# Singularities of nilpotent Slodowy slices and collapsing levels of W-algebras

aa r X i v : . [ m a t h . R T ] F e b SINGULARITIES OF NILPOTENT SLODOWY SLICES ANDCOLLAPSING LEVELS OF W -ALGEBRAS TOMOYUKI ARAKAWA , JETHRO VAN EKEREN , AND ANNE MOREAU Abstract.

We apply results from the geometry of nilpotent orbits and nilpo-tent Slodowy slices, together with modularity and asymptotic analysis of char-acters, to prove many new isomorphisms between aﬃne W -algebras and aﬃneKac-Moody vertex algebras and their ﬁnite extensions at speciﬁc admissiblelevels. In particular we identify many new collapsing levels for W -algebras. Introduction

Let G be a complex connected, simple algebraic group of adjoint type with Liealgebra g , and let O be a nilpotent orbit in g and ( e, h, f ) an sl -triple in g . The nilpotent Slodowy slice associated with these data is the intersection S O ,f := O ∩ S f where S f is the Slodowy slice of the sl -triple, that is, S f ∼ = f + g e where g e is thecentraliser in g of e (for f = 0, we set S f := g ). Normalisations of nilpotent Slodowyslices are known to have symplectic singularities in the sense of Beauville [23] and,just as nilpotent orbit closures in simple Lie algebras, nilpotent Slodowy slices play asigniﬁcant role in representation theory and in the theory of symplectic singularities.They are best understood in the case that G.f is a minimal degeneration of O , thatis, for f such that G.f is open in the boundary of O in O . The best known casesare nilpotent Slodowy slices associated with the principal nilpotent orbit O prin and a subregular nilpotent element f subreg for types A, D, E . As was shown byBrieskorn and Slodowy ([29, 100]) conﬁrming a conjecture of Grothendieck, thesehave simple singularities of the same type as G . Motivated by the normality problemfor nilpotent orbit closures, Kraft and Procesi determined nilpotent Slodowy slicesat minimal degenerations in the classical types [83, 84]. Recently, Fu, Juteau,Levy and Sommers [52] have completed that work and determined the genericsingularities of O when g is of exceptional type by studying the various nilpotentSlodowy slices S O ,f at minimal degenerations G.f .In another direction, it is known that nilpotent Slodowy slices appear in vertexalgebras and string theory. Firstly, as associated varieties of W -algebras [10, 20](see also Section 4). It is also known that they appear as the Higgs branches of theArgyres-Douglas theories ([21, 22]) in four-dimensional N = 2 superconformal ﬁeldtheories, see [99, 106] and references therein. These two facts are connected by thefact that the Higgs branch of a four-dimensional N = 2 superconformal ﬁeld theory T is conjecturally [25] isomorphic to the associated variety of the vertex algebra Date : March 1, 2021. The boundary of O in O is precisely the singular locus of O as was shown by Namikawa [94]using results of Kaledin and Panyushev [78, 95]; this can also be deduced from [83, 84, 52]. corresponding to T via the 4D/2D-duality discovered in [24]. In particular, thevertex algebras coming from 4D theories are expected to be quasi-lisse [16], thatis, their associated varieties have ﬁnitely many symplectic leaves. The reader isreferred to [11] for a survey on this conjecture.Given a nilpotent element f of g , the universal W -algebra associated with ( g , f )at level k ∈ C is by deﬁnition the vertex algebra W k ( g , f ) := H DS,f ( V k ( g )) , where H • DS,f (?) is the BRST cohomology functor of the quantized Drinfeld-Sokolovreduction associated with ( g , f ) ([46, 72]). Let g ♮ be the centraliser of the sl -triple( e, h, f ) in g . The W -algebra W k ( g , f ) contains the universal aﬃne vertex algebra V k ♮ ( g ♮ ) as its vertex subalgebra, where k ♮ is a certain number determined by k (see Sect. 5). The level k is said to be collapsing [3] if W k ( g , f ) admits a surjectivevertex algebra homomorphism W k ( g , f ) −! L k ♮ ( g ♮ ) , where L k ♮ ( g ♮ ) is the simple quotient of V k ♮ ( g ♮ ). Note that k is collapsing if andonly if the simple quotient vertex algebra W k ( g , f ) of W k ( g , f ) is isomorphic to L k ♮ ( g ♮ ). In [4] it was shown that collapsing levels have remarkable applications tothe representation theory of aﬃne vertex algebras. In view of the 4D/2D duality,the collapsing levels are particularly interesting when the W -algebras are quasi-lisse. Recently it has been observed [106] that many collapsing levels for quasi-lisse W -algebras should come from non-trivial isomorphisms of 4D theories.There is a full classiﬁcation of collapsing levels for the case that f is a minimalnilpotent element f min , including the case where g is a simple aﬃne Lie superalgebra([2, 3]). However, little or nothing is known for collapsing levels for non minimalnilpotent elements. The main reason is that for an arbitrary nilpotent element f ,the commutation relations in W k ( g , f ) are unknown, and so it is extremely diﬃcultto predict which levels are collapsing.In this context, the notion of associated variety and the singularities of nilpotentSlodowy slices are proving to be very useful to ﬁnd new collapsing levels. Let usoutline the main idea.If k is collapsing for W k ( g , f ), then obviously X W k ( g ,f ) ∼ = X L k♮ ( g ♮ ) , (1)where X V is the associated variety of a vertex algebra V . However, the associatedvariety X W k ( g ,f ) is a C ∗ -invariant subvariety of the Slodowy slice S f , while X L k♮ ( g ♮ ) is a G ♮ -invariant conic subvariety of g ♮ . Hence the fact that k is collapsing immedi-ately implies a non-trivial isomorphism between such varieties. Conversely, if oneﬁnds an isomorphism between a C ∗ -invariant subvariety of S f and a G ♮ -invariantconic subvariety of g ♮ , one may hope that it comes from a collapsing level of a W -algebra associated with ( g , f ).Further, if k is collapsing for a quasi-lisse W k ( g , f ), then X W k ( g ,f ) and X L k♮ ( g ♮ ) must be contained in S f ∩ N g and N g ♮ , respectively. Here N g and N g ♮ are thenilpotent cone of g and g ♮ . Since the associated variety of a quasi-lisse vertex algebrais conjecturally irreducible [20], we conclude that the isomorphism (1) should bean isomorphism between a nilpotent Slodowy slice and a nilpotent orbit closure. OLLAPSING LEVELS OF W -ALGEBRAS 3 Although the determination of the varieties X W k ( g ,f ) and X L k♮ ( g ♮ ) , are wide openin general, there are cases when one has a good understanding of these varieties.First, W k ( g , f ) is conjectured to be isomorphic to H DS,f ( L k ( g )) whenever the latteris nonzero ([72]), and this conjecture has been veriﬁed in many cases. In such casesone may replace (1) by the condition X H DS,f ( L k ( g )) ∼ = X L k♮ ( g ♮ ) , (2)to facilitate the search for collapsing levels. Second, there exists an isomorphism X H DS,f ( L k ( g )) ∼ = S f ∩ X L k ( g ) ([10]), which translates (2) into the condition S f ∩ X L k ( g ) ∼ = X L k♮ ( g ♮ ) . (3)Third, the associated varieties of simple aﬃne vertex algebras at admissible levelhave been completely characterized [10] as certain nilpotent orbit closures. There-fore, if both k and k ♮ are admissible, the condition (3) asserts an isomorphismbetween a certain nilpotent Slodowy slice and a nilpotent orbit closure. Such vari-eties are relatively well understood and so (3) can be used to place restrictions onpossible collapsing levels. In particular, the singularity of the nilpotent Slodowyslice on the left-hand-side should be of the same type as that of the nilpotent or-bit closure on the right-hand-side. Therefore we may apply the results of Kraftand Procesi [83, 84] and Fu, Juteau, Levy and Sommers [52] to ﬁnd candidates forcollapsing levels.With such candidate collapsing levels in hand, it remains to prove that theyare indeed collapsing, since (1) is only a necessary condition. In order to do so,we study the asymptotic behaviour of the normalised character of vertex algebras,which goes back to Kac and Wakimoto [73]. The asymptotic behaviour deﬁnes asymptotic data ; an important invariant of vertex algebras, generalizing the notionof the quantum dimension [37] (or the Frobenius-Perron dimension ) for rationalvertex algebras (see Section 2).

