align} &\mathbf{W}^k(\mathfrak{g})\simeq \mathbf{W}^{{}^Lk}({}^L\mathfrak{g}),\;\; r^\vee(k+h^\vee)({}^Lk+{}^Lh^\vee)=1,\;\;\mathfrak{g}=X_r^{(1)},\tag{48}\\ &\mathbf{W}^k(\mathfrak{osp}(1|2r))\simeq \mathbf{W}^{{}^Lk}(\mathfrak{osp}(1|2r)),\;\; 4(k+h^\vee)({}^Lk+h^\vee)=1\tag{49} \end{align}\] in our cases (see [45], [46], [47]). More generally, for \(\check{\mu}\in\check{P}\) in the coweight lattice \(\check{P}=Q^\ast\), we have the twisted Drinfeld–Sokolov reduction \(H^0_{\mathrm{DS},\check{\mu}}(-)\) and the Arakawa–Frenkel module \[\begin{align} \mathbf{T}^{k+h^\vee}_{\beta,\check{\mu}} :=H^0_{\mathrm{DS},\check{\mu}}(\hat{V}_{\beta,k}) \end{align}\] over \(\mathbf{W}^{k}(\mathfrak{g})\). For the Langlands dual \({}^L\mathfrak{g}\), we can define \(\check{\mathbf{T}}^{{}^Lk+{}^Lh^\vee}_{\check{\mu},\beta}=H^0_{\mathrm{DS},\beta}(\hat{V}_{\beta,{}^Lk})\) in the same manner. Then for \(k\not\in Q\) or the case in [4], the Feigin–Frenkel duality \(\mathbf{T}^{k+h^\vee}_{\beta,\check{\mu}}\simeq \check{\mathbf{T}}^{{}^Lk+{}^Lh^\vee}_{\check{\mu},\beta}\) under the identification \(\mathbf{W}^k(\mathfrak{g})\simeq \mathbf{W}^{{}^Lk}({}^L\mathfrak{g})\) is known to hold (see [4], [24]). Let \(\chi_{\mu}:Z(\mathfrak{g})\rightarrow \mathbb{C}\) with \(\mu\in\mathfrak{h}^\ast\) be the map in [44], where \(Z(\mathfrak{g})\) is the center of the universal enveloping algebra of \(U(\mathfrak{g})\). Then one can define the Verma module \(\mathbf{M}_k(\chi_\mu):=U(\mathbf{W}^k(\mathfrak{g}))\otimes_{U(\mathbf{W}^k(\mathfrak{g})_{\geq 0})}\mathbb{C}_{\chi_\mu}\) over \(\mathbf{W}^k(\mathfrak{g})\) with highest weight \(\chi_\mu\) and its simple quotient \(\mathbf{L}_k(\chi_\mu)\) in the same manner above (in the Langlands dual case, denote \(\check{\mathbf{M}}_k(\chi_\mu)\) and \(\check{\mathbf{L}}_k(\chi_\mu)\), respectively).

