2 Asymptotic data of vertex algebras

A vertex algebra is a complex vector space V equipped with a distinguished vector ${|0\rangle } \in V$ , an endomorphism $T \in {\mathrm End}\,V$ and a linear map

$$ \begin{align*}V \to (\mathrm{End}\,V)[[z,z^{-1}]], \quad a \mapsto a(z)=\sum_{n\in \mathbb{Z}} a_{(n)} z^{-n-1}\end{align*} $$

satisfying the following axioms:

• • $a(z)b \in V((z))$ for all $a,b \in V$ ,

• • (vacuum axiom) ${|0\rangle } (z) = \mathrm {Id}_V$ and $a(z) {|0\rangle } \in a+ zV[[z]]$ for all $a \in V$ ,

• • (translation invariance axiom) $[T, a(z)] = \tfrac {\partial }{\partial z} a(z)$ ,

• • (locality axiom) $(z-w)^{N_{a,b}} [a(z),b(w)]=0$ for a sufficiently large integer $N_{a,b}$ , for all $a,b \in V$ .

A vertex algebra V is called conformal if there exists a vector $\omega $ called the conformal vector such that $L(z) = \omega (z) =\sum _{n\in \mathbb {Z}}L_nz^{-n-2}$ satisfies

1. (a) $[L_m,L_n]=(m-n)L_{m+n}+\dfrac {m^3-m}{12}\delta _{m+n,0}c_V$ , where $c_V$ is a constant called the central charge of V,

2. (b) $L_0$ acts semisimply on V,

3. (c) $L_{-1}=T$ on V.

For a conformal vertex algebra V and a V-module M, we set $M_d=\{m\in M\colon L_0 m=dm\}$ . The $L_0$ -eigenvalue of a nonzero $L_0$ -eigenvector $m \in M$ is called its conformal weight. For an element $a \in V$ of conformal weight $\Delta $ , we write $a(z) = \sum _{n \in \mathbb {Z}} a_n z^{-n-\Delta }$ . Note that in general $a_0 : M_d \rightarrow M_d$ .

A finitely generated V-module M is called ordinary if $L_0$ acts semisimply, $\dim M_d<\infty $ for all d, and the conformal weights of M are bounded from below. The minimum conformal weight of a simple ordinary V-module M is called the conformal dimension of M. More generally, a V-module M will be said to be of positive energy (also called admissible) if it possesses a $\mathbb {Z}_+$ -grading $M = \bigoplus _{k \in \mathbb {Z}_+} M_k$ such that $a_n M_k \subset M_{k-n}$ for all $a \in V$ . The normalised character of an ordinary representation M is defined by

$$ \begin{align*} \chi_M(\tau)=\mathrm{tr}_M q^{L_0-c_V/24}=q^{-c_V/24}\sum_{d\in \mathbb{C}}(\dim M_d)q^d,\quad q=e^{2\pi {\mathrm{i}} \tau} {\; \text{ with }\; \tau \in \mathbb{C}}. \end{align*} $$

A conformal vertex algebra is called conical if $V=\bigoplus _{\Delta \in \frac {1}{r}\mathbb {Z}_{\ge 0}}V_{\Delta }$ for some $r\in \mathbb {Z}_{\ge 0}$ and $V_0=\mathbb {C}$ . A $\mathbb {Z}$ -graded conical vertex algebra is said to be of CFT-type. Let V be a vertex algebra of CFT-type. Then V is called self-dual if $V \cong V'$ as V-modules, where $M'$ denotes the contragredient dual [Reference Frenkel, Huang and Lepowsky46] of the V-module M. Equivalently, V is self-dual if and only if it admits a non-degenerate symmetric invariant bilinear form.

The following definition goes back to [Reference Kac and Wakimoto58, Conjecture 1].Footnote ²

Definition 2.1. A conformal vertex algebra V is said to admit an asymptotic datum if there exist $\mathbf {A}_V\in \mathbb {R}$ , $\mathbf {w}_V \in \mathbb {R}$ , $\mathbf {g}_V\in \mathbb {R}$ such that

$$ \begin{align*}\chi_V(\tau) \sim \mathbf{A}_V (-{\mathrm{i}} \tau)^{{\frac{\mathbf{w}_V}{2}}}e^{\frac{\pi {\mathrm{i}} }{12 \tau}\mathbf{g}_V}\quad \text{as }\tau\downarrow 0.\end{align*} $$

The numbers $\mathbf {A}_V$ , $\mathbf {w}_V$ and $\mathbf {g}_V$ are called the asymptotic dimension of V, the asymptotic weight, and the asymptotic growth, respectively. Similarly, an ordinary V-module M is said to admit an asymptotic datum if there exist $\mathbf {A}_M\in \mathbb {C}$ , $\mathbf {w}_M,\mathbf {g}_M\in \mathbb {R}$ such that

$$ \begin{align*}\chi_M(\tau) \sim \mathbf{A}_M (-{\mathrm{i}} \tau)^{\frac{\mathbf{w}_M}{2}}e^{\frac{\pi {\mathrm{i}} }{12 \tau}\mathbf{g}_M}\quad \text{as }\tau\downarrow 0.\end{align*} $$

For a conformal vertex algebra V and an ordinary V-module M,

is called the quantum dimension of M if it exists ([Reference Dong, Jiao and Xu35]). If both V and M admit asymptotic data, $\mathbf {g}_V=\mathbf {g}_M$ and $\mathbf {w}_V=\mathbf {w}_M$ , then the quantum dimension of M exists and is equal to the ratio of the asymptotic dimensions:

(4) $$ \begin{align} \operatorname{qdim}_V M=\frac{\mathbf{A}_M}{\mathbf{A}_V}. \end{align} $$

Given a vertex algebra V, one naturally defines a Poisson algebra $R_V$ , called the Zhu $C_2$ -algebra, as follows ([Reference Zhu83]). Let $C_2(V)$ be the subspace of V spanned by the elements $a_{(-2)}b$ , where $a,b \in V$ , and set $R_V =V/C_2(V)$ . Then $R_V$ acquires a Poisson algebra structure via

$$ \begin{align*}1= \overline{{|0\rangle}}, \qquad \bar{a} \cdot \bar{b} = \overline{a_{(-1)}b} \quad \text{ and }\quad \{\bar{a},\bar{b}\} =\overline{a_{(0)}b},\end{align*} $$

where $\bar {a}$ denotes the image of $a\in V$ in the quotient $R_V$ .

The associated variety [Reference Arakawa7] $X_V$ of a vertex algebra V is the affine Poisson variety defined by

$$ \begin{align*}X_V = \mathrm{Specm}\, R_V.\end{align*} $$

A vertex algebra V is called lisse if $\dim X_V=0$ . It is called quasi-lisse [Reference Arakawa, Kawasetsu, Kac and Popov17] if $X_V$ has finitely many symplectic leaves.

A vertex algebra V is called rational if any finitely generated positively graded V-module is completely reducible. For a lisse conformal vertex algebra V, any finitely generated V-module is ordinary, and there exist finitely many simple V-modules ([Reference Abe, Buhl and Dong1]).

The following fact is well known.

Proof. We include a proof for completeness. By [Reference Abe, Buhl and Dong1], any simple V-module is ordinary, and there exist finitely many simple V-modules – say, . Let $h_i$ be the conformal dimension of $L_i$ . Then

$$ \begin{align*}\chi_{L_i}(\tau)=q^{h_i-c/24}\sum_{d\ge 0}(\dim (L_i)_{h_i+d})q^d.\end{align*} $$

By Zhu’s theorem [Reference Zhu83], the vector space spanned by $\chi _{L_i}(\tau )$ , $i=0,\dots , r$ is invariant under the natural action of the modular group $\operatorname {SL}_2(\mathbb {Z})$ . Hence,

$$ \begin{align*} \chi_{L_i}(\tau)=\sum_{j=1}^r S_{i,j}\chi_{L_j}(-1/\tau) \end{align*} $$

for some $S_{i,j}\in \mathbb {C}$ , $j=0,\dots , r$ . The assertion follows.

The following assertion is widely believed.

In Theorem 4.11, we confirm the uniqueness of the simple module with minimal conformal dimension for exceptional W-algebras ([Reference Arakawa, Futorny and Ramirez12, Reference Kac and Wakimoto60]). See also Conjecture 4.6 and the subsequent remarks.

Proposition 2.4. Let V be as in Conjecture 2.3, with simple modules $\{L_0,\dots , L_r\}$ ordered so that $L_0 = V$ , and assume that there exists a unique simple module $L_{i_{\circ }}$ satisfying the assertion of Conjecture 2.3. Then

$$ \begin{align*} \mathbf{g}_{L_i}=c_V-24 h_{\text{min}},\quad \mathbf{A}_{L_i}=\dim(L_{i_\circ})_{h_{\text{min}}} S_{i,i_{\circ}} \end{align*} $$

for all i. Moreover,

$$ \begin{align*} \operatorname{qdim}_V(L_i)=\frac{S_{i, i_{\circ}}}{S_{0, i_{\circ}}}, \end{align*} $$

and

(5) $$ \begin{align} \operatorname{qdim}_V(L_i\boxtimes L_j)=\operatorname{qdim}_V(L_i)\operatorname{qdim}_V(L_j), \end{align} $$

where $\boxtimes $ is the fusion product ([Reference Huang and Lepowsky49, Reference Huang and Lepowsky51, Reference Huang and Lepowsky52, Reference Huang and Lepowsky53]). In particular, the quantum dimension is well defined for all simple V-modules.

The number $c_V-24 h_{\text {min}}$ is called the effective central charge of V in the literature ([Reference Dong and Mason36]).

Proof. By [Reference Arakawa, Kawasetsu, Kac and Popov17], the set $\{L_i\}$ of simple ordinary V-modules is finite, and the characters $\chi _{L_i}(\tau )$ are solutions of a modular linear differential equation. Since the space spanned by the solutions of a modular linear differential equation is invariant under the natural action of $\operatorname {SL}_2(\mathbb {Z})$ , the assertion follows in a similar manner as Proposition 2.2, except that a solution of a modular linear differential equation may have logarithmic terms – that is, it has the form

Let $\operatorname {Vir}^c$ be the universal Virasoro vertex algebra at central charge $c\in \mathbb {C}$ , $\operatorname {Vir}_c$ the unique simple quotient of $\operatorname {Vir}^c$ .

Proof. It is well known that $\operatorname {Vir}^c$ has length two if $c=1-6(p-q)^2/pq$ for some $p,q\in \mathbb {Z}_{\geq 2}$ , $(p,q)=1$ , and otherwise $\operatorname {Vir}^c=\operatorname {Vir}_c$ . Hence, a quotient V of $\operatorname {Vir}^c$ is either $\operatorname {Vir}^c$ or $\operatorname {Vir}_c$ . If $V=\operatorname {Vir}^c$ , then

$$ \begin{align*} \chi_V(\tau)=\frac{(1-q)q^{(1-c_V)/24}}{\eta(q)}, \end{align*} $$

where $\eta (q)=q^{1/24}\prod _{j\geq 1}(1-q^j)$ . Hence (indicating by $+ \cdots $ terms of lower growth),

where we have used l’Hopital’s rule. So V admits an asymptotic datum with $\mathbf {A}_V=2\pi $ , $\mathbf {w}_V=3$ , $\mathbf {g}_V=1$ .

If $V=\operatorname {Vir}_c$ with $c=1-6(p-q)^2/pq$ , $p,q\in \mathbb {Z}_{\geq 2}$ , $(p,q)=1$ , then as it is well known [Reference Feĭgin and Fuchs43, Reference Kac and Wakimoto58] V admits an asymptotic datum with $\mathbf {w}_V=0$ ,

(6)

where $(a, b)$ is the unique solution of $pa-qb=1$ in integers and , and

(7) $$ \begin{align} \mathbf{g}_V=1-\frac{6}{pq}. \end{align} $$

The simple Virasoro vertex algebra $\operatorname {Vir}_c$ with $c=1-6(p-q)^2/pq$ , $p,q\in \mathbb {Z}_{\geq 2}$ , $(p,q)=1$ is known to be rational and lisse ([Reference Wang79]). The simple $\operatorname {Vir}_c$ -modules are the $(p,q)$ -minimal series representations of the Virasoro algebra, and for each simple $\operatorname {Vir}_c$ -module L, we have $\mathbf {w}_L=0$ , $\mathbf {g}_L=\mathbf {g}_{\operatorname {Vir}_c}$ and $\mathbf {A}_L>0$ ([Reference Feĭgin and Fuchs43, Reference Kac and Wakimoto58]).

Recall that a homomorphism $f\colon V\rightarrow W$ of conformal vertex algebras is called conformal if $\omega _W=f(\omega _V)$ .