Theorem 1.1.

Assume that k and k ♮ are admissible levels for g and g ♮ , respectively,that f ∈ X L k ( g ) and that χ H DS,f ( L k ( g )) ( τ ) ∼ χ L k♮ ( g ♮ ) ( τ ) , as τ . Then k is acollapsing level, that is, W k ( g , f ) ∼ = L k ♮ ( g ♮ ) . Note that if k is an admissible level, and if V is either the simple aﬃne vertexalgebra L k ( g ) or its Drinfeld-Sokolov reduction H DS,f ( L k ( g )), then its normalisedcharacter χ V admits “nice” asymptotic behaviour [74] (see Proposition 2.6, togetherwith Theorems 3.1 and 4.3).In this way we discovered a large number of collapsing levels for W k ( g , f ). SeeTheorems 8.6, 8.7, 9.7, 9.10, 9.13, 9.15 for statements in the classical types, andTheorems 10.1, 10.4, 10.7, 10.12, 10.10 for the exceptional types.We observe that many of our examples of collapsing levels are of the form k = − h ∨ g + h ∨ g /q , with ( h ∨ g , q ) = ( r ∨ , q ) = 1, or of the form k = − h ∨ g + ( h g + 1) /q , with( h g , q ) = 1, ( r ∨ , q ) = r ∨ , where h g is the Coxeter number, h ∨ g is the dual Coxeternumber, r ∨ is the lacing number of g , respectively. A level of the ﬁrst form is calleda boundary principal admissible level ([72], see also [77]).It is known that the vertex algebras L − h ∨ g + h ∨ g /q ( g ), W − h ∨ g + h ∨ g /q ( g , f ) at bound-ary principal admissible levels appear as the vertex algebras corresponding to the TOMOYUKI ARAKAWA, JETHRO VAN EKEREN, AND ANNE MOREAU

Argyres-Douglas theories in the 4D/2D duality mentioned above ([99, 108, 104]).Collapsing levels which are boundary principal admissible levels have been studiedby Xie and Yan [106] in view of the 4D/2D duality, and our results conﬁrm theirconjectures [106, 3.4].We also ﬁnd some cases where W k ( g , f ) is not isomorphic to L k ♮ ( g ♮ ) but is a ﬁnite extension of L k ♮ ( g ♮ ). Here, we say a vertex algebra V is a ﬁnite extension ofa vertex algebra W if W is a conformal vertex subalgebra of V and V is a ﬁnitedirect sum of simple W -modules. Conjecture 1.2. If V is a ﬁnite extension of W , then X V is birationally equivalentto X W . Note that the validity of Conjecture 1.2 in particular implies the widely-believedfact that if V is a ﬁnite extension of W and V is lisse, then W is lisse as well.We ﬁnd examples where W k ( g , f ) is a ﬁnite extension of L k ♮ ( g ♮ ), X W k ( g ,f ) is notisomorphic to X L k♮ ( g ♮ ) but is birationally equivalent to X L k♮ ( g ♮ ) , see Remarks 10.2,10.5, 10.8, 10.11, 10.13. Acknowledgements.

TA is partially supported by JSPS KAKENHI Grant Num-bers JP17H01086, JP17K18724. JvE is supported by the Serrapilheira Institute(grant number Serra – 1912-31433) and by CNPq grant 303806/2017-6. AM ispartially supported by ANR Project GeoLie Grant number ANR-15-CE40-0012.TA and AM wish to express their gratitude to Daniel Juteau for the numerousfruitful discussions on nilpotent Slodowy slices. The authors are grateful to WenbinYan for several helpful comments concerning collapsing levels in 4D superconformaltheory. They also warmly thank Thomas Creutzig and Kazuya Kawasetsu whosequestion about the isomorphism W − / ( sl ) ∼ = L − / ( sl ) and corresponding asso-ciated varieties was the starting point of this work. TA and JvE wish to thank theUniversity of Lille and the Laboratoire Painlev´e for the organisation and the ﬁnan-cial support of the conference Geometric and automorphic aspects of W-algebras in2019 where the collaboration between the three authors started. AM gratefully ac-knowledges the RIMS of Kyoto, where part of this paper was written, for ﬁnancialsupport and its hospitality during her stay in autumn 2018.