2.0.3.2 The non-super case [non-super32case]

Let us consider our case [non-super32case]. By the injectivity of the Miura map [47] and the consideration in [48], we have \[\begin{align} \mathbf{W}^k(\mathfrak{g})\hookrightarrow\bigcap_{i\in I}\ker e^{-\frac{1}{{k+h^\vee}}b_i}_{(0)}|_{M(0)}, \end{align}\] where \(b_i\) satisfies \((b_i,b_j)=(k+h^\vee)(\alpha_i,\alpha_j)\). In particular, Fock module \(M(\mu)\) is also a \(\mathbf{W}^k(\mathfrak{g})\)-module. By comparing the conformal vector 34 and [24], for \(k=\frac{1}{p}-h^\vee\) and \(b_i=-\frac{1}{\sqrt{p}}\alpha_i\), we have \[\begin{align} \label{embedding32of32W-algebra32into32W950} \mathbf{W}^{m-{}^Lh^\vee}({}^L\mathfrak{g}) \overset{\text{\eqref{FF32duality32for32non-super32case}}}{\simeq} \mathbf{W}^{\frac{1}{p}-h^\vee}(\mathfrak{g}) \hookrightarrow \bigcap_{i\in I}\ker e^{\sqrt{p}\alpha_i}_{(0)}|_{M(0)}=\mathcal{W}_0. \end{align}\tag{50}\] On the other hands, by [24], we have \[\begin{align} \label{character32of32Arakawa-Frenkel32module} \operatorname{ch}_q T^{m}_{p\lambda_\bullet,\alpha+\lambda^\bullet} = \operatorname{ch}_q T^{\frac{1}{p}}_{\alpha+\lambda^\bullet,p\lambda_\bullet} = \operatorname{ch}_q\mathcal{W}_{-\alpha+\lambda} \end{align}\tag{51}\] for \(\alpha\in P_+\cap Q\) and \(\lambda\in\Lambda\) satisfying the strong condition Lemma 12[lemm:32strong32condition32for32non-super32case]. In particular, by applying \(\alpha=\lambda=0\), we have \(\operatorname{ch}_q\mathbf{W}^{m-{}^Lh^\vee}({}^L\mathfrak{g})=\operatorname{ch}_q\mathbf{W}^{\frac{1}{p}-h^\vee}(\mathfrak{g})=\operatorname{ch}_q\mathcal{W}_0\), and thus the injection in 50 is isomorphic. By Corollary 3, \(\mathcal{W}_{-\beta}\) is simple as \(\mathbf{W}^{m-{}^Lh^\vee}({}^L\mathfrak{g})\)-module for any \(\beta\in P_+\). In particular, for \(\beta=0\), we have \[\begin{align} \label{W95032and32W-algebra32coincides32in32non-super32case} \mathcal{W}_0 \simeq \mathbf{W}^{m-{}^Lh^\vee}({}^L\mathfrak{g}) \simeq \mathbf{W}_{m-{}^Lh^\vee}({}^L\mathfrak{g}) \simeq \mathbf{W}^{\frac{1}{p}-h^\vee}(\mathfrak{g}) \simeq \mathbf{W}_{\frac{1}{p}-h^\vee}(\mathfrak{g}). \end{align}\tag{52}\] In the remainder of this subsection, we use \[\begin{align} \label{k32and32check-k32in32non-super32case} k=\tfrac{1}{p}-h^\vee,\;\;\check{k}=m-{}^Lh^\vee. \end{align}\tag{53}\] On the other hand, in the same manner as [4], we have the following.