Proof. Our aim is to show that $\omega =\omega _W-f(\omega _V)$ vanishes. Now, if $\omega \ne 0$ , then [Reference Lepowsky and Li67, Theorem 3.11.12] asserts that the commutant subalgebra

$$\begin{align*}\operatorname{Com}(W, f(V)) = \{w \in W \colon f(v)_{(j)}w\ \text{for all}\ v \in V\ \text{and}\ j \in \mathbb{Z}_+ \} \end{align*}$$

is conformal with Virasoro vector $\omega $ of some central charge c. We note that $\omega \notin f(V)$ , for otherwise we would have $\omega _{(0)}\omega = 0$ , which contradicts that $\omega $ is a conformal vector. We thus proceed to show that $\omega \in f(V)$ and deduce that $\omega = 0$ by contradiction.

Consider the homomorphism $\tilde {f} :\operatorname {Vir}^c \otimes V \rightarrow W$ of conformal vertex algebras that sends the conformal vector $\omega ^{(0)}$ of $\operatorname {Vir}^c $ to $\omega $ , and such that $\tilde {f}({|0\rangle } \otimes v) = f(v)$ for all $v \in V$ . We now have the vertex subalgebra $U=\tilde {f}(\operatorname {Vir}^c \otimes V)\subset W$ whose simple quotient is isomorphic to $\operatorname {Vir}_c\otimes L$ . We thus have $\mathbf {g}_{\tilde {W}}\ge \mathbf {g}_L+\mathbf {g}_{\operatorname {Vir}_c}$ , and hence, $\mathbf {g}_{\operatorname {Vir}_c}=0$ by our hypotheses. From Lemma 2.7, it follows that $c=0$ .

We recall that $\operatorname {Vir}_0 = \mathbb {C}$ , that the maximal proper ideal of $\operatorname {Vir}^{0}$ is generated by $\omega ^{(0)}$ and that this ideal is simple as a $\operatorname {Vir}^0$ -module.

Let N be the maximal ideal of V, so that $L = V / N$ . We now consider the surjection

$$\begin{align*}\tilde{f}_1 : \operatorname{Vir}^0 \otimes (V / N) \rightarrow U / \tilde{f}(\operatorname{Vir}^c \otimes N), \end{align*}$$

and we define

$$\begin{align*}K=\{v\in L \colon \tilde{f}_1( \omega^{(0)} \otimes v) \in \tilde{f}(\operatorname{Vir}^c \otimes N)\}. \end{align*}$$

Since $K \subset L$ is a V-submodule, we have either $K=L$ or $K=0$ .

If $K = 0$ , then $\tilde {f}_1$ is an isomorphism because of simplicity of the maximal proper ideal in $\operatorname {Vir}^0$ . But then we would have $\mathbf {g}_{\tilde {W}} \ge \mathbf {g}_L+\mathbf {g}_{\operatorname {Vir}^0}> \mathbf {g}_L$ since $\mathbf {g}_{\operatorname {Vir}^0}>0$ , and this contradicts our hypotheses.

Therefore, $K = L$ . Since $(\operatorname {Vir}^0 \otimes V)_2 = \mathbb {C} \omega ^{(0)} \otimes {|0\rangle } \oplus {|0\rangle } \otimes V_2$ , it follows that

$$\begin{align*}\tilde{f}(\omega^{(0)} \otimes {|0\rangle}) \subset \tilde{f}({|0\rangle} \otimes N) = f(N) \subset f(V) \end{align*}$$

and in particular that $\omega \in f(V)$ . Thus, $\omega = 0$ , and we are done.

3 Admissible affine vertex algebras

Let $\mathfrak {g}$ be a complex simple Lie algebra of rank $\ell $ . Let $\mathfrak {g}=\mathfrak {n}_- \mathop {\oplus } \mathfrak {h}\mathop {\oplus } \mathfrak {n}_+$ be a triangular decomposition with a Cartan subalgebra $\mathfrak {h}$ , $\Delta $ the root system of $(\mathfrak {g},\mathfrak {h})$ and $\Delta _{+}$ a set of positive roots for $\Delta $ , $\Pi =\{\alpha _1,\dots ,\alpha _{\ell }\}$ the set of simple roots. Let $\theta $ be the highest root, and $\theta _s$ the highest short root. We also have $\Delta ^\vee $ the set of coroots. Let P be the weight lattice, Q the root lattice and $Q^\vee $ the coroot lattice. The lattice P is dual to $Q^\vee $ , and we write $P^\vee $ for the dual of Q. Recall that the Coxeter number and the dual Coxeter number of $\mathfrak {g}$ are denoted by $h_{\mathfrak {g}}$ and $h_{\mathfrak {g}}^\vee $ , respectively. Identifying $\mathfrak {h}$ with $\mathfrak {h}^*$ using the inner product

$$ \begin{align*}(~|~)_{\mathfrak{g}}=\displaystyle{\frac{1}{2 h_{\mathfrak{g}}^\vee}}\times\ \text{Killing form of }\ \mathfrak{g},\end{align*} $$

we view $Q^\vee $ as a sub-lattice of both P and Q. We denote by $\rho $ the Weyl vector (i.e., the half-sum of positive roots).

For , let be the irreducible highest weight representation of $\mathfrak {g}$ with highest weight , and let

(8)

Let $\widetilde {\mathfrak {g}} = \mathfrak {g}[t,t^{-1}] \oplus \mathbb {C} K \mathop {\oplus } \mathbb {C} D$ be the affine Kac-Moody algebra, with the commutation relations

$$ \begin{align*}[x t^m,y t^n] = [x,y] t^{m+n} + m \delta_{m+n,0} (x|y)_{\mathfrak{g}} K, \quad [D,xt^n]=nxt^n,\quad [K,\widehat{\mathfrak{g}}]=0,\end{align*} $$

for all $x,y \in \mathfrak {g}$ and all $m,n\in \mathbb {Z}$ . Let $\widetilde {\mathfrak {g}}=\widehat {\mathfrak {n}}_-\mathop {\oplus } \widetilde {\mathfrak {h}}\mathop {\oplus }\widehat {\mathfrak {n}}_+$ be the standard triangular decomposition – that is, $\widetilde {\mathfrak {h}} = \mathfrak {h} \oplus \mathbb {C} K\mathop {\oplus } \mathbb {C} D $ is the Cartan subalgebra of $\widetilde {\mathfrak {g}}$ , $\widehat {\mathfrak {n}}_+=\mathfrak {n}_+ +t\mathfrak {g}[t]$ , $\widehat {\mathfrak {n}}_-=\mathfrak {n}_-+t^{-1}\mathfrak {g}[t^{-1}]$ .

Let $\widehat {\mathfrak {g}} =[\widetilde {\mathfrak {g}},\widetilde {\mathfrak {g}}]= \mathfrak {g}[t,t^{-1}] \oplus \mathbb {C} K$ , and let $\widehat {\mathfrak {h}}=\mathfrak {h}\mathop {\oplus } \mathbb {C} K\subset \widehat {\mathfrak {g}}$ , so that $\widehat {\mathfrak {g}}=\widehat {\mathfrak {n}}_-\mathop {\oplus } \widehat {\mathfrak {h}}\mathop {\oplus } \widehat {\mathfrak {n}}_+$ . The Cartan subalgebra $\widetilde {\mathfrak {h}}$ is equipped with a bilinear form extending that on $\mathfrak {h}$ by

$$ \begin{align*} (K | D) = 1, \quad (\mathfrak{h} | \mathbb{C} K \oplus \mathbb{C} D) = (K | K) = (D | D) = 0. \end{align*} $$

We write $\delta $ and for the elements of $\widetilde {\mathfrak {h}}^*$ orthogonal to $\mathfrak {h}^*$ and dual to K and D, respectively. We have the (real) root system

$$ \begin{align*} \widehat{\Delta}^{\text{re}} &= \{\alpha+n\delta \colon n \in \mathbb{Z}, \alpha \in \Delta\}= \widehat{\Delta}^{\text{re}}_+\sqcup (-\widehat{\Delta}^{\text{re}}_+),\\ \widehat{\Delta}^{\text{re}}_+ &=\{\alpha+n\delta\colon \alpha\in \Delta_+, \ n\ge 0\}\sqcup \{\alpha+n\delta\colon \alpha\in \Delta, \ n> 0\}, \end{align*} $$

and the affine Weyl group $\widehat {W}$ generated by reflections $r_\alpha $ for $\alpha \in \widehat {\Delta }^{\text {re}}$ . For $\alpha \in \mathfrak {h}^*$ , the translation $t_\alpha : \widetilde {\mathfrak {h}}^* \rightarrow \widetilde {\mathfrak {h}}^*$ is defined by

For $\alpha \in Q^\vee $ , we have $t_\alpha \in \widehat {W}$ and, in fact, . The extended affine Weyl group, which is the group of isometries of $\widehat {\Delta }$ , is . Here, for R a subset of P, $t_R$ stands for the set $\{t_\alpha \colon \alpha \in R\}$ .

Let $\tilde {\mathcal {O}}_k$ be the category $\mathcal {O}$ of $\widetilde {\mathfrak {g}}$ at level k ([Reference Kac54]), and let $\tilde {\mathbf {KL}}_k$ be the full subcategory of $\tilde {\mathcal {O}}_k$ consisting of objects on which the action of $\mathfrak {g}$ is integrable. The simple objects of $\tilde {\mathcal {O}}_k$ are the irreducible highest weight representations with such that , while the simple objects of $\tilde {\operatorname {KL}}_k$ are those with , where $P_+$ is the set of dominant integral weights of $\mathfrak {g}$ .

For a weight , the corresponding integral root system is

where $\alpha ^{\vee }=2\alpha /(\alpha |\alpha )$ as usual, and the subgroup of $\widehat {W}$ generated by $r_\alpha $ with is called the integral Weyl group of .

Given any $k\in \mathbb {C}$ , let

$$ \begin{align*} V^k(\mathfrak{g}) = U(\widehat{\mathfrak{g}})\otimes_{U(\mathfrak{g}[t]\oplus \mathbb{C} K)}\mathbb{C}_k, \end{align*} $$

where $\mathbb {C}_k$ is the one-dimensional representation of $\mathfrak {g}[t] \oplus \mathbb {C} K$ on which $\mathfrak {g}[t]$ acts by 0 and K acts as a multiplication by the scalar k. There is a unique vertex algebra structure on $V^k(\mathfrak {g})$ such that ${|0\rangle }$ is the image of $1\otimes 1$ in $V^k(\mathfrak {g})$ and

$$ \begin{align*}x(z) :=(x_{(-1)}{|0\rangle})(z)= \sum_{n\in\mathbb{Z}} (xt^n) z^{-n-1}\end{align*} $$

for all $x\in \mathfrak {g}$ , where we regard $\mathfrak {g}$ as a subspace of V through the embedding $x \in \mathfrak {g} \hookrightarrow x_{(-1)}{|0\rangle } \in V^k(\mathfrak {g})$ . The vertex algebra $V^k(\mathfrak {g})$ is called the universal affine vertex algebra associated with $\mathfrak {g}$ at level k.

The vertex algebra $V^k(\mathfrak {g})$ has a conformal structure given by the Sugawara construction provided that k is noncritical – that is, $k\ne -h^{\vee }_{\mathfrak {g}}$ , with central charge

$$ \begin{align*}c_{V^{k}(\mathfrak{g})}= \dfrac{k \dim \mathfrak{g} }{k + h_{\mathfrak{g}}^\vee}.\end{align*} $$

A $V^k(\mathfrak {g})$ -module is the same as a smooth $\widehat {\mathfrak {g}}$ -module of level k. Recall that a $\widehat {\mathfrak {g}}$ -module M is smooth if, for all $a \in \mathfrak {g}$ and $m \in M$ , one has $at^n m = 0$ for all sufficiently large positive $n \in \mathbb {Z}$ . The highest weight $\widetilde {\mathfrak {g}}$ -module (regarded as a $\widehat {\mathfrak {g}}$ -module and thus $V^k(\mathfrak {g})$ -module) acquires conformal dimension

For a noncritical level k, we consider a $V^k(\mathfrak {g})$ -module M as a $\widetilde {\mathfrak {g}}$ -module by letting D act as the semisimplification of $-L_0$ . In particular, we identify the categories $\mathcal {O}_k$ and $\mathbf {KL}_k$ of $\widehat {\mathfrak {g}}$ -modules of level k with the respective full subcategories of $\tilde {\mathcal {O}}_k$ and $\tilde {\mathbf {KL}}_k$ consisting of modules on which the universal Casimir element C [Reference Kac54, Section 2.5] of $\widetilde {\mathfrak {g}}$ acts nilpotently. Accordingly, the affine space is identified with an affine subvariety of $\widetilde {\mathfrak {h}}^*$ by the correspondence

preserving the corresponding linkage relations in $\mathcal {O}_k$ .

Let $L_k(\mathfrak {g})$ be the unique simple graded quotient of $V^k(\mathfrak {g})$ . For any graded quotient V of $V^k(\mathfrak {g})$ , we have $R_V=V/t^{-2}\mathfrak {g}[t^{-1}] V$ . In particular, $R_{V^k(\mathfrak {g})} \cong \mathbb {C}[\mathfrak {g}^*]$ , and hence, $X_{V^k(\mathfrak {g})}=\mathfrak {g}^*$ . Furthermore, $X_{L_k(\mathfrak {g})}$ is a subvariety of $\mathfrak {g}^* \cong \mathfrak {g}$ , which is G-invariant and conic.