Contents

1. Introduction 12. Asymptotic data of vertex algebras 53. Admissible aﬃne vertex algebras 94. Asymptotic data of W -algebras 205. Collapsing levels for W -algebras 266. The main strategy 297. Some useful product formulas 348. Collapsing levels for type sl n sp n and so n sl -triples in simple exceptional Lie algebras 85References 90 OLLAPSING LEVELS OF W -ALGEBRAS 5 Asymptotic data of vertex algebras A vertex algebra is a complex vector space V equipped with a distinguishedvector | i ∈ V and a linear map V ! (End V )[[ z, z − ]] , a a ( z ) = X n ∈ Z a ( n ) z − n − satisfying the following axioms: • a ( z ) b ∈ V (( z )) for all a, b ∈ V , • (vacuum axiom) | i ( z ) = Id V and a ( z ) | i ∈ a + zV [[ z ]] for all a ∈ V , • (locality axiom) ( z − w ) N a,b [ a ( z ) , b ( w )] = 0 for a suﬃciently large integer N a,b for all a, b ∈ V .The linear map T : V ∋ a a ( − | i ∈ V is called the translation operator .A vertex algebra V is called conformal if there exists a vector ω called the con-formal vector such that L ( z ) = P n ∈ Z L n z − n − := Y ( ω, z ) satisﬁes (a) [ L m , L n ] =( m − n ) L m + n + ( m − m ) / δ m + n, c V , where c V is a constant called the centralcharge of V , (b) L acts semisimply on V , and (c) L − = T on V . For a conformalvertex algebra V and a V -module M , we set M d = { m ∈ M : L m = dm } . The L -eigenvalue of a nonzero L -eigenvector m is called its conformal weight . A ﬁnitelygenerated V -module M is called ordinary if L acts semisimply, dim M d < ∞ forall d , and the conformal weights of M are bounded from below. The minimum con-formal weight of a simple ordinary V -module M is called the conformal dimension of M . The normalised character of an ordinary representation M is deﬁned by χ M ( τ ) = tr M q L − c V / = q − c V / X d ∈ C (dim M d ) q d , q = e π i τ . A conformal vertex algebra is called conical if V = L ∆ ∈ r Z > V ∆ for some r ∈ Z > and V = C . A Z -graded conical vertex algebra is said to be of CFT-type . Let V bea vertex algebra of CFT-type. Then V is called self-dual if V ∼ = V ′ as V -modules,where M ′ denotes the contragredient dual [51] of the V -module M . Equivalently V is self-dual if and only if it admits a non-degenerate symmetric invariant bilinearform.The following deﬁnition goes back to [73, Conjecture 1] . Deﬁnition . A conformal vertex algebra V is said to admit an asymptotic datum if there exist A V ∈ R , w V ∈ R , g V ∈ R such that χ V ( τ ) ∼ A V ( − i τ ) w V e π i τ g V as τ . The numbers A V , w V and g V are called the asymptotic dimension of V , the asymptotic weight , and the asymptotic growth , respectively. Similarly, an ordi-nary V -module M is said to admit an asymptotic datum if there exist A M ∈ C , w M , g M ∈ R such that χ M ( τ ) ∼ A M ( − i τ ) w M e π i τ g M as τ . For a conformal vertex algebra V and an ordinary V -module M ,qdim V M := lim τ χ M ( τ ) χ V ( τ ) In [73] the triple ( A V , w V , g V ) was called the asymptotic dimension. TOMOYUKI ARAKAWA, JETHRO VAN EKEREN, AND ANNE MOREAU is called the quantum dimension of M if it exists ([37]). If both V and M admitasymptotic data, g V = g M and w V = w M , then the quantum dimension of M exists and is equal to the ratio of the asymptotic dimensions:qdim V M = A M A V . (4)With every vertex algebra V one associates a Poisson algebra R V , called the Zhu C -algebra , as follows ([109]). Let C ( V ) be the subspace of V spanned by theelements a ( − b , where a, b ∈ V , and set R V = V /C ( V ). Then R V is naturally aPoisson algebra by1 = | i , ¯ a · ¯ b = a ( − b and { ¯ a, ¯ b } = a (0) b, where ¯ a denotes the image of a ∈ V in the quotient R V .The associated variety [6] X V of a vertex algebra V is the aﬃne Poisson varietydeﬁned by X V = Specm R V . A vertex algebra V called lisse if dim X V = 0. It is called quasi-lisse if X V hasﬁnitely 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 ﬁnitelygenerated V -module is ordinary, and there exist ﬁnitely many simple V -modules([1]).The following fact is well-known. Proposition 2.2.