We aim to apply the discussion in Section 2.0.1.2 (see the last two paragraphs). First, let us give a KL-type decomposition in Lemma 9[formal32KL32decomposition]. Unless otherwise noted, all symbols (e.g., \(Q\), \(\alpha_i\),...) are the ones of \(\mathfrak{g}\). Denote \(h_\mu\) and \(\check{h}_\mu\) the top conformal weight of \(\mathbf{M}_{k}(\chi_\mu)\) and \(\check{\mathbf{M}}_{\check{k}}(\chi_\mu)\), respectively (see [44]). By 30 , for any \(\beta\in(\sqrt{p}Q)^\ast\) (in other words, \(\sqrt{p}\beta\in P_{{}^L\mathfrak{g}}\)), we have \[\begin{align} \label{compare32the32Delta32and32h} \Delta_{\beta} =h_{-\frac{1}{\sqrt{p}}\beta} =\check{h}_{\sqrt{p}\beta}. \end{align}\tag{54}\] Let us take \(\beta=\sqrt{p}(-\alpha+\lambda)\) (\(\alpha\in P_+\cap Q\), \(\lambda\in\Lambda\)). Then the corresponding highest weight of \(V^{\check{k}}({}^L\mathfrak{g})\) is \[\begin{align} -p(\alpha+\lambda^\bullet+\rho)+p\lambda_\bullet+\check{k}\Lambda_{0,{}^L\mathfrak{g}}. \end{align}\] Let us recall the notation in [21]. In our case (i.e., \(\widehat{{}^L\mathfrak{g}}\)), we have \[\begin{align} &\hat{W}_{{}^L\mathfrak{g}} =W_{{}^L\mathfrak{g}}\ltimes(Q_{{}^L\mathfrak{g}})^\vee =W\ltimes r^\vee Q,\\ &{\Delta}^{\mathrm{re}}_{+,\widehat{{}^L\mathfrak{g}}} = \{\alpha_{\pm}+n\delta\;|\; \alpha_{\pm}\in\Delta_{+,{}^L\mathfrak{g}},\;n\in\mathbb{Z}_{>0}\} \sqcup \Delta_{+,{}^L\mathfrak{g}} = \{\alpha_{\pm}^\vee+n\delta\;|\; \alpha_{\pm}\in\Delta_+,\;n\in\mathbb{Z}_{>0}\} \sqcup \Delta_{+}^\vee \\ & ({\Delta}^{\mathrm{re}}_{+,\widehat{{}^L\mathfrak{g}}})^\vee =\{r^\vee\alpha_{\pm}^l+r^\vee n\delta\;|\;\alpha_{\pm}^l\in\Delta_{\pm}^l,\;n\in\mathbb{Z}_{>0}\} \sqcup \{r^\vee\alpha_{\pm}^s+n\delta\;|\;\alpha_{\pm}^s\in\Delta_{\pm}^s,\;n\in\mathbb{Z}_{>0}\} \sqcup r^\vee\Delta_{+}, \\ & \begin{aligned} {\mathcal{C}}^{+}_{\widehat{{}^L\mathfrak{g}}}&= \{\nu\;|\;(\gamma^\vee,\nu+\hat{\rho}_{{}^L\mathfrak{g}})_{{}^LQ}\geq 0\;\text{for any \gamma\in\Delta^{\mathrm{re}}_{+,\widehat{{}^L\mathfrak{g}}}} \} \\ &= \{\nu\;|\;(\gamma^\vee,\nu+\rho^\vee+{}^Lh^\vee\Lambda_{0,{}^L\mathfrak{g}})_{{}^LQ}\geq 0\;\text{for any \gamma^\vee\in(\Delta^{\mathrm{re}}_{+,\widehat{{}^L\mathfrak{g}}})^\vee} \}. \end{aligned} \end{align}\] By [21], there exists unique \(\sigma t_{r^\vee\beta}\in\hat{W}_{{}^L\mathfrak{g}}=W\ltimes r^\vee Q\) such that \[\begin{align} \sigma t_{r^\vee\beta} \circ_{\widehat{{}^L\mathfrak{g}}} (-p(\alpha+\lambda^\bullet+\rho)+p\lambda_\bullet+\check{k}\Lambda_{0,{}^L\mathfrak{g}})\in{\mathcal{C}}^{+}_{\widehat{{}^L\mathfrak{g}}}. \end{align}\] In other words, we have \[\begin{align} &0\leq (\sigma^{-1}(\Delta_+^l),p(\beta-(\alpha+\lambda^\bullet+\rho))+p\lambda_\bullet+\rho^\vee)_Q \leq r^\vee m,\\ \land\; &0\leq (\sigma^{-1}(\Delta_+^s),p(\beta-(\alpha+\lambda^\bullet+\rho))+p\lambda_\bullet+\rho^\vee)_Q \leq m. \end{align}\] We show that if \(\lambda\) satisfies the condition Lemma 12[lemm:32strong32condition32for32non-super32case], i.e. \((p\lambda_\bullet+\rho^\vee,\theta_s)\leq m\), such \(\sigma t_{r^\vee\beta}\) is independent of the choice of \(\lambda_\bullet\). Let us consider the case \(\mathfrak{g}=B_r\) (other cases are similar). Set \(\beta-(\alpha+\lambda^\bullet+\rho)=\sum_{i\in I}n_i\varpi_i\) (\(n_i\in\mathbb{Z}\)). For \(\alpha_{+}\in\Delta_+\), if there exists, denote \(a_{\pm\alpha_{+}}\in\Delta_{+}\) such that \(\sigma^{-1}(a_{\pm\alpha_{+}})=\pm\alpha_{+}\). For any \(\alpha_+\in\Delta_+\), one of \(a_{\alpha_+}\) or \(a_{-\alpha_+}\) always exists. Set \(I_{\pm}=\{i\;|\;\exists a_{\pm\alpha_i}\}\). Clearly, we have \(I=I_+\sqcup I_{-}\). For \(i\in I_+\), we have \[\begin{align} 0\leq (\alpha_i,p\sum_{i\in I}n_i\varpi_i+p\lambda_\bullet+\rho^\vee)=pn_i(\alpha_i,\varpi_i)+\lambda_i\leq \begin{cases} r^\vee m&(\alpha_i\in\Delta_+^l),\\ m&(\alpha_i\in\Delta_+^s), \end{cases} \end{align}\] and thus \(n_i=0\) for \(i\in I_+\). Similarly, we have \(n_i=-1\) for \(i\in I_{-}\), and thus \(\sum_{i\in I}n_i\varpi_i=-\sum_{i\in I_-}\varpi_i\). Next, let us consider \(a_{\pm\theta_s}\). If \(a_{\theta_s}\) exists, then we have \(I_{-}=\phi\). On the other hand, if \(a_{-\theta_s}\) exists, then we have \(I_{-}=\{r\}\). Each of these cases corresponds to the case \(\alpha+\lambda^\bullet+\rho\in Q\) or not, which is independent of the choice of \(\lambda_\bullet\). Therefore, we can use the notation \(y_{\alpha,\lambda^\bullet}\in \hat{W}\) for the inverse of the unique element \(\sigma t_\beta\) in the above discussion. By [21], we obtain the following:

Let us apply the discussion in Section 2.0.1.2 to our case. In Lemma 9, set \[\begin{align} \label{setup32for32lem322466} \begin{aligned} &\beta=\alpha+\lambda^\bullet, \;\; (\lambda_0,\lambda_1)=(0,\lambda_\bullet), \;\; a_{y,\beta}=Q_{y,y_{\alpha,\lambda^\bullet}}(1),\\ &\text{\mu_{\lambda_i} is \eqref{the32element32in32C4332in32on-super32case} for each case}, \;\; y_\sigma=t_{r^\vee(\alpha+\lambda^\bullet-\sigma\circ(\alpha+\lambda^\bullet))}y_{\alpha,\lambda^\bullet}\;\;(\sigma\in W),\\ &{\mathbb{M}}(y,\mu_{\lambda_i}) = \check{\mathbf{M}}_{\check{k}}(\chi_{y\circ_{\widehat{{}^L\mathfrak{g}}}\mu_\lambda+p\rho-\check{k}\Lambda_{0,{}^L\mathfrak{g}}}), \;\; {\mathbb{L}}(-\beta+\lambda_i) = \check{\mathbf{L}}_{\check{k}}(\chi_{p(-\alpha+\lambda)}), \end{aligned} \end{align}\tag{55}\] respectively. Then the conditions Lemma 9[formal32KL32decomposition],[simplicity32at32lambda0] are already satisfied. On the other hand, the condition Lemma 9[Verma32coincidence32condition],[Verma32and32staggered32B-module] follows from 54 and [44]. Then by Lemma 9 and Lemma [lemma:32top32component], \(\mathcal{W}_{-\alpha+\lambda}\) is simple as \(\mathcal{W}_0\)-module. In particular, each \(\mathcal{W}_{-\alpha+\lambda}\) is a cyclic \(\mathcal{W}_0\)-module. Finally, by applying Lemma 11, the simplicity of \(H^0(G\times_BV_\lambda)\) also holds. In conclusion, we have the following:

2.0.3.3 The super case [super32case]

In this subsection, set \[\begin{align} \label{k32and32check-k32in32super32case} {k}=\tfrac{1}{2(2m-1)}-(r+\tfrac{1}{2}),\;\; \check{k}=m-r-1,\;\; h^\vee=h^\vee_{\mathfrak{osp}(1|2r)}=r+\tfrac{1}{2}. \end{align}\tag{56}\] Then \(4(k+h^\vee)(\check{k}+h^\vee)=1\) and by [47] and Corollary 5, we have \[\begin{align} \label{W95032and32W-algebra32coincides32in32super32case} \mathcal{W}_0 \simeq \mathbf{W}^{k}(\mathfrak{osp}(1|2r)) \simeq \mathbf{W}_{k}(\mathfrak{osp}(1|2r)) \simeq \mathbf{W}^{\check{k}}(\mathfrak{osp}(1|2r)) \simeq \mathbf{W}_{\check{k}}(\mathfrak{osp}(1|2r)) \end{align}\tag{57}\] and each \(\mathcal{W}_{-\alpha}\) (\(\alpha\in P_+\cap Q\)) is simple as \(\mathcal{W}_0\)-module. In the same manner as above, we have the following:

We aim to apply the discussion in Section 2.0.1.2 (see the last two paragraphs). First let us consider Lemma 9 in our case (\(\lambda_0=0\), \(\lambda_1=\lambda\) such that \(\lambda^\bullet=0\) and \((p\lambda_\bullet+\rho,{}^L\theta)\leq p\)). The condition Lemma 9[simplicity32at32lambda0] is already checked in the above discussion. By [49] and the results in Section 2.0.1.3, we have \[\begin{align} \operatorname{ch}_q\mathbf{M}_{\check{k}}(\chi_{\mu}) = \tfrac{\operatorname{ch}_qF}{\eta(q)^r}{q^{\frac{1}{2p}|\mu-p\rho+\frac{1}{p}\rho|^2}},\;\; \text{in particular, \operatorname{ch}_q\mathbf{M}_{\check{k}}(\chi_{p(-\alpha+\lambda)}) = \operatorname{ch}_qV_{\lambda}^{h=\sigma\circ(\alpha+\lambda^\bullet)} } \end{align}\] and the condition Lemma 9[Verma32coincidence32condition],[Verma32and32staggered32B-module] are satisfied.

Lastly, we want to check the Lemma 9[formal32KL32decomposition]. However, to the authors’ knowledge, the KL type decomposition as [21], [23] has not been proven in our case \(\mathfrak{osp}(1|2r)\). In this paper, we will prove the KL type decomposition (and Lemma 9[formal32KL32decomposition]) in the case of twisted affine Lie algebra of type \(A^{(2)}_{2r}\), and prove the simplicity theorem for super-case [super32case] under the assumption that the characters in these two types are the same under the coordinate change. Let us recall the data in \(\hat{\mathfrak{g}}'=A_{2r}^{(2)}\):⁹ \[\begin{align} &\hat{W}_{\hat{\mathfrak{g}}'}=W\ltimes Q, \;\; h^\vee_{A_{2r}^{(2)}}=2r+1=2h^\vee_{\mathfrak{osp}(1|2r)}, \\ &{\Delta}^{\mathrm{re}}_{+,\hat{\mathfrak{g}}'} =\{ \alpha_{\pm}+n\delta \;|\; \alpha_{\pm}\in\Delta_{\pm},\;n\in\mathbb{Z}_{>0} \} \sqcup \{2\alpha_{\pm}^s+(2n+1)\delta \;|\; \alpha_{\pm}^s\in\Delta_{\pm}^s,\;n\in\mathbb{Z}_{\geq 0}\} \sqcup \Delta_{+},\\ &{\mathcal{C}}^+_{\hat{\mathfrak{g}}'} = \{ \mu\in\hat{\mathfrak{h}}^\ast \;|\; (\gamma^\vee,\mu+\hat{\rho}_{A_{2r}^{(2)}})\in \mathbb{Z}_{\geq 0}\;\text{for any \gamma\in{\Delta}^{\mathrm{re}}_{+,\hat{\mathfrak{g}}'}} \}, \;\; \hat{\rho}_{A_{2r}^{(2)}}=\rho+h^\vee_{A_{2r}^{(2)}}\Lambda_0^c, \end{align}\] where all classical terms are of type \(B_n\) (see [31]), and \(\Lambda_0^c=\frac{1}{2}\Lambda_0\). By [21], there exists unique \(\sigma t_{\beta}\in\hat{W}_{\hat{\mathfrak{g}}'}=W\ltimes Q\) such that \[\begin{align} &\sigma t_\beta\circ_{\hat{\mathfrak{g}}'}(-p(\alpha+\lambda^\bullet+\rho^\vee)+p\lambda_\bullet+\check{k}\Lambda_0)\\ &= \sigma(p(\beta-\alpha-\lambda^\bullet-\rho^\vee)+p\lambda_\bullet+\rho)+p\Lambda_0^c-\hat{\rho}_{A_{2r}^{(2)}} \in {\mathcal{C}}^+_{\hat{\mathfrak{g}}'}. \end{align}\] In other words, we have \[\begin{align} 0\leq (p(\beta-\alpha-\lambda^\bullet-\rho^\vee)+p\lambda_\bullet+\rho,\sigma^{-1}(\alpha_+^\vee))\leq p\;\;\text{for any \alpha_+\in\Delta_+}. \end{align}\] In the same manner as Section 2.0.3.2, we can determine such a (\(\lambda_\bullet\)-independent) \(\sigma t_\beta\) explicitly as follows: \[\begin{align} \begin{cases} (\sigma,\beta)=(\operatorname{id},\alpha+\rho^\vee), \;\; \mu_\lambda=p\lambda+\check{k}\Lambda_0 &(\lambda^\bullet=0)\\ (\sigma,\beta)=(\sigma_r,\alpha+\rho^\vee), \;\; \mu_\lambda=\sigma_r\circ_{\hat{\mathfrak{g}}'}(p\lambda+\check{k}\Lambda_0) &(\lambda^\bullet=\varpi_r) \end{cases} \end{align}\] By combining it with [21], we obtain the KL type decomposition in \(A_{2r}^{(2)}\) case.