More generally, let $\mathfrak {a}$ be a Lie algebra endowed with a symmetric invariant bilinear form $\kappa $ , and

$$ \begin{align*}\widehat{\mathfrak{a}}_\kappa= \mathfrak{a}[t,t^{-1}]\oplus \mathbb{C} \mathbf{1}\end{align*} $$

be the Kac-Moody affinisation of $\mathfrak {a}$ . It is a Lie algebra with commutation relations

$$ \begin{align*}[x t^m, y t^n] = [x,y] t^{m+n} + m \delta_{m+n,0} \kappa(x,y) \mathbf{1}, \quad [\mathbf{1},\widehat{\mathfrak{a}}_\kappa]=0,\end{align*} $$

for all $x,y \in \mathfrak {a}$ and all $m,n \in \mathbb {Z}$ . Then the $\widehat {\mathfrak {a}}_\kappa $ -module

$$ \begin{align*} V^\kappa(\mathfrak{a}) = U(\mathfrak{a})\otimes_{U(\mathfrak{a}[t]\oplus \mathbb{C} {\mathbf 1})}\mathbb{C}, \end{align*} $$

where $\mathbb {C}$ is the one-dimensional representation of $\mathfrak {a}[t] \oplus \mathbb {C} {\mathbf 1}$ on which $\mathfrak {a}[t]$ acts by 0 and ${\mathbf 1}$ acts as the identity, has a unique vertex algebra structure such that ${|0\rangle }$ is the image of $1\otimes 1$ in $V^\kappa (\mathfrak {a})$ and

$$ \begin{align*}x(z) :=(x_{(-1)}{|0\rangle})(z)= \sum_{n\in\mathbb{Z}} (xt^n) z^{-n-1}\end{align*} $$

for all $x\in \mathfrak {a}$ . We have $X_{V^\kappa (\mathfrak {a})} \cong \mathfrak {a}^*$ , and, letting $L_\kappa (\mathfrak {a})$ be the unique simple graded quotient of $V^\kappa (\mathfrak {a})$ , $X_{L_\kappa (\mathfrak {a})}$ is a subvariety of $\mathfrak {a}^*$ , which is Poisson and conic.

For $M\in \mathcal {O}_k$ on which $L_0$ acts semisimply, we consider the multivariable character $\chi _M$ of M defined by

$$ \begin{align*} \chi_{M}(\tau, z, t) = e^{2\pi \mathrm{i} k t} \mathrm{tr}_{M} (e^{2\pi \mathrm{i} z} e^{2\pi \mathrm{i} \tau (L_0 - c/24)}), \qquad (\tau, z, t) \in \mathbb{H} \times \mathfrak{h} \times \mathbb{C}. \end{align*} $$

We also write, in particular, .

For admissible weight , a closed form for was given in [Reference Kac and Wakimoto59]. It is convenient to write $v = 2 \pi \mathrm {i} (-\tau D+z+t K) \in \widetilde {\mathfrak {h}}^*$ as in [Reference Kac54]. Then the character formula is

where

The complex number k is said to be admissible for $\widehat {\mathfrak {g}}$ if is admissible. If this is the case, $L_k(\mathfrak {g})$ is called a simple admissible affine vertex algebra. By [Reference Kac and Wakimoto61, Proposition 1.2], k is admissible if and only if

(9) $$ \begin{align} k +h_{\mathfrak{g}}^\vee = \dfrac{p}{q} \text{ with } p,q \in \mathbb{Z}_{\ge 1}, \; (p,q)=1, \; p \ge \begin{cases} h_{\mathfrak{g}}^\vee & \text{if } (r^\vee,q)=1 \\ h_{\mathfrak{g}} & \text{if } (r^\vee,q)=r^\vee, \end{cases} \end{align} $$

where $r^\vee $ is the lacing numberFootnote ³ of $\mathfrak {g}$ .

If k is admissible with $(r^\vee ,q)=1$ , we say that k is principal. If k is admissible with $(r^\vee ,q)=r^\vee $ , we say that k is coprincipal.

In particular, the associated variety of an admissible affine vertex algebra is contained in the nilpotent cone $\mathcal {N}_{\mathfrak {g}}$ of $\mathfrak {g}$ . We note that the converse is not true, and there are affine vertex algebras at non-admissible levels whose associated variety is contained in $\mathcal {N}_{\mathfrak {g}}$ [Reference Arakawa and Moreau19].

For a principal admissible number, let $\operatorname {Pr}^k$ be the set of admissible weights such that for some $y\in \widetilde {W}$ . Similarly, for a coprincipal admissible number, let $\operatorname {CoPr}^k$ be the set of admissible weights such that for some $y\in \widetilde {W}$ . An element of $\operatorname {Pr}^k$ (resp. $\operatorname {CoPr}^k$ ) is called a principal admissible weight (resp. coprincipal admissible weight) of level k.

Occasionally, we shall use $\operatorname {Adm}^k$ to refer to the set $\operatorname {Pr}^k$ or $\operatorname {CoPr}^k$ , according to whether k is a principal or coprincipal admissible number.

For , let us denote by the restriction of to $\mathfrak {h}$ . For (resp. ), the primitive ideal is an maximal ideal of $U(\mathfrak {g})$ , and if and only if for (resp. for ) ([Reference Arakawa, Futorny and Ramirez12, Proposition 2.4]). Here and throughout, $\circ $ denotes the ‘dot’ action . Set

(10) $$ \begin{align} [\operatorname{Pr}^{k}]=\operatorname{Pr}^{k}/\sim,\quad [\operatorname{CoPr}^{k}]=\operatorname{CoPr}^{k}/\sim, \end{align} $$

where .

If V is a $\frac {1}{2}\mathbb {Z}$ -graded vertex algebra, we denote by $\operatorname {Zhu}(V)$ the Ramond twisted Zhu algebra of V, briefly recalling its construction from ([Reference De Sole and Kac34, Reference Zhu83]). Bilinear products $*_n : V \otimes V \rightarrow V$ are defined by

$$\begin{align*}a *_n b = \sum_{j \in \mathbb{Z}_{\ge 0}} \binom{\Delta(a)}{j} a_{(n+j)}b, \end{align*}$$

for a of conformal weight $\Delta (a)$ , $b \in V$ arbitrary. We write in particular $a * b = a *_{-1} b$ and $a \circ b = a *_{-2} b$ . Then the Zhu algebra is a quotient of V by the subspace $V \circ V$ spanned by all elements of the form $a \circ b$ for $a, b \in V$ . The operation $a \otimes b \mapsto a * b$ is well defined in the quotient and turns it into an associative unital algebra with unit $[{|0\rangle }]$ .

The fundamental property satisfied by $\operatorname {Zhu}(V)$ is the existence of a bijection between the set of irreducible $\operatorname {Zhu}(V)$ -modules and the set of irreducible positive energy Ramond twisted V-modules, sending a V-module M to its lowest graded piece $M_{\text {low}}$ equipped with the $\operatorname {Zhu}(V)$ -action $[a] \cdot m = a_0 m$ for $a \in V$ and $m \in M_{\text {low}}$ .

For the simple affine vertex algebra $L_k(\mathfrak {g})$ , we have

(11) $$ \begin{align} \operatorname{Zhu}(L_{k}(\mathfrak{g}))\cong U(\mathfrak{g})/I_k \end{align} $$

for some two-sided ideal $I_k$ of $U(\mathfrak {g})$ .

The statement of Theorem 3.3 is strengthened by the following assertion.

Theorem 3.4 [Reference Arakawa and van Ekeren14].

Let k be admissible. We have

where the product is taken over .

Proof. It is known [Reference Kac and Wakimoto58] that the maximal proper submodule $\widetilde {N}$ of $V^k(\mathfrak {g})$ is generated by a singular vector of weight – say, . Therefore, $N = \varphi (\widetilde {N})$ is generated as a $\widehat {\mathfrak {g}}$ -module by , and we shall show that .

Let M be an ordinary V-module. Since $\varphi $ is conformal, the restricted dual $M^*=\bigoplus _{d\in \mathbb {C}}\operatorname {\mathrm {Hom}}_{\mathbb {C}}(M_d,\mathbb {C})$ ([Reference Frenkel, Huang and Lepowsky46]) as a V-module is the same as the restricted dual of M as a $V^k(\mathfrak {g})$ -module. In particular, V is self-dual as a $V^k(\mathfrak {g})$ -module.

Assume that . Then we have a non-splitting exact sequence $0\rightarrow N\rightarrow V\rightarrow V/N\rightarrow 0$ of $\widehat {\mathfrak {g}}$ -modules. By the self-duality of V, this gives a non-splitting exact sequence

(12) $$ \begin{align} 0\rightarrow (V/N)^*\rightarrow V\rightarrow N^*\rightarrow 0 \end{align} $$

of $V^k(\mathfrak {g})$ -modules.

Let be a weight vector of V of weight that is mapped to the highest weight vector of $N^*$ . Since $V_0=\mathbb {C}|0{\rangle }$ and (12) is non-split, we have . It follows that is not a singular vector and also that . Thus, and are linearly independent vectors of , which are both primitive in the sense of [Reference Moody and Pianzola72, 2.6]. This implies that .

Let be the projective cover of in $\mathcal {O}_k$ and consider the homomorphism that sends the generator of of weight to . Since $\varphi $ is conformal, the universal Casimir element C of $\widetilde {\mathfrak {g}}$ acts as zero on V. It follows that g factors through a homomorphism . But then Lemma 3.6 below says that the multiplicity of in is at most one. Since this is a contradiction, we obtain , as desired. The assertion on complete reducibility now follows from Theorem 3.3.

The following assertion is a direct consequence of [Reference Arakawa and Fiebig16, Lemma 6.9].

If is an ordinary module over the simple affine vertex algebra $L_k(\mathfrak {g})$ of principal (resp. coprincipal) admissible level k, then (resp. ). We have for , while for , where $\Delta ^{\mathrm {long}}$ is the set of long roots of $\Delta $ and $\Delta ^{\mathrm {short}}$ is the set of short roots of $\Delta $ . The set of simple roots of , both for and for , is given by , where $\gamma _i = \alpha _i$ for and

$$ \begin{align*} \gamma_0 = \begin{cases} - \theta +q\delta& \quad \text{if}\ (r^\vee, q) = 1, \\ - \theta_{s}+q\delta /r^{\vee}& \quad \text{if}\ (r^\vee, q) = r^\vee, \end{cases} \end{align*} $$

where $\theta _s$ is the highest short root of $\Delta $ .

Let $\phi : \widetilde {\mathfrak {h}}^* \rightarrow \widetilde {\mathfrak {h}}^*$ be the isometry defined to act as the identity on the finite part $\mathfrak {h}^*$ and to act by

The adjoint $\phi ^* : \widetilde {\mathfrak {h}} \rightarrow \widetilde {\mathfrak {h}}$ acts by

$$ \begin{align*} \phi^*(K) = (1/q) K, \qquad \phi(D) = q D. \end{align*} $$

We have [Reference Kac and Wakimoto59]

$$ \begin{align*} \operatorname{Pr}^k &=\bigcup_{\substack{y\in \widetilde{W} \\ y(S_{(q)} )\subset \widehat{\Delta}^{\text{re}}_+}}\operatorname{Pr}^k_y, \quad \operatorname{Pr}^k_y= y\phi(\widehat{P}^{p-h^{\vee}_{\mathfrak{g}}}_{{+}}+\widehat\rho)-\widehat\rho,\\ \operatorname{CoPr}^k &=\bigcup_{\substack{y\in \widetilde{W} \\ y(S_{(q)} )\subset \widehat{\Delta}^{\text{re}}_+}} \operatorname{Pr}^k_y, \quad \operatorname{CoPr}^k_y= y\phi({}^\circ\widehat{P}^{p-h_{\mathfrak{g}}}_{{+}}+\widehat\rho)-\widehat\rho. \end{align*} $$

We have denoted by $\widehat {P}^{k}_{{+}}$ the set of dominant integral weights of $\widehat {\mathfrak {g}}$ of level k – that is, the set of weights such that for $i=1,\dots , \ell $ , and . Similarly, $\widehat {P}^{k}_{{++}}$ will denote the set of regular dominant integral weights, given by for $i=1,\dots , \ell $ , and . Furthermore, ${}^\circ \widehat {P}^{k}_{{+}}$ is defined similarly to $\widehat {P}^{k}_{{+}}$ , the condition being replaced by . In particular, we have

$$ \begin{align*} \operatorname{Pr}^k_{\mathbb{Z}}=\operatorname{Pr}^k_{1}=\phi(\widehat{P}^{p-h^{\vee}_{\mathfrak{g}}}_{{+}}+\widehat{\rho})-\widehat{\rho},\quad \operatorname{CoPr}^k_{\mathbb{Z}}=\operatorname{CoPr}^k_{1}=\phi({}^\circ \widehat{P}^{p-h_{\mathfrak{g}}}_{{+}}+\widehat{\rho})-\widehat{\rho}. \end{align*} $$

Proposition 3.7 [Reference Kac and Wakimoto59].