Let V be a ﬁnitely strongly generated, rational, lisse self-dualsimple vertex operator algebra of CFT-type. Then any simple V -module M admitsan asymptotic datum with w M = 0 .Proof. We include a proof for completeness. By [1], any simple V -module is ordi-nary, and there exist ﬁnitely many simple V -modules, say, { L i : i = 0 , . . . , r } . Let h i be the conformal dimension of L i . Then χ L i ( τ ) = q h i − c/ X d > (dim( L i ) h i + d ) q d . By Zhu’s theorem [109], the vector space spanned by χ L i ( τ ), i = 0 , . . . , r , is invari-ant under the natural action of the modular group SL ( Z ). Hence, χ L i ( τ ) = r X j =1 S i,j χ L j ( − /τ )for some S i,j ∈ C , j = 0 , . . . , r . The assertion follows. (cid:3) It seems that the following assertion is widely believed.

Conjecture 2.3.

Let V be a rational, lisse, simple, self-dual conformal vertexalgebra V , { L , . . . , L r } the complete set of simple V -modules. There exists a uniquesimple module L i o with conformal dimension h min := h i o such that h i > h min forall i = i o , and that S ii o = 0 for all i . OLLAPSING LEVELS OF W -ALGEBRAS 7 In Theorem 4.11 below, we conﬁrm the uniqueness of the simple module withminimal conformal dimension for exceptional W -algebras ([75, 12]) that are lisse[10] and conjecturally rational. Proposition 2.4.

Let V be as in Conjecture 2.3, and assume that there exists aunique simple module L i o satisfying the condition of Conjecture 2.3. Then g L i = c V − h min , A L i = dim( L i o ) h min S i,i o for all i . Moreover, identifying V with L , we have qdim V ( L i ) = S i,i o S ,i o , and qdim V ( L i ⊠ L j ) = qdim V ( L i ) qdim V ( L j ) , (5) where ⊠ is the fusion product ( [60, 62, 63, 64] ). In particular, the quantum dimen-sion is well-deﬁned for all simple V -modules.Proof. The assertions except for the last follows from the proof of Proposition 2.2.For the assertion (5), see [37, Remark 4.10]. (cid:3)

The number c V − h min is called the eﬀective central charge of V in the literature([39]). Remark . Let V be as in Proposition 2.4. By a result of Huang [61], the category V -mod of ﬁnitely generated V -modules form a modular tensor category. In thiscontext, the quantum dimension of a simple V -module is the same as the Frobenius-Perron dimension ([44]) of V in V -mod. Proposition 2.6.

Let V be a ﬁnitely strongly generated, quasi-lisse vertex operatoralgebra of CFT-type. Then any simple ordinary V -module L admits an asymptoticdatum.Proof. By [16], there exist only ﬁnitely many simple ordinary V -modules { L i } and χ L i ( τ ) is a solution of a modular linear diﬀerential equation. Since the spacespanned by the solutions of a modular linear diﬀerential equation is invariant underthe natural action of SL ( Z ), the assertion follows in a similar manner as Proposi-tion 2.2, except that a solution of a modular linear diﬀerential equation may havelogarithmic terms, that is, it has the form q β i e X i =1 f i ( q )(log q ) e − i , f i ( q ) ∈ C [[ q ]] . (cid:3) Let Vir c be the universal Virasoro vertex algebra at central charge c ∈ C , Vir c the unique simple quotient of Vir c . Lemma 2.7 ([48, 73]) . A quotient of a universal Virasoro vertex algebra admitsan asymptotic datum. The rationality of exceptional W -algebras has been proved for the principal W -algebras ([8]),type A W -algebras and subregular W -algebras of type ADE ([13]).