Let us relate type \(A_{2r}^{(2)}\) and \(\mathfrak{osp}(1|2r)\). Because the second description of \(A_{2n}^{(2)}\) in [50] has the same Dynkin diagram as the one of type \(B(0,r)=\mathfrak{osp}(1|2r)\), replacing a black node with a white node, these two types are closely related. In particular, according to [51], the character formulas for the modules of affine Lie (super)algebras of type \(A_{2r}^{(2)}\) and of type \(B(0,r)\) are the same up to coordinate change. In fact, choosing the coordinate of \(h\in \hat{\mathfrak{h}}\) of twisted affine Kac-Moody Lie algebra of type \(A_{2r}^{(2)}\): \[\begin{align} h:=2\pi i(-\tau\Lambda_0^c+z+u\delta). \end{align}\] Then by proper coordinate transformation \(t\), one can obtain that \[\begin{align} \label{transformation32formula} \operatorname{ch}_q[\hat{M}(A_{2r}^{(2)})](h)=\operatorname{ch}_q[\hat{M}(B(0,r)](t(h)). \end{align}\tag{58}\] Meanwhile, for type \(A_{2n}^{(2)}\), the KL type decomposition was already proved in Lemma 20 above. Therefore, we also have the same KL type decomposition on the \(\mathfrak{osp}(1|2r)\)-side.

To apply Lemma 9[formal32KL32decomposition] to our super case [super32case], one must move from affine to the principal \(W\)-algebra. We denote the category \(\mathcal{O}\) for \(L_\kappa(\mathfrak{osp}(1|2r))\) by \(\mathcal{O}_\kappa\) (\(\kappa\in\mathbb{C}\)). Let \(\mathcal{O}_{\kappa}^{[\lambda]}\) be the full subcategory of \(\mathcal{O}_\kappa\) whose objects have their local composition factors isomorphic to \(\hat{L}_{\kappa}(w\circ \lambda)\). The exactness of the \(+\)-reduction \(H^0_{\mathrm{DS},+}(\cdot)\) is already proved in [23] (and thus the KL type character formula of \(\mathbf{W}^\kappa(\mathfrak{g})\)-module is derived from [21]), but in our case it has not known yet to the authors’ knowledge.

If we assume the above conjecture, then we have the KL-type decomposition in our case \(\mathbf{W}^{\check{k}}(\mathfrak{osp}(1|2r))\) and the last condition Lemma 9[formal32KL32decomposition] is also satisfied. The remaining discussion is the same as Section 2.0.3.2, and the conclusion is as follows:

References↩︎