Let k be principal admissible of the form (9), and let

with $y = t_\beta \overline y$ , $\beta \in P^\vee $ , $\overline y \in W$ , and $\nu \in {}^\circ \widehat {P}_+^{p-h^\vee }+\widehat \rho $ . Let $T \in \mathbb {R}_{>0}$ and $z \in \mathfrak {h}$ such that $\alpha (z) \notin \mathbb {Z}$ for all $\alpha \in \Delta $ . Then as $T \rightarrow 0^+$ , one has

where

The proof of the proposition below closely follows the arguments of [Reference Kac and Wakimoto59], with adaptations to the coprincipal case.

Proposition 3.8. Let k be coprincipal admissible of the form (9), and let

with $y = t_\beta \overline y$ , $\beta \in P^\vee $ , $\overline y \in W$ , and $\nu \in {}^\circ \widehat {P}_+^{p-h^\vee }+\widehat {\rho }$ . Let $T \in \mathbb {R}_{>0}$ and $z \in \mathfrak {h}$ such that $\alpha (z) \notin \mathbb {Z}$ for all $\alpha \in \Delta $ . Then as $T \rightarrow 0^+$ , we have

where

Proof. The denominator of

(13)

is the standard Weyl denominator. Its asymptotic behaviour is well known [Reference Kac54, Proposition 13.13] to be

(14) $$ \begin{align} A_{\widehat{\rho}}({\mathrm{i}}T, -{\mathrm{i}}T z, 0) \sim b(\rho, z) T^{-\ell/2} e^{-\pi\dim(\mathfrak{g})/12T}, \end{align} $$

where

$$ \begin{align*} b(\rho, z) = \prod_{\alpha \in \Delta_+} 2 \sin\ {\pi (\alpha | z)}. \end{align*} $$

We analyse the numerator by writing it in terms of theta functions. Let m be a positive integer for which the lattice $\sqrt {m}Q$ is integral, and let , where $\mu \in P^\vee $ . Then we define

The modular transformation of these theta functions is given by

where in the sum, .

In the sum defining , the action of the Weyl group is intertwined with the action of the group via the automorphism y. The numerator of (13) is thus expressed in terms of theta functions as

Now we compute the asymptotic behaviour of

$$ \begin{align*} \varepsilon(\overline y) \sum_{w \in W} \varepsilon(w) \Theta_{q w(\widehat\nu), Q}({\mathrm{i}}T, -{\mathrm{i}}Tz, t). \end{align*} $$

Since is coprincipal, the lattice $pq Q$ is integral and we may use the modular transformation formula to write

(The summands corresponding to nonregular $\widehat {\mu }$ cancel out in the sum over W.) In the limit $T \rightarrow 0^+$ , the dominant terms in the sum above come from the shortest regular $\mu \in P^\vee $ . Such $\mu $ consist precisely of the W-orbit of $\rho ^\vee $ . We therefore obtain the asymptotic

The Weyl denominator formula asserts that

(15)

(There is a more standard version of the formula, from which (15) is obtained by applying to the Langlands dual Lie algebra and noting that the duality exchanges $\rho $ with $\rho ^\vee $ up to a factor of $\sqrt {r^\vee }$ .) Using (15), we reduce the double sum over W to

$$ \begin{align*} \prod_{\alpha \in \Delta_+} 4 \sin\ \frac{\pi (\alpha^\vee | \nu)}{p} \sin\ \pi (\alpha^\vee | z). \end{align*} $$

To deduce the asymptotic behaviour of , we note that

It then follows that

(16)

and, combining with (14), finally

The second form for $\mathbf {g}$ given in the theorem statement follows from the Freudenthal-de Vries strange formula and the fact that Langlands duality exchanges $\rho $ and $\rho ^\vee $ .

Proof. The assertion follows from Propositions 3.7 and 3.8 by taking the limit $z \rightarrow 0$ with $\beta =0$ and $\bar y=1$ . Let us explain the details for the coprincipal case. The normalised character is just the specialization . To prove the first assertion, we therefore apply Proposition 3.8 with $\beta = 0$ , $\overline y = 1$ , , and we let z tend to $0$ along a ray disjoint from the hyperplanes $(\alpha | z) = 0$ . Comparing the definition of the asymptotic dimension $\mathbf {A}$ (Definition 2.1 above) with that of yields

Here, we used l’Hopital’s rule.

In the above, we have since for , $\alpha \in \Delta _+$ , and for , $\alpha \in \Delta _+$ .

Note that $|P /(pq) Q^\vee | = (pq)^\ell |P /Q^\vee |$ and if $r^\vee | q$ . The values of $|P /Q^\vee |$ and $|P^\vee / r^\vee Q|$ , as well as other useful data, are collected in Table 1 for each simple Lie algebra.

Table 1

Some data for simple Lie algebras.

Theorem 3.10. Let k be admissible, and let V be a conical self-dual conformal vertex algebra equipped with a vertex algebra homomorphism $\varphi :V^k(\mathfrak {g})\rightarrow V$ such that $\varphi (\omega _{V^k(\mathfrak {g})})\in V_2$ and $(\omega _V)_{(2)}\varphi (\omega _{V^k(\mathfrak {g})})=0$ . Assume that V is a quotient of a vertex algebra $\tilde {V}$ that admits an asymptotic datum such that

$$ \begin{align*} \mathbf{w}_{\tilde{V}}=0 \quad \text{ and }\quad \mathbf{g}_{\tilde{V}} = \mathbf{g}_{L_{k}(\mathfrak{g})}. \end{align*} $$

Then $\varphi $ factors through a finite extension $L_k(\mathfrak {g})\hookrightarrow V$ . If moreover,

$$ \begin{align*} \chi_{\tilde{V}}(\tau) \sim \chi_{L_{k}(\mathfrak{g})}(\tau), \quad \text{as }\tau\downarrow 0, \end{align*} $$

that is,

$$ \begin{align*} \mathbf{A}_{\tilde{V}} = \mathbf{A}_{L_{k}(\mathfrak{g})}, \quad \mathbf{w}_{\tilde{V}}=0 \quad \text{ and }\quad \mathbf{g}_{\tilde{V}} = \mathbf{g}_{L_{k}(\mathfrak{g})}. \end{align*} $$

Then $\varphi $ factors through an isomorphism $L_k(\mathfrak {g})\cong V$ .

Proof. By Proposition 2.9, $\varphi $ is conformal. Hence, by Theorem 3.5, V is a direct sum of admissible representations with if k is principal (resp. if k is coprincipal). So we can write

with . In particular, V admits an asymptotic datum with $\mathbf {g}_V=\mathbf {g}_{L_k(\mathfrak {g})}=\mathbf {g}_{\tilde {V}}$ , $\mathbf {w}_V=0$ , (resp. ). Since and for all (resp. ), we get the first assertion. If, in addition, $\mathbf {A}_{\tilde {V}}=\mathbf {A}_{L_k(\mathfrak {g})}$ , then , which forces and all others are zero, whence the statement.

We now briefly discuss some relationships between primitive ideals, associated varieties and conformal dimensions. For a two-sided ideal I of $U(\mathfrak {g})$ , we denote $\operatorname {Var}(I)$ the zero locus of $\operatorname {gr}I\subset \operatorname {gr}U(\mathfrak {g})=\mathbb {C}[\mathfrak {g}^*]$ in $\mathfrak {g}^{*}$ . Now let k be an admissible number with denominator q. We distinguish the following weight:

(17)

By [Reference Arakawa, van Ekeren and Moreau15, Proposition 2.4],

(18)

and

Remark 3.12. Let k be an admissible number, and let such that . Since $\operatorname {Var}(I_k)=\overline {\mathbb {O}}_k$ by [Reference Arakawa9, Theorem 9.5] (recall (11)) and is a $\operatorname {Zhu}(L_k(\mathfrak {g}))$ -module (Theorem 3.3), we have

The connection with conformal dimensions is given by the following:

Proof. By [Reference Arakawa8], a simple $L_k(\mathfrak {g})$ -module which belongs to $\mathcal {O}_k$ is an irreducible highest weight module $L(\widehat {\mu })$ of highest weight .

First, we consider the principal admissible case $(q,r^{\vee })=1$ . We have, cf. Proposition 3.7 above and preceding remarks, $\widehat {\mu } = y \phi (\widehat {\nu }) - \widehat {\rho }$ where and $\nu \in P_{++}^{p}$ , where $y = \overline y t_{-\eta }$ with $\eta \in P_+^{\vee , q}$ .

The conformal weight of $L(\widehat \mu )$ is

$$\begin{align*}h_{L(\mu)} = \frac{(\mu | \mu + 2\rho)}{2(k+h^\vee_{\mathfrak{g}})} = \frac{|\mu + \rho|^2 - |\rho|^2}{2(k+h^\vee_{\mathfrak{g}})}. \end{align*}$$

Now we consider

$$ \begin{align*} |\mu + \rho|^2 = |\overline{y \phi(\nu)}|^2 = |\nu - \frac{p}{q} \eta|^2 = \frac{1}{q^2} |q\nu - p\eta|^2. \end{align*} $$

We show that $q\nu - p\eta \in P$ is a regular weight. Indeed, suppose for some and put and . Then $qm - pn = 0$ and so p divides m, but this is a contradiction since $0 < m < p$ because $\nu $ is regular.

The regular elements of P of minimal norm are $\rho $ and its images under the finite Weyl group, so the claim is proved in the principal admissible case. The coprincipal admissible case is similar. Since $(q, r^\vee ) = r^\vee $ , we have $q\nu \in qP \subset P^\vee $ , and hence, $q\nu - p\eta \in P^\vee $ . The regular elements of $P^\vee $ of minimal norm are $\rho ^\vee $ and its images under the finite Weyl group, thus proving the claim.

4 Asymptotic data of W-algebras

Let f be a nilpotent element of $\mathfrak {g}$ . Recall [Reference Kac, Roan and Wakimoto57] that a $\frac {1}{2}\mathbb {Z}$ -grading

(19) $$ \begin{align} \mathfrak{g} = \bigoplus_{j \in \frac{1}{2}\mathbb{Z}} \mathfrak{g}_\Gamma^j \end{align} $$

is called good for f if $f \in \mathfrak {g}_\Gamma ^{-1}$ , $\operatorname {\mathrm {ad}} f \colon \mathfrak {g}_\Gamma ^{j} \to \mathfrak {g}_\Gamma ^{j-1}$ is injective for $j \ge \frac {1}{2}$ and surjective for . The grading is called even if $\mathfrak {g}_\Gamma ^j = 0$ for $j\not \in \mathbb {Z}$ . By the Jacobson-Morosov Theorem, the nilpotent element f embeds into an ${\mathfrak {sl}}_2$ -triple $(e,h,f)$ , and $\mathfrak {g}$ thereby inherits a $\frac {1}{2}\mathbb {Z}$ -grading induced by the eigenvalues of $\operatorname {\mathrm {ad}}(h/2)$ . Such a grading is called a Dynkin grading. All Dynkin gradings are good, but not all good gradings are Dynkin.

Let $x^0_\Gamma $ be the semisimple element of $\mathfrak {g}$ defining the grading (19) – that is,

$$ \begin{align*}\mathfrak{g}_\Gamma^{i} = \{y \in \mathfrak{g} \colon [x^0_\Gamma,y] = i y\}.\end{align*} $$

We can assume that $x^0_\Gamma $ is contained in the Cartan subalgebra $\mathfrak {h}$ and that $\alpha (x^0_\Gamma ) \in \frac {1}{2}\mathbb {Z}_{\ge 0}$ for all $\alpha \in \Delta _+$ . For $j \in \frac {1}{2}\mathbb {Z}$ , we set

$$ \begin{align*}\Delta_\Gamma^{j} := \{\alpha \in \Delta \colon (\alpha | x^0_\Gamma ) = j \} =\{\alpha \in \Delta \colon \mathfrak{g}^\alpha \subset \mathfrak{g}_\Gamma^j \}, \end{align*} $$

where $\mathfrak {g}^\alpha $ is the $\alpha $ -root space.

Since the bilinear form $(x,y) \mapsto (f | [x,y])$ is non-degenerate on $\mathfrak {g}_\Gamma ^{1/2} \times \mathfrak {g}_\Gamma ^{1/2}$ , the set $\Delta _\Gamma ^{1/2}$ has even cardinality. We set $\Delta ^0_{\Gamma ,+}= \Delta _\Gamma ^0 \cap \Delta _{+}$ . It is a set of positive roots for $\Delta _\Gamma ^0$ .