TOMOYUKI ARAKAWA, JETHRO VAN EKEREN, AND ANNE MOREAU

Proof.

It is well-known that Vir c = Vir c unless c = 1 − p − q ) /pq for some p, q ∈ Z > , ( p, q ) = 1, and Vir c has length two if c = 1 − p − q ) /pq for some p, q ∈ Z > , ( p, q ) = 1. Hence, a quotient V of Vir c is either Vir c or Vir c . If V = Vir c , then χ V ( τ ) = (1 − q ) q (1 − c V ) / η ( q ) , where η ( q ) = q / Q j > (1 − q j ). Hence (indicating by + · · · terms of lower growth) χ V ( e π i τ ) = (1 − e π i τ ) e π i τ (1 − c V ) / ( − i τ ) (cid:16) e π i ( − τ )( − ) + · · · (cid:17) , ∼ ( − π i τ )( − i τ ) e π i12 τ , where we have used l’Hopital’s rule. So V admits an asymptotic datum with A V =2 π , w V = 3, g V = 1.If V = Vir c with c = 1 − p − q ) /pq , p, q ∈ Z > , ( p, q ) = 1, then as it iswell-known [48, 73] V admits an asymptotic datum with w V = 0, A V = (cid:18) pq (cid:19) / sin (cid:18) πa ( p − q ) q (cid:19) sin (cid:18) πb ( p − q ) p (cid:19) , (6)where ( a, b ) is the unique solution of pa − qb = 1 in integers 1 a q and 1 b p ,and g V = 1 − pq . (7) (cid:3) The simple Virasoro vertex algebra Vir c with c = 1 − p − q ) /pq , p, q ∈ Z > ,( p, q ) = 1, is known to be rational and lisse ([103]). The simple Vir c -modules arethe ( p, q )-minimal series representations of the Virasoro algebra, and for each simpleVir c -module L we have w L = 0, g L = g Vir c and A L > Lemma 2.8.

Let V be a conformal vertex algebra with central charge c = 1 − p − q ) /pq , p, q ∈ Z > , ( p, q ) = 1 , and suppose that V admits an asymptotic datum with g V < . Then V is a direct sum of simple ( p, q ) -minimal series representations ofthe Virasoro algebra. If further A V = A Vir c , then V ∼ = Vir c .Proof. The vertex algebra homomorphism Vir c ! V , ω Vir c ! ω V , factors throughthe embedding Vir c ֒ ! V because otherwise g V > g Vir c = 1. Thus the rationalityof Vir c proves the ﬁrst statement, and so we have V = L i L ⊕ mi i , where { L i } is the set of simple ( p, q )-minimal series representations of the Virasoro algebraand m i ∈ Z > ∪ {∞} . It follows that A V = P i m i A L i , and we get the secondassertion. (cid:3) Recall that a homomorphism f : V ! W of conformal vertex algebras is called conformal if ω W = f ( ω V ). Proposition 2.9.

Let f : V ! W be a homomorphism of conformal vertex alge-bras. Suppose that • f ( ω V ) ∈ W and ( ω W ) (2) f ( ω V ) = 0 , • the simple quotient L of V admits an asymptotic datum, • W is a quotient of a conformal vertex algebra ˜ W that admits an asymptoticdatum. OLLAPSING LEVELS OF W -ALGEBRAS 9 If g L = g ˜ W , then f is conformal.Proof. Let ω = ω W − f ( ω V ). We wish to show that ω = 0.Suppose that ω = 0. Then by [85, Theorem 3.11.12], ω generates a Virasorovertex algebra of central charge, say, c , and there is a homomorphism ˜ f : Vir c ⊗ V ! W of conformal vertex algebras that sends the conformal vector of Vir c to ω . Let U = ˜ f (Vir c ⊗ V ) ⊂ W . Then U is a vertex subalgebra of W whose simple quotientis isomorphic to Vir c ⊗ L . It follows that g ˜ W > g L + g Vir c . Thus, g Vir c = 0 and byLemma 2.7, this happens if and only if c = 0 so that Vir c = C .Let N be the maximal ideal of V and consider the quotient vertex algebra U = U/ ˜ f (Vir ⊗ N ). The map ˜ f induces a surjection ˜ f : Vir ⊗ L ! U .Let us denote by ω (0) the conformal vector of Vir . In order to show ω = 0it therefore suﬃces to show that ˜ f ( ω (0) ⊗ v ) = 0 in U for all v ∈ L . Deﬁne K = { v ∈ L | ˜ f ( ω (0) ⊗ v ) = 0 in U } , then K ⊂ L is a V -submodule, and henceeither K = L or K = 0.The maximal ideal of Vir is generated by ω (0) and is simple as a Vir -module.So if K = 0 then ˜ f is an isomorphism. But since g Vir > g ˜ W > g L + g Vir > g L , which contradicts our hypotheses. Therefore K = L . Butnow we have ˜ f ( ω (0) ⊗ | i ) ∈ ˜ f (Vir (0) ⊗ N ), which implies that ω = 0 and we aredone. (cid:3) Admissible affine vertex algebras

Let g be a complex simple Lie algebra as in the introduction. Let g = n − ⊕ h ⊕ n + be a triangular decomposition with a Cartan subalgebra h , ∆ the root system of( g , h ) and ∆ + a set of positive roots for ∆, Π = { α , . . . , α ℓ } the set of simpleroots. Let θ be the highest root, θ s the highest short root. We also have ∆ ∨ theset of coroots. Let P be the weight lattice, Q the root lattice and Q ∨ the corootlattice. The lattice P is dual to Q ∨ and we write P ∨ for the dual of Q . Recall thatthe Coxeter number and the dual Coxeter number of g are denoted by h g and h ∨ g ,respectively. Identifying h with h ∗ using the inner product( | ) g = 12 h ∨ g × Killing form of g , we view Q ∨ as a sub-lattice of both P and Q . We write ℓ for the rank of g , and wedenote by ρ the half-sum of positive roots.For λ ∈ h ∗ , let L g ( λ ) be the irreducible highest weight representation of g withhighest weight λ , and let J λ = Ann U ( g ) L g ( λ ) . (8)Let e g = g [ t, t − ] ⊕ C K ⊕ C D be the aﬃne Kac-Moody algebra, with the commu-tation relations:[ xt m , yt n ] = [ x, y ] t m + n + mδ m + n, ( x | y ) g K, [ D, xt n ] = − nxt n , [ K, b g ] = 0 , for all x, y ∈ g and all m, n ∈ Z Let e g = b n − ⊕ e h ⊕ b n + be the standard triangulardecomposition, that is, e h = h ⊕ C K ⊕ C D is the Cartan subalgebra of e g , b n + = n + + t g [ t ], b n − = n − + t − g [ t − ]. Let b g = [ e g , e g ] = g [ t, t − ] ⊕ C K , and let b h = h ⊕ C K ⊂ b g , so that b g = b n − ⊕ b h ⊕ b n + .The Cartan subalgebra e h is equipped with a bilinear form extending that on h by( K | D ) = 1 , ( h | C K ⊕ C D ) = ( K | K ) = ( D | D ) = 0 . We write δ and Λ for the elements of e h ∗ orthogonal to h ∗ and dual to K and D ,respectively. We have the (real) root system b ∆ re = { α + nδ : n ∈ Z , α ∈ ∆ } = b ∆ re+ ⊔ ( − b ∆ re+ ) , b ∆ re+ = { α + nδ : α ∈ ∆ + , n > } ⊔ { α + nδ : α ∈ ∆ , n > } , and the aﬃne Weyl group c W , generated by reﬂections r α for α ∈ b ∆ re . For α ∈ h ∗ the translation t α : e h ∗ ! e h ∗ is deﬁned by t α ( λ ) = λ + λ ( K ) α − (cid:20) ( α | λ ) + | α | λ ( K ) (cid:21) δ. For α ∈ Q ∨ we have t α ∈ c W and in fact c W ∼ = W ⋉ t Q ∨ . The extended aﬃne Weylgroup, which is the group of isometries of b ∆, is f W = W ⋉ t P . Here, for R a subsetof P , t R stands for the set { t α : α ∈ R } .Let ˜ O k be the category O of e g at level k ([69]), and let ˜KL k be the full subcategoryof ˜ O k consisting of objects on which g acts as semisimply. The simple objects of ˜ O k are the irreducible highest weight representations L ( λ ) with λ ∈ e h ∗ with λ ( K ) = k ,while the simple objects of ˜KL k are those L ( λ ) with λ ∈ P + + k Λ + C δ , where P + is the set of dominant integral weights of g .For a weight λ ∈ b h ∗ the corresponding integral root system is b ∆( λ ) = { α ∈ b ∆ re : h λ, α ∨ i ∈ Z } , where α ∨ = 2 α/ ( α | α ) as usual, and the subgroup c W ( λ ) of c W generated by r α with α ∈ b ∆( λ ) is called the integral Weyl group of λ .A weight λ ∈ e h ∗ is said to be admissible if(1) λ is regular dominant, that is, h λ + ˆ ρ, α ∨ i > α ∈ b ∆ + ( λ ) := b ∆( λ ) ∩ b ∆ re+ ,(2) Q b ∆ re = Q b ∆( λ ).Here ˆ ρ = ρ + h ∨ Λ with ρ = P α ∈ ∆ + α/