We denote by $\mathscr {W}^k(\mathfrak {g},f)$ the universal affine W-algebra associated with $\mathfrak {g},f$ at level k and a good grading $\mathfrak {g}_\Gamma = \bigoplus _{j \in \frac {1}{2}\mathbb {Z}} \mathfrak {g}_\Gamma ^j$ :

$$ \begin{align*}\mathscr{W}^k(\mathfrak{g},f) = H_{DS,f}^{0}(V^k(\mathfrak{g})),\end{align*} $$

where $H_{DS,f}^\bullet (?)$ is the BRST cohomology functor of the quantized Drinfeld-Sokolov reduction associated with $\mathfrak {g},f$ ([Reference Feigin and Frenkel42, Reference Kac, Roan and Wakimoto57]). The BRST complex of a $\widehat {\mathfrak {g}}$ -module M takes the form

$$ \begin{align*} C^\bullet(M) = M \otimes F^{\text{ch}} \otimes F^{\text{ne}}, \end{align*} $$

where $F^{\text {ch}}$ denotes the Clifford vertex algebra associated with $\mathfrak {g}_\Gamma ^{>0} \oplus (\mathfrak {g}_\Gamma ^{>0})^*$ and its canonical symmetric bilinear form, or in the terminology of [Reference Kac, Roan and Wakimoto57] the charged free fermions vertex algebra associated with $\mathfrak {g}_\Gamma ^{>0} \oplus (\mathfrak {g}_\Gamma ^{>0})^*$ made into a purely odd vector superspace and canonical skew-supersymmetric form, and $F^{\text {ne}}$ denotes the neutral free fermion vertex algebra associated with $\mathfrak {g}_\Gamma ^{1/2}$ and bilinear form $(x,y) \mapsto (f | [x,y])$ . We omit the detailed definition, referring the reader to ([Reference Kac, Roan and Wakimoto57]).

The W-algebra $\mathscr {W}^k(\mathfrak {g},f)$ is conformal provided that $k\ne -h_{\mathfrak {g}}^{\vee }$ and its central charge $c_{\mathscr {W}^k(\mathfrak {g},f)}$ is given by

(20) $$ \begin{align} c_{\mathscr{W}^k(\mathfrak{g},f)}&=c_{V^k(\mathfrak{g})}-\operatorname{dim}G.f-\frac{3}{2}\dim \mathfrak{g}_{1/2}+24(\rho|x^0_{\Gamma})-12(k+h^{\vee}_{\mathfrak{g}})|x^0_{\Gamma}|^2 \end{align} $$

(21) $$ \begin{align}& =\dim \mathfrak{g}_\Gamma^0 -\dfrac{1}{2} \dim \mathfrak{g}_\Gamma^{1/2} -\dfrac{12}{k+h_{\mathfrak{g}}^\vee} | \rho - (k+h_{\mathfrak{g}}^\vee) x^0_\Gamma |^2\qquad \end{align} $$

([Reference Kac, Roan and Wakimoto57, Theorem 2.2]). Although the vertex algebra structure of $\mathscr {W}^k(\mathfrak {g},f)$ does not depend on the choice of the good grading of $\mathfrak {g}$ ([Reference Arakawa, Kuwabara and Malikov18, §3.2.5]), its conformal structure does. In general, we denote by $\mathscr {W}_k(\mathfrak {g},f)$ the unique simple graded quotient of $\mathscr {W}^k(\mathfrak {g},f)$ .

We now discuss the associated varieties of affine W-algebras. The associated variety $X_{\mathscr {W}^k(\mathfrak {g},f) }$ of the universal W-algebra is isomorphic to the Slodowy slice

$$ \begin{align*}\mathcal{S}_f=f+\mathfrak{g}^{e},\end{align*} $$

where $\mathfrak {g}^{e}$ is the centraliser of e in $\mathfrak {g}$ ([Reference De Sole and Kac31]). There is a natural $\mathbb {C}^*$ -action on $\mathcal {S}_f$ that contracts to f ([Reference Slodowy77]), and the associated variety $X_{\mathscr {W}_k(\mathfrak {g},f)}$ of the simple quotient is a $\mathbb {C}^*$ -invariant subvariety of $\mathcal {S}_f$ .

Regarding the relationship between the simple W-algebra $\mathscr {W}_k(\mathfrak {g},f)$ and the reduction $H_{DS,f}^0(L_k(\mathfrak {g}))$ of which it is a quotient, there is the following natural conjecture.

This has been proved in many cases. Indeed, we have the following:

Let k be an admissible level. By Theorems 3.2 and 4.3, $\mathscr {W}_k(\mathfrak {g},f)$ is lisse if $f\in {{\mathbb {O}}}_k$ and is quasi-lisse if $f\in \overline {{\mathbb {O}}}_k$ . Moreover, under the hypotheses of Theorem 4.5, the associated variety of $ \mathscr {W}_k(\mathfrak {g},f)$ is equal to the nilpotent Slodowy slice

(22) $$ \begin{align} \mathcal{S}_{{\mathbb{O}}_k,f}:=\mathcal{S}_f\cap \overline{{\mathbb{O}}}_k. \end{align} $$

Conjecture 4.6 has been settled positively by different methods in a number of cases – namely, for f principal [Reference Arakawa9], for $\mathfrak {g}$ of type A [Reference Arakawa and van Ekeren14], for $\mathfrak {g}$ of type $ADE$ and f subregular [Reference Arakawa and van Ekeren14], for $\mathfrak {g}$ of type $B_2$ and f subregular [Reference Fasquel40], for $\mathfrak {g}$ of type $B_n$ and f subregular [Reference Creutzig and Linshaw33], for $\mathfrak {g}$ of type $C_n$ and f minimal [Reference Creutzig and Linshaw33]. More recently, a general proof has been announced in [Reference McRae71].

By [Reference De Sole and Kac34], the associative algebra $\operatorname {Zhu}(\mathscr {W}^k(\mathfrak {g},f))$ is naturally isomorphic to the finite W-algebra [Reference Premet75] $U(\mathfrak {g},f)$ associated with $(\mathfrak {g},f)$ . More generally, we have the following assertion.

Theorem 4.7 [Reference Arakawa9].

There exists an isomorphism

$$ \begin{align*}\operatorname{Zhu}(H_{DS,f}^{0}(L_k(\mathfrak{g}))\cong H_{DS,f}^{0}(\operatorname{Zhu}(L_{k}(\mathfrak{g})),\end{align*} $$

where on the right-hand side $H_{DS,f}^{0}(?)$ is the finite-dimensional analogue of the Drinfeld-Sokolov reduction functor that is denoted by $M\mapsto M_{\dagger }$ in [Reference Losev69].

By [Reference Ginzburg48, Reference Losev70], see also [Reference Arakawa9, Section 2], we have the following equivalence:

(23) $$ \begin{align} H_{DS,f}^{0}(U(\mathfrak{g})/I)\ne 0\iff G.f\subset \operatorname{Var}(I). \end{align} $$

From this, together with Theorems 3.4 and 4.7, we obtain the following:

Corollary 4.8. Let k be an admissible number. We have

where the product is taken over such that .

The image $[\omega ]$ of the conformal vector of $\mathscr {W}^k(\mathfrak {g},f)$ in $\operatorname {Zhu}(H_{DS,f}^{0}(L_k(\mathfrak {g}))$ acts on as the constant multiplication by

(24)

In particular, a simple module of $H_{DS,f}^{0}(L_k(\mathfrak {g}))$ corresponding to a simple module of via Corollary 4.8 has the conformal dimension .

For a $V^k(\mathfrak {g})$ -module M, we consider the character of its reduction – that is, we set

$$ \begin{align*} \chi_{H^{\bullet}_{DS, f} (M)}(\tau,z)=\sum_{i\in {\mathbb{Z}}}(-1)^i{\mathrm{tr}}_{H^{i}_{DS, f}( M)} (e^{2\pi {\mathrm{i}} z}q^{L_0 - c/24}), \quad (\tau,z)\in {\mathbb{H}}\times {\mathfrak{h}}^{f} \end{align*} $$

when it is well defined. Here,

$$\begin{align*}L_0 = L_0^{\text{Sug}} - (x_\Gamma^0)_0 + L_0^{\text{ch}} + L_0^{\text{ne}}. \end{align*}$$

Note that is well defined for finitely generated objects M of $\operatorname {KL}_k$ , by Theorem 4.3, and we have

$$\begin{align*}\chi_{H^{\bullet}_{DS, f} (M)}(\tau,z) = {\mathrm{tr}}_{H^0_{DS, f}( M)} (e^{2\pi {\mathrm{i}} z}q^{L_0 - c/24}). \end{align*}$$

Proposition 4.9. Let $M\in \mathcal {O}_k$ be a $V^k(\mathfrak {g})$ -module whose character has asymptotic behaviour

$$ \begin{align*}\chi_M(\tau,-\tau z,0)\sim \mathbf{A}_M(z) (-{\mathrm{i}} \tau)^{\mathbf{w}_M}e^{\frac{\pi {\mathrm{i}} }{12 \tau}\mathbf{g}_M},\end{align*} $$

and suppose that $\chi _{H^{\bullet }_{DS, f} (M)}(\tau ,z)$ is well defined. Then

$$ \begin{align*} \chi_{H^{\bullet}_{DS, f} (M)}(\tau,-\tau z)\sim \mathbf{A}_{H^{\bullet}_{DS, f} (M)}(z)(-{\mathrm{i}} \tau)^{\mathbf{w}_M}e^{\frac{\pi {\mathrm{i}} }{12 \tau}(\mathbf{g}_M-\dim \mathbb{O}_f)}, \end{align*} $$

where

Proof. Let us write

$$ \begin{align*} \chi_M(\tau,z,t)=\frac{(A_{\widehat{\rho}}\chi_M)(\tau,z,t)}{A_{\widehat{\rho}}(\tau,z,t)}. \end{align*} $$

The asymptotic behaviour of $A_{\widehat {\rho }}(\tau ,z,t)$ is well known [Reference Kac54, Proposition 13.13] to be

(25) $$ \begin{align} A_{\widehat{\rho}}(\tau, -\tau z, 0) \sim b(\rho, z)(-{\mathrm{i}} \tau)^{-\ell/2} e^{-\frac{\pi{\mathrm{i}} }{12\tau} \dim(\mathfrak{g}),} \end{align} $$

where

$$ \begin{align*} b(\rho, z) = \prod_{\alpha \in \Delta_+} 2 \sin\ {\pi (\alpha | z)}. \end{align*} $$

Hence,

(26) $$ \begin{align} &\chi_M(\tau,-\tau z,0)\sim \mathbf{A}_M(z) (-{\mathrm{i}} \tau)^{\mathbf{w}_M}e^{\frac{\pi {\mathrm{i}} }{12 \tau}\mathbf{g}_M} \\ &\iff (A_{\widehat{\rho}}\chi_M)(\tau,-\tau z,0)\sim b(\rho,z)A(\rho,z) (-{\mathrm{i}} \tau)^{\mathbf{w}_M-\frac{\ell}{2}} e^{\frac{\pi {\mathrm{i}} }{12 \tau}(\mathbf{g}_M-\dim \mathfrak{g})}. \nonumber \end{align} $$

Now by Theorem 4.3 and the Euler-Poincaré principle (see the discussion in [Reference Kac and Wakimoto61, Section 2]),

Here, $L_0^{\text {Sug}}$ denotes the zero mode of the Sugawara conformal vector in $V^k(\mathfrak {g})$ while $L_0^{\text {ch}}$ and $L_0^{\text {ne}}$ denote the conformal vectors in $F^{\text {ch}}$ and $F^{\text {ne}}$ given in [Reference Kac, Roan and Wakimoto57, Section 2.2], or in [Reference Kac and Wakimoto61, Section 2.1]. We thus have

(27)

in which

We now describe the proof of (27) briefly, referring the reader to [Reference Kac and Wakimoto61, Section 2.2] for details. The numerator of (27) comes directly from the numerator of the expression (13) for . The modification $z \mapsto z - \tau x_\Gamma ^0$ corresponds to the shift of Virasoro operator $L_0^{\text {Sug}} \mapsto L_0^{\text {Sug}} - (x_\Gamma ^0)_0$ . The Weyl denominator is $\eta (\tau )^\ell $ times a product of theta functions indexed by $\alpha \in \Delta _+$ . At the level of characters, the effect of the tensor product with $F^{\text {ch}}$ is to cancel those factors associated with $\alpha \in \Delta _{\Gamma }^j$ for $j> 0$ . The tensor product with $F^{\text {ne}}$ reintroduces those factors associated with $\alpha \in \Delta _{\Gamma , +}^{1/2}$ (whose cardinality is half that of $\Delta _{\Gamma }^{1/2}$ ). Ultimately, this yields the denominator $\psi (\tau , z - \tau x_\Gamma ^0, t)$ of (27).

The function $\psi $ satisfies the asymptotic

(28) $$ \begin{align} \psi(\tau, -\tau z, 0) \sim \prod_{\alpha \in \Delta_{\Gamma, +}^{0} \cup \Delta^{1/2}_{\Gamma, +}} 2 \sin\ \pi(\alpha | z) (-{\mathrm{i}} \tau)^{-\ell/2} e^{\pi {\mathrm{i}} \dim(\mathfrak{g}^f) / 12\tau} \end{align} $$

as $\tau \downarrow 0$ . The asymptotic behaviour of the numerator $A_{\nu +\widehat {\rho }}(\tau , z - \tau x_\Gamma ^0, t)$ was established in the proof of Proposition 3.8. The required asymptotic follows from (26), (27) and (28), together with the fact that

$$ \begin{align*} \dim \overline{\mathbb{O}}_f=\dim \mathfrak{g}-\dim \mathfrak{g}^f. \end{align*} $$

Proposition 4.10. Let $k =-h_{\mathfrak {g}}^\vee +\dfrac {p}{q}$ be an admissible level, and let $f \in X_{L_k(\mathfrak {g})}$ . Then admits an asymptotic datum for all simple ordinary $L_k(\mathfrak {g})$ -modules with . Moreover,

1. 1. ([Reference Kac and Wakimoto61]) For ,

   

2. 2. For ,

   

In particular,

for any simple ordinary $L_k(\mathfrak {g})$ -module .

Proof. Let k be coprincipal and let . By Proposition 3.8 and Proposition 4.9, we have

(29)

where $\nu $ , $\bar y$ are related with as in Proposition 3.8.

We are interested in the case which corresponds to $\overline y = 1$ , $\beta = 0$ and . We now simplify the products appearing in the asymptotic above to

The second product here becomes simply $2^{-|\Delta _{\Gamma }^{1/2}|/2}$ . In the third product, the terms $x_\Gamma ^0$ are irrelevant since $\alpha \in \Delta _{\Gamma , +}^0$ . In the limit $z \rightarrow 0$ , this product reduces, by l’Hopital’s rule, to

$$ \begin{align*} \prod_{\alpha \in \Delta_{\Gamma, +}^{0}} \frac{|\alpha^\vee|/q}{|\alpha|} = \frac{(r^\vee)^{|\Delta_{\Gamma, +}^0 \cap \Delta^{\text{short}}|}}{q^{|\Delta_{\Gamma, +}^0|}}. \end{align*} $$

In the fourth product, the limit obtains by simply putting $z=0$ . Thus, we have obtained the stated formula.

Proof. First, we treat (1). By Theorem 3.11, , which contains the orbit $G.f$ by the assumption. Hence, is nonzero by (23). It follows from Corollary 4.8 that there exists a simple $\mathscr {W}_k(\mathfrak {g},f)$ -module, corresponding to a simple -module, which has the minimal conformal dimension among simple positive energy representations of $\mathscr {W}_k(\mathfrak {g},f)$ by Proposition 3.13 and (24). Next, we have

where $\Gamma $ is a good even grading. Hence, the assertion follows from (21) and the fact that $\dim \mathfrak {g}_{1/2}=0$ , which is true by hypothesis.

Now we turn to (2). By [Reference Arakawa and van Ekeren14], each factor in Corollary 4.8 is a simple algebra, which has a unique simple module. Hence, the assertion follows from Remark 3.14.

In view of Proposition 2.4, Theorem 4.11 gives a supporting evidence for Conjecture 4.6.

5 Collapsing levels for W-algebras

In general, if $\mathfrak {s}$ is a semisimple Lie algebra, we write $\kappa _{\mathfrak {s}}$ for the Killing form of $\mathfrak {s}$ . For now, we assume that $\mathfrak {s}$ is simple. We denote by $h_{\mathfrak {s}}^\vee $ its dual Coxeter number and by $(~|~)_{\mathfrak {s}}$ the normalised inner product $\dfrac {1}{2h_{\mathfrak {s}}^\vee } \kappa _{\mathfrak {s}}$ so that $(\theta | \theta )=2$ for $\theta $ the highest positive root of $\mathfrak {s}$ .

As in Section 4, we fix an ${\mathfrak {sl}}_2$ -triple $(e,h,f)$ of $\mathfrak {g}$ , with related notation. In particular, for $j \in \frac {1}{2}\mathbb {Z}$ , recall that

$$ \begin{align*}\mathfrak{g}^{j} = \{x \in \mathfrak{g} \colon [h,x] = 2 j x\},\end{align*} $$

and $\mathfrak {g}^0$ is the centraliser of h in $\mathfrak {g}$ . The centraliser $\mathfrak {g}^\natural $ of the ${\mathfrak {sl}}_2$ -triple $(e,h,f)$ in $\mathfrak {g}$ is given by

(30) $$ \begin{align} \mathfrak{g}^\natural = \mathfrak{g}^0 \cap \mathfrak{g}^{f} = \mathfrak{g}^{e}\cap \mathfrak{g}^{f}. \end{align} $$

The Lie algebra $\mathfrak {g}^\natural $ is a reductive subalgebra of $\mathfrak {g}$ . The centraliser $G^\natural $ of the ${\mathfrak {sl}}_2$ -triple $(e,h,f)$ in G is a maximal reductive subgroup of the centraliser of f in G whose Lie algebra is $\mathfrak {g}^\natural $ . We can write $\mathfrak {g}^\natural = \mathfrak {g}_0^\natural \oplus [\mathfrak {g}^\natural ,\mathfrak {g}^\natural ]$ , with $\mathfrak {g}_0^\natural $ the centre of the reductive Lie algebra $\mathfrak {g}^\natural $ . Denoting by the simple factors of $[\mathfrak {g}^\natural ,\mathfrak {g}^\natural ]$ , we get

$$ \begin{align*}\mathfrak{g}^\natural=\bigoplus_{i=0}^s \mathfrak{g}^\natural_i.\end{align*} $$

More generally, for $\mathfrak {g}_\Gamma = \bigoplus _{j \in \frac {1}{2}\mathbb {Z}} \mathfrak {g}_\Gamma ^j$ a good grading for f, we set

(31) $$ \begin{align} \mathfrak{g}_\Gamma^\natural:=\mathfrak{g}_\Gamma^{0} \cap \mathfrak{g}^f. \end{align} $$

We note that $\mathfrak {g}_\Gamma ^\natural $ is not a reductive Lie algebra in general. In fact, $\mathfrak {g}^\natural $ is a maximal reductive subalgebra of $\mathfrak {g}_\Gamma ^\natural $ . Indeed, by [Reference Elashvili, Kac and Vinberg38, Lemma 1.2], $\mathfrak {g}^\natural $ is a maximal reductive subalgebra of $\mathfrak {g}^f$ while, on the other hand, since $\mathfrak {g}^\natural $ is reductive and contained in $\bigoplus _{j \ge 0} \mathfrak {g}_\Gamma ^j$ , we have $\mathfrak {g}^\natural \subset \mathfrak {g}_\Gamma ^0$ .

Let . For any element $x \in \mathfrak {g}_i^\natural \subset \mathfrak {g}_\Gamma ^\natural $ , the adjoint action $\operatorname {\mathrm {ad}} x$ restricts to an endomorphism of $\mathfrak {g}_\Gamma ^{j}$ , which we denote $\rho _x^{j}$ , for any j. Setting $\kappa _{\mathfrak {g}_\Gamma ^{j}}(x,x)=\mathrm {tr}(\rho _x^{j} \circ \rho _x^{j} )$ , one defines an invariant bilinear form on $\mathfrak {g}_{i}^\natural $ by (see [Reference Kac and Wakimoto60, Theorem 2.1] for the case where $\Gamma $ is the Dynkin grading)

$$ \begin{align*} \phi_{\Gamma,i}^\natural(x,x) := k (x|x)_{\mathfrak{g}} + \dfrac{1}{2} (\kappa_{\mathfrak{g}}(x,x) - \kappa_{\mathfrak{g}_\Gamma^0}(x,x)-\kappa_{\mathfrak{g}_\Gamma^{1/2}}(x,x)). \end{align*} $$

Thus, for $i \not =0$ , there exists a scalar $k_{\Gamma ,i}^\natural $ such that

(32) $$ \begin{align} \phi_{\Gamma,i}^\natural = k_{\Gamma,i}^\natural (~|~)_{i} \quad \text{where}\quad (~|~)_i :=(~|~)_{\mathfrak{g}_i^\natural}. \end{align} $$

Note that $V^{\phi _{\Gamma ,0}^\natural } (\mathfrak {g}_0^\natural ) \cong M(1)^{\otimes \mathrm {rank}\, \phi _{\Gamma ,0}^\natural }$ , where $M(1)$ is the Heisenberg vertex algebra of central charge 1.

Following [Reference Brundan and Goodwin30], we say that two good gradings $\Gamma ,\Gamma '$ are adjacent if

where $i^-$ denotes the largest half-integer strictly smaller than i and $i^+$ denotes the smallest half-integer strictly greater than i.

Proof. According to [Reference Brundan and Goodwin30, Theorem 2 and Lemma 26], there are Lagrangian subspaces $\mathfrak {l}_{\Gamma }$ and $\mathfrak {l}_{\Gamma '}$ of $\mathfrak {g}_{\Gamma }^{1/2}$ and $\mathfrak {g}_{\Gamma '}^{1/2}$ , respectively, with respect to the non-degenerate bilinear form $ (f | [ \cdot ,\cdot ])$ such that

$$ \begin{align*} \mathfrak{l}_\Gamma \oplus \mathfrak{g}_\Gamma^ {>1/2} = \mathfrak{l}_{\Gamma'} \oplus \mathfrak{g}_{\Gamma'}^ {>1/2} \quad \text{ and }\quad \mathfrak{l}_\Gamma^- \oplus \mathfrak{g}_\Gamma^ {<-1/2} = \mathfrak{l}_{\Gamma'}^- \oplus \mathfrak{g}_{\Gamma'}^ {<-1/2}, \end{align*} $$

where $\mathfrak {a}^-$ denotes the dual space to the subspace $\mathfrak {a} \subset \mathfrak {g}$ with respect to the Killing form of $\mathfrak {g}$ . Then for any $x \in \mathfrak {g}^\natural $ ,

$$ \begin{align*} \kappa_{\mathfrak{g}}(x,x) - \kappa_{\mathfrak{g}_{\Gamma}^0}(x,x)-\kappa_{\mathfrak{g}_{\Gamma}^{1/2}}(x,x) & =2 \kappa_{\mathfrak{l}_\Gamma \oplus \mathfrak{g}_\Gamma^ {>1/2}}(x,x) \\ & = \kappa_{\mathfrak{g}}(x,x) - \kappa_{\mathfrak{g}_{\Gamma'}^0}(x,x)-\kappa_{\mathfrak{g}_{\Gamma'}^{1/2}}(x,x). \end{align*} $$

Indeed, from the decompositions

$$ \begin{align*} \mathfrak{g} & = (\mathfrak{l}_\Gamma^- \oplus \mathfrak{g}_\Gamma^ {<-1/2}) \oplus (\mathfrak{k}_{\Gamma}^- \oplus \mathfrak{g}_{\Gamma}^0 \oplus \mathfrak{k}_{\Gamma}) \oplus (\mathfrak{l}_\Gamma \oplus \mathfrak{g}_\Gamma^ {>1/2}), \\ \mathfrak{g} & = (\mathfrak{l}_{\Gamma'}^- \oplus \mathfrak{g}_{\Gamma'}^ {<-1/2}) \oplus (\mathfrak{k}_{\Gamma'}^- \oplus \mathfrak{g}_{\Gamma'}^0 \oplus \mathfrak{k}_{\Gamma'}) \oplus (\mathfrak{l}_{\Gamma'} \oplus \mathfrak{g}_{\Gamma'}^ {>1/2}), \end{align*} $$

where $\mathfrak {k}_\Gamma $ (resp. $\mathfrak {k}_{\Gamma '}$ ) is a Lagrangian complement in $\mathfrak {g}_\Gamma ^{1/2}$ (resp. $\mathfrak {g}_{\Gamma '}^{1/2}$ ) to $\unicode{x0142} _\Gamma $ (resp. $\mathfrak {l}_{\Gamma '}$ ), we deduce that

$$ \begin{align*} \kappa_{\mathfrak{g}_{\Gamma}^{1/2}}(x,x) = \kappa_{\mathfrak{k}_\Gamma \oplus \mathfrak{k}_\Gamma^{-}}(x,x) \quad \text{ and } \quad \kappa_{\mathfrak{g}_{\Gamma'}^{1/2}}(x,x) = \kappa_{\mathfrak{k}_{\Gamma'} \oplus \mathfrak{k}_{\Gamma'}^{-}}(x,x). \end{align*} $$

This completes the proof.

Denoting by $\Gamma _D$ the Dynkin grading, we set

$$ \begin{align*}\phi_i^\natural:= \phi_{\Gamma_D,i}^\natural, \quad \text{ and }\quad k_i^\natural:= k_{\Gamma_D,i}^\natural \quad \text{ for } \quad i \not=0. \end{align*} $$

Set

(33) $$ \begin{align} V^{k^\natural}(\mathfrak{g}^\natural) := V^{\phi_0^\natural} (\mathfrak{g}_0^\natural) \otimes \bigotimes_{i=1}^s V^{k_i^\natural} (\mathfrak{g}_i^\natural). \end{align} $$

By [Reference Kac and Wakimoto60, Theorem 2.1], there exists an embedding

(34)

of vertex algebras. We have

(35) $$ \begin{align} \iota(\omega_{V^{k^\natural}(\mathfrak{g}^\natural)})\in \mathscr{W}^k(\mathfrak{g},f)_2,\quad (\omega_{\mathscr{W}^k(\mathfrak{g},f)})_{(2)}\iota(\omega_{V^{k^\natural}(\mathfrak{g}^\natural)})=0. \end{align} $$

We denote by $\mathcal {V}^k(\mathfrak {g}^\natural )$ the image in $\mathscr {W}^k(\mathfrak {g},f)$ of the embedding $\iota $ , and by $\mathcal {V}_k(\mathfrak {g}^\natural )$ the image of $\mathcal {V}^k(\mathfrak {g}^\natural )$ by the canonical projection $\pi \colon \mathscr {W}^k(\mathfrak {g},f) \twoheadrightarrow \mathscr {W}_k(\mathfrak {g},f)$ .

Lemma 5.4. The level k is collapsing if and only if

$$ \begin{align*}\mathscr{W}_k(\mathfrak{g},f) \cong L_{k^\natural}(\mathfrak{g}^\natural),\end{align*} $$

where $L_{k^\natural }(\mathfrak {g}^\natural )$ stands for $\displaystyle {\bigotimes _{i=0}^s L_{k_i^\natural } (\mathfrak {g}_i^\natural )}$ . Equivalently, k is collapsing for $\mathscr {W}^k(\mathfrak {g},f)$ if and only if there exists a surjective vertex algebra homomorphism

$$ \begin{align*} \mathscr{W}^k(\mathfrak{g},f)\twoheadrightarrow L_{k^\natural}(\mathfrak{g}^\natural). \end{align*} $$

For example, if $ \mathscr {W}_k(\mathfrak {g},f) \cong \mathbb {C}$ , then k is collapsing.

We close this section with a lemma which is used above in the proof of Proposition 4.2.

Proof. As previously mentioned, $\mathfrak {g}^\natural $ is a maximal reductive subalgebra of $\mathfrak {g}^{f}$ ([Reference Elashvili, Kac and Vinberg38, Lemma 1.2]). Let $\mathfrak {t}$ be a maximal torus of $\mathfrak {g}^{0}=\mathfrak {g}^{h}$ containing a maximal torus of $\mathfrak {g}^\natural $ , and set

$$ \begin{align*}\mathfrak{t}^{\natural}:=\mathfrak{t} \cap \mathfrak{g}^\natural= \mathfrak{t}\cap \mathfrak{g}^{f}.\end{align*} $$

We intend to show that $\mathfrak {z}(\mathfrak {g}^\natural )$ is contained in $\mathfrak {t}^{\natural }$ .

We use the decomposition in $\mathfrak {t}^{\natural }$ -weight spaces of $\mathfrak {g}^{\natural }$ following [Reference Brundan and Goodwin30, Section 2]. For $\alpha \in (\mathfrak {t}^{\natural })^*$ and $n \ge 0$ , let $L(\alpha ,n)$ denote the irreducible $\mathfrak {t}^{\natural }\oplus \mathfrak {s}$ -modules of dimension $(n+1)$ on which $\mathfrak {t}^{\natural }$ acts by the weight $\alpha $ , where $\mathfrak {s}\cong {\mathfrak {sl}}_2$ is the Lie algebra generated by $e,h,f$ . Since each $L(\alpha ,n)$ contains a nonzero vector annihilated by f, the set of weights of $\mathfrak {t}^{\natural }$ on $\mathfrak {g}$ is also the set of weights of $\mathfrak {t}^\natural $ on $\mathfrak {g}^{f}$ . Let $\Phi _f \subset (\mathfrak {t}^{\natural })^*$ be the set of all nonzero weights of $\mathfrak {t}^{\natural }$ on $\mathfrak {g}^f$ . We have the following decomposition:

(36)

where $\mathfrak {g}^{\mathfrak {t}^{\natural }}$ is the centraliser of $\mathfrak {t}^{\natural }$ in $\mathfrak {g}$ (it is a Levi subalgebra of $\mathfrak {g}$ ) and

$$ \begin{align*}\mathfrak{g}^f (\alpha,i)= \{x \in \mathfrak{g}^f \colon [h,x]=i x \text{ and } [t,x] = \alpha(t)x \text{ for all } t \in \mathfrak{t}^{\natural}\}.\end{align*} $$

This decomposition is compatible with the decomposition $\mathfrak {g}^f = \mathfrak {g}^\natural \oplus \mathfrak {g}^{f}_{<0}$ , and we have

where $\Phi _f^\circ $ denotes the set of all nonzero element of $\Phi _f$ .

Let now $x \in \mathfrak {z}(\mathfrak {g}^\natural )$ that we write $x=x_{0}+\sum _{\alpha \in \Phi _f^\circ } x_\alpha $ relatively to the above decomposition, with $x_0 \in \mathfrak {t}^{\natural } $ and $x_\alpha \in \mathfrak {g}^f (\alpha ,0)$ for $\alpha \in \Phi _f^\circ $ . Let $\alpha ' \in \Phi _f^\circ $ and pick $t' \in \mathfrak {t}^{\natural }$ such that $\alpha '(t')\not =0$ . From the equalities

$$ \begin{align*} 0 &= [y,x] = [y,x_0]+\sum_{\alpha \in \Phi_f^\circ} [y,x_\alpha] = 0 + \sum_{\substack{\alpha \in \Phi_f^\circ \\ \alpha \not=\alpha'}} \alpha(t) x_\alpha + \alpha'(t') x_{\alpha'}, \end{align*} $$

we deduce that $x_{\alpha '}=0$ . Since this is true for each $\alpha \in \Phi _f^\circ $ , we get that $x=x_0 \in \mathfrak {t}^{\natural }$ . In particular, x is a semisimple elements of $\mathfrak {g}$ .

To show the second assertion, first note that we have the following decomposition:

(37) $$ \begin{align} \mathfrak{g}= \bigoplus_{\alpha \in \Phi_f} \bigoplus_{n \ge 0} m(\alpha,n) L(\alpha,n), \end{align} $$

where $m(\alpha ,n) \not =0$ for all $\alpha \in \Phi _f$ . Since $\Phi _f$ is a restricted root system in the sense of [Reference Brundan and Goodwin30, Section 2], for $\alpha \in (\mathfrak {t}^\natural )^*$ , $\alpha \in \Phi _f$ implies that $-\alpha \in \Phi ^f$ and the corresponding root vector spaces have the same dimension. So the second assertion follows from the decompositions (36) and (37).

6 The main strategy

We describe in this section our general strategy to find new (admissible) collapsing levels using Theorem 3.10. The main difficulty is to find potential candidates for k and f. We explain below our strategy to achieve this.

6.1 Associated variety

If k is collapsing for $\mathscr {W}^k(\mathfrak {g},f)$ , then certainly

(38) $$ \begin{align} X_{\mathscr{W}_k(\mathfrak{g},f)} \cong X_{L_{k^\natural}(\mathfrak{g}^\natural)} \end{align} $$

as Poisson varieties. As we saw in Theorem 4.3, the vertex algebra $\mathscr {W}_k(\mathfrak {g},f)$ is quasi-lisse whenever $f \in \overline {\mathbb {O}}_k$ .

Lemma 6.1. Suppose that $\mathscr {W}_k(\mathfrak {g},f)$ is quasi-lisse. If k is collapsing, then $\phi _0^\natural $ is identically zero on $\mathfrak {g}_0^\natural $ . In particular, if k is admissible and $f\in \overline {\mathbb {O}}_k$ , then k can only be collapsing if $\phi _0^\natural =0$ .

Our convention is that $\phi _0^\natural =0$ when $\mathfrak {g}_0^\natural =\{0\}$ .

Proof. The associated variety $X_{L_{k^\natural }(\mathfrak {g}^\natural )} $ is a subvariety of $(\mathfrak {g}_0^\natural )^*\times (\mathfrak {g}_1^\natural )^*\times \dots \times (\mathfrak {g}_s^\natural )^*$ , and the symplectic leaves of $(\mathfrak {g}_0^\natural )^*\times (\mathfrak {g}_1^\natural )^*\times \dots \times (\mathfrak {g}_s^\natural )^*$ are the coadjoint orbits of $G_0^\natural \times \cdots \times G_s^\natural $ , where $G_i^\natural $ is the adjoint group of $\mathfrak {g}_i^\natural $ . Recall that $V^{\phi _0^{\natural }}(\mathfrak {g}_0^\natural )$ is a Heisenberg vertex algebra of rank $\dim \mathfrak {g}_0^\natural $ . The associated variety of its simple quotient is $\mathbb {C}^{ \mathrm {rank}\,\phi _0^\natural }$ , provided that $\phi _0^{\natural }\not =0$ . Hence, $X_{L_{k^\natural }(\mathfrak {g}^\natural )}$ has finitely many symplectic leaves if and only if it is contained in $\mathcal {N}_{\mathfrak {g}_1}\times \dots \times \mathcal {N}_{\mathfrak {g}_s}$ , where $\mathcal {N}_{\mathfrak {g}_i}$ is the nilpotent cone of $\mathfrak {g}_i^\natural $ . In particular, we must have $X_{L_{\phi _0^{\natural }}(\mathfrak {g}_0^\natural )}=\{0\}$ . This happens if and only if $\phi _0^{\natural }=0$ .

We are thus led to consider levels k for which $\phi _0^\natural = 0$ (further conditions on k will appear later). Now, for each $i>0$ , the bilinear form $\phi ^\natural _i$ equals the standard form on the simple component $\mathfrak {g}_i^\natural $ , times a well-defined level $k_i^\natural $ , as per (32). These levels $k_i^\natural $ can then be expressed as functions of the level k, as is done below in Tables 2, 3 and 4 for classical types, and Tables 11–17 for exceptional types. In the tables, we also list $k_0^\natural $ for those cases in which $\mathfrak {g}_0^\natural \neq 0$ , and since there is no standard normalised form on $\mathfrak {g}_0^\natural $ , we now explain what we mean by this. If the centre $\mathfrak {g}_0^\natural $ is one-dimensional, then we choose nonzero $x\in \mathfrak {g}_0^\natural $ and set $k_{0}^\natural = \phi _{0}^\natural (x,x)$ . Then $k_0^\natural $ is well defined, as a function of k, up to multiplication by a nonzero scalar. This is sufficient for our purposes since, by Lemma 6.1, we require $k_0^\natural =0$ . For the vast majority of pairs $(\mathfrak {g}, f)$ relevant for us, we have in fact. Only two exceptions occur (one in type $E_6$ and one in type $E_7$ ), and in both these cases, $\mathfrak {g}^\natural =\mathfrak {g}_0^\natural \cong \mathbb {C}^2$ . These cases are easily handled separately, and in the tables, we abuse notation by also writing $k_{0}^\natural $ for the value of $\phi _{0}^\natural (x,x)$ for some nonzero $x\in \mathfrak {g}_0^\natural $ .

We now call $k^\natural $ admissible if $k_0^\natural =0$ and $k_i^\natural $ is admissible for $\mathfrak {g}_i^\natural $ , for each .

Remark 6.2. It may happen that k is admissible while $k^\natural $ is not, even if $\mathfrak {g}_0^\natural =0$ . For example, for $\mathfrak {g}=G_2$ , f in the nilpotent orbit labelled $\tilde {A}_1$ (of dimension 8) and $k = -4+4/7$ which is a principal admissible level for $G_2$ , then $k^{\natural } = k +3/2=-2+1/14$ is not admissible for $\mathfrak {g}^\natural \cong {\mathfrak {sl}}_2$ (see Table 11). However, under additional conditions on f and k, we expect that $k^\natural $ is admissible if k is admissible. See Conjecture 6.4 below.

Recall

$$ \begin{align*} \mathcal{S}_{{\mathbb{O}}_k,f}=\mathcal{S}_f\cap \overline{{\mathbb{O}}}_k, \end{align*} $$

the nilpotent Slodowy slice. Now we assume k and $k^\natural $ are both admissible, where $f\in \overline {\mathbb {O}}_k$ , and we assume that $\mathscr {W}_k(\mathfrak {g},f)=H_{DS,f}^0(L_k(\mathfrak {g}))$ (verified if f has a good even grading, and in general if Conjecture 4.4 holds). Then according to Theorem 3.2 and Theorem 4.3,

(39) $$ \begin{align} \mathcal{S}_{{\mathbb{O}}_k,f}= \overline{{\mathbb{O}}}_{k_1^\natural} \times \cdots \times \overline{{\mathbb{O}}}_{k_s^\natural}, \end{align} $$

where are nilpotent orbits in , respectively. This motivates the following definition:

Definition 6.3. We say that a nilpotent Slodowy slice $\mathcal {S}_{{\mathbb {O}},f}$ is collapsing if $\mathcal {S}_{{\mathbb {O}},f}$ is isomorphic to a product of nilpotent orbit closures in $\mathfrak {g}^\natural $ .

Based on the above analysis, we consider pairs $({\mathbb {O}},{\mathbb {O}}')$ such that

1. 1. $\overline {{\mathbb {O}}}=\overline {{\mathbb {O}}}_k$ is the associated variety of some simple admissible affine vertex algebra $L_k(\mathfrak {g})$ ,

2. 2. ${\mathbb {O}}' \subset \overline {{\mathbb {O}}}$ ,

3. 3. for $f \in {\mathbb {O}}'$ , $\mathcal {S}_{{\mathbb {O}},f}$ is a collapsing nilpotent Slodowy slice which is the associated variety X of the simple admissible affine vertex algebra $L_{k^\natural }(\mathfrak {g}^\natural )$ . (In particular, X is a product of nilpotent orbit closures in $\mathfrak {g}^{\natural }$ .)

The nilpotent orbit ${\mathbb {O}}_k$ , for admissible $k= -h_{\mathfrak {g}}^\vee +p/q$ , is given by [Reference Arakawa10, Tables 2–10] and depends only on the denominator q: it is described in terms of the corresponding partition of n for $\mathfrak {g}$ simple of classical type – that is, $\mathfrak {g}={\mathfrak {sl}}_n$ , $\mathfrak {sp}_n$ or ${\mathfrak {so}}_n$ – and in the Bala-Carter classification for $\mathfrak {g}$ simple of exceptional type.

Motivated by the determination of generic singularities of the nilpotent closures $\overline {{\mathbb {O}}}$ in simple Lie algebras $\mathfrak {g}$ of classical types, Kraft and Procesi [Reference Kraft and Procesi64, Reference Kraft and Procesi66] described the smooth equivalences of singularities between Slodowy slices $\mathcal {S}_{{\mathbb {O}}_1,f_1}$ and $\mathcal {S}_{{\mathbb {O}}_2,f_2}$ , where $\mathcal {S}_{{\mathbb {O}}_2,f_2}$ is obtained from $\mathcal {S}_{{\mathbb {O}}_1,f_1}$ by the row/column removal rule (see Lemmas 8.2 and 9.2). It turns out that these smooth equivalences actually yield isomorphisms of varieties; see [Reference Li68, Proposition 7.3.2].Footnote ⁴ Thus, the row/column removal rule, combined with [Reference Arakawa10], yields many pairs $({\mathbb {O}},{\mathbb {O}}')$ satisfying the conditions above in classical types. We consider all such pairs.

For the simple Lie algebras of exceptional types, the authors of [Reference Fu, Juteau, Levy and Sommers47] determine the isomorphism type of most of the Slodowy slices $\mathcal {S}_{{\mathbb {O}},f}$ for which $G.f$ is a minimal degeneration of ${\mathbb {O}}$ (i.e., those for which $G.f$ is a maximal orbit in $\overline {{\mathbb {O}}} \setminus {\mathbb {O}}$ ) and also for some of the Slodowy slices for which it is not a minimal degeneration. For the remaining cases, they obtain some weaker information. From their work, together with [Reference Arakawa10], one possibility would be to proceed by finding pairs $({\mathbb {O}},{\mathbb {O}}')$ satisfying the above conditions, as in the classical cases.

In fact, our work is more exhaustive in a sense since in exceptional types, we are able to directly exploit the description of the $k_i^\natural $ in Tables 11–17 and the central charge (see §6.2 below) to detect all possible collapsing levels. In this way, we find many nontrivial isomorphisms between a Slodowy slice $\mathcal {S}_{{\mathbb {O}},f}$ and a product of nilpotent orbit closures in $\mathfrak {g}^\natural $ . Many of these were observed in [Reference Fu, Juteau, Levy and Sommers47] already, though others seem to be new. However, of course, many of isomorphisms between Slodowy slices obtained in [Reference Fu, Juteau, Levy and Sommers47] do not correspond to collapsing levels and so do not appear in our work.

We remark that all the isomorphisms between nilpotent orbits and nilpotent Slodowy slices that we obtain from collapsing levels are automatically isomorphisms of Poisson varieties.

Roughly speaking, a collapsing Slodowy slice will yield a collapsing level if the asymptotic data of the vertex algebras corresponding to the two sides of (39) can be shown to coincide. In fact, asymptotic data is rather difficult to compute, so the next step in our strategy is to rule out spurious cases by comparing central charges. This is described in §6.2 below.

We close this subsection with a discussion of admissibility of the levels $k_i^\natural $ .

Conjecture 6.4. If k is admissible and if $f \in \overline {{\mathbb {O}}}_k$ is such that $\mathcal {S}_{{\mathbb {O}}_k,f}$ is collapsing, then $k^\natural $ is admissible, provided that $k_0^\natural =0$ .

We will verify the conjecture in the classical cases in the cases where $\mathcal {S}_{{\mathbb {O}}_k,f}$ is collapsing and the isomorphism between $\mathcal {S}_{{\mathbb {O}}_k,f}$ and a product of nilpotent orbit closures in $\mathfrak {g}^\natural $ is obtained from the row/column removal rule of Kraft-Procesi (see Lemma 8.4 and Lemma 9.5). We feel that these cases exhaust all possible cases of collapsing nilpotent Slodowy slices so that, together with Lemma 6.5 below, it would complete the proof of the conjecture.

Lemma 6.5. Conjecture 6.4 is true if $\mathfrak {g}$ is simple of exceptional type.

Proof. Assume that k is admissible. It is known that $\mathcal {S}_{{\mathbb {O}}_k,f}$ is equidimensional of dimension $\dim {\mathbb {O}}_k -\dim G.f$ ([Reference Ginzburg48, Corollary 1.3.8]). Moreover, by the main result of [Reference Arakawa and Moreau21], $\mathcal {S}_{{\mathbb {O}}_k,f}$ is irreducible since k is admissible.

If $\mathcal {S}_{{\mathbb {O}}_k,f}$ is collapsing, then, in particular, it is contained in the nilpotent cone $\mathcal {N}_{\mathfrak {g}^\natural }$ of $\mathfrak {g}^\natural $ , and so .

The semisimple type of $\mathfrak {g}^\natural $ and the values of the $k_i^\natural $ ’s are computed in Tables 11–17. Hence, fixing f, we first consider nilpotent orbits ${\mathbb {O}}$ containing $\overline {G.f}$ such that and such that ${\mathbb {O}}={\mathbb {O}}_k$ for some admissible level k for $\mathfrak {g}$ . This heavily restricts the possibilities for the denominator q of k. Then we can ask whether for such q, the corresponding level $k^\natural $ is admissible for $\mathfrak {g}^\natural $ . Let us illustrate with an example.

Assume that $\mathfrak {g}=F_4$ and that f belongs to the nilpotent orbit labelled $B_3$ in the Bala-Carter classification (dimension 42). According to Table 12, $\mathfrak {g}^\natural \cong A_1$ . Hence, , so from which we see that ${\mathbb {O}}_k$ must be $B_3$ , $C_3$ or $F_4(a_2)$ .

Recall that $k=-9+p/q$ with $(p,q)=1$ and $p\ge 9$ if q is odd, and $p\ge 12$ if q is even. By [Reference Arakawa10], if ${\mathbb {O}}_k = B_3$ , then $q=8$ , if ${\mathbb {O}}_k = F_4(a_2)$ , then $q=7$ or $q=10$ , and finally the orbit $C_3$ cannot occur as ${\mathbb {O}}_k$ . Now we examine admissibility of

$$ \begin{align*}k^\natural = 8k+60 =-2+\frac{2 (4 p - 5 q)}{q}\end{align*} $$

in each case.

• * If $q=7$ , then $2 (4 p - 5 q) \ge 2 (4 \times 9 - 5\times 7) =2$ , so $k^\natural $ is admissible.

• * If $q=8$ , then $2 (4 p - 5 q)/q = (4 p - 5 \times 8)/4$ and $4 p - 5 q \ge 4 \times 13 - 5\times 8 =12$ so $k^\natural $ is admissible.

• * If $q=10$ , then $2 (4 p - 5 q)/q = (4 p - 5 \times 10)/5$ and $4 p - 5 q \ge 4 \times 13 - 5\times 10 =2$ so $k^\natural $ is admissible.

In most cases, considerations of orbit dimension, like those above, suffice to reach the desired conclusion. In some cases, however, a more detailed analysis of the Hasse diagrams of $\mathfrak {g}$ and $\mathfrak {g}^\natural $ is necessary (we use the diagrams of [Reference Fu, Juteau, Levy and Sommers47]). This is the case, for example, for the minimal nilpotent orbits of $F_4$ and $E_7$ . We explain the verification for these two cases.

Let us first consider the minimal nilpotent orbit of $F_4$ (dimension $16$ ). In this case, $\mathfrak {g}^\natural \cong C_3$ . Since the nilpotent cone of $C_3$ has dimension $18$ , we have . These inequalities are satisfied by $6$ nilpotent orbits, but of these, only the orbits $A_1$ (the minimal orbit itself) and $A_2 + \widetilde {A}_1$ can occur as ${\mathbb {O}}_k$ .

If ${\mathbb {O}}_k = A_2+\tilde {A}_1$ , then $q=4$ (here, $k = -9 + p/q$ as above), and if ${\mathbb {O}}_k = A_1$ , then $q=2$ . Now we examine admissibility of the level

$$ \begin{align*}k^\natural = k+5/2 =-4+\frac{2 p - 5 q}{2 q}\end{align*} $$

in each case.

• * If $q=2$ , then $(2p-5q)/(2q) = (p - 5)/2$ and $p - 5 \ge 13 - 5 = 8 \ge 6=h_{C_3}$ so $k^\natural $ is admissible.

• * If $q=4$ , then $(2p - 5 q)/(2q) = (p - 10)/4$ , but $p \ge 13$ gives us only $p - 10 \ge 3$ , not $p - 10 \ge 6$ as required for admissibility.

To exclude the second case, we observe instead that $\dim {\mathbb {O}}_{A_2+\tilde {A}_1}-\dim {\mathbb {O}}_{A_1}=18$ . So if $\mathcal {S}_{{\mathbb {O}}_{A_2+\tilde {A}_1},A_1}$ were collapsing, then necessarily $\mathcal {S}_{{\mathbb {O}}_{A_2+\tilde {A}_1},A_1}$ would be isomorphic to the nilpotent cone $\mathcal {N}_{C_3}$ of $C_3$ since it is irreducible. In particular, it should contain a nilpotent $G^\natural $ -orbit of dimension 16 (corresponding to the subregular nilpotent orbit of $C_3\cong \mathfrak {sp}_6$ ). Here, $G^\natural $ denotes the centraliser in $F_4$ of the ${\mathfrak {sl}}_2$ -triple associated with f in the minimal nilpotent orbit $A_1$ of $F_4$ whose Lie algebra is $\mathfrak {g}^\natural $ . But from the Hasse diagram of the $F_4$ , we see that this is not possible. Indeed, the Hasse diagram of the $G^\natural $ -action on $\mathcal {S}_{{\mathbb {O}}_{A_2+\tilde {A}_1},A_1}$ is just the interval between the orbits labelled ${A_2+\tilde {A}_1}$ and $A_1$ in the Hasse diagram of $F_4$ , and the dimension of the corresponding $G^\natural $ -orbits would be $18, 14, 12, 6, 0$ . Since it does not coincide with the Hasse diagram of $\mathcal {N}_{C_3}$ , this case is ruled out.

Consider now the minimal nilpotent orbit $A_1$ of $E_7$ (dimension 34). In this case, $\mathfrak {g}^\natural \cong D_6$ . Since the nilpotent cone of $D_6$ has dimension $60$ , we have . Only the orbits $4 A_1$ and $2 A_2 + A_1$ can satisfy this conditions and occur as ${\mathbb {O}}_k$ .

If ${\mathbb {O}}_k = 4 A_1$ , then $q=2$ (here, $k = -18 + p/q$ ), and if ${\mathbb {O}}_k = 2 A_2+A_1$ , then $q=3$ . Now we examine admissibility of the level

$$ \begin{align*}k^\natural = k+4 =-10+(p-4q)/q\end{align*} $$

in each case.

• * If $q=2$ , then $(p-4q)/q = (p - 8)/2$ and $p - 8 \ge 19 - 8 = 11 \ge 19=h_{D_6}^\vee $ so $k^\natural $ is admissible.

• * If $q=3$ , then $(p-4q)/q= (p - 12)/3$ , but $p \ge 19$ gives us only $p - 12 \ge 7$ , not $p - 12 \ge 10$ as required for admissibility.

To exclude the second case, we observe instead that $\dim {\mathbb {O}}_{2 A_2+A_1}-\dim {\mathbb {O}}_{A_1}=56$ . In $\mathcal {N}_{D_6}$ , there are exactly two nilpotent orbits of dimension 56: there are associated with the partition $(7,5)$ and $(9,1^3)$ . So if $\mathcal {S}_{{\mathbb {O}}_{A_2+\tilde {A}_1},A_1}$ were collapsing, then necessarily $\mathcal {S}_{{\mathbb {O}}_{A_2+\tilde {A}_1},A_1}$ would be isomorphic to one of the nilpotent orbit closures of $D_6$ of dimension 56 since it is irreducible. Both of them contain the nilpotent $G^\natural $ -orbit of dimension 54, associated with the partition $(7,3,1^2)$ while, looking at the Hasse diagram of the $G^\natural $ -action on $\mathcal {S}_{{\mathbb {O}}_{2 A_2+A_1},A_1}$ in $E_7$ , we see that this is not possible.

The rest of the verifications are left to the reader.