2. An admissible e g -module is one of theform L ( λ ) for λ admissible.Given any k ∈ C , let V k ( g ) = U ( b g ) ⊗ U ( g [ t ] ⊕ C K ) C k , where C k is the one-dimensional representation of g [ t ] ⊕ C K on which g [ t ] acts by0 and K acts as a multiplication by the scalar k . There is a unique vertex algebrastructure on V k ( g ) such that | i is the image of 1 ⊗ V k ( g ) and x ( z ) := ( x ( − | i )( z ) = X n ∈ Z ( xt n ) z − n − for all x ∈ g , where we regard g as a subspace of V through the embedding x ∈ g ֒ ! x ( − | i ∈ V k ( g ). The vertex algebra V k ( g ) is called the universal aﬃnevertex algebra associated with g at level k . OLLAPSING LEVELS OF W -ALGEBRAS 11 The vertex algebra V k ( g ) has a conformal structure given by Sugawara construc-tion provided that k is non-critical, that is, k = − h ∨ g , with central charge c V k ( g ) = k dim g k + h ∨ g . A V k ( g )-module is the same as a smooth b g -module of level k . For a non-criticallevel k , we consider a V k ( g )-module M as a e g -module by letting D act as thesemisimpliﬁcation of − L . In particular, we identify the category O k of b g -module oflevel k with the full subcategory of ˜ O k consisting of modules on which the universalCasimir element C [69, Section 2.5] of e g acts nilpotently. Accordingly, the aﬃnespace b h ∗ k := h ∗ + k Λ is considered as a subset of e h ∗ by the correspondence λ λ − ( λ | λ + 2 ρ )2( k + h ∨ g ) δ when the linkage in O k is described.Let L k ( g ) be the unique simple graded quotient of V k ( g ). For any graded quo-tient V of V k ( g ), we have R V = V /t − g [ t − ] V . In particular, R V k ( g ) ∼ = C [ g ∗ ]and, hence, X V k ( g ) = g ∗ . Furthermore, X L k ( g ) is a subvariety of g ∗ ∼ = g , which is G -invariant and conic.More generally, let a be a Lie algebra endowed with a symmetric invariant bilinearform κ , and b a κ = a [ t, t − ] ⊕ C be the Kac-Moody aﬃnisation of a . It is a Lie algebra with commutation relations[ xt m , yt n ] = [ x, y ] t m + n + mδ m + n, κ ( x, y ) , [ , b a κ ] = , for all x, y ∈ a and all m, n ∈ Z . Then the b a κ -module V κ ( a ) = U ( a ) ⊗ U ( a [ t ] ⊕ C ) C , where C is the one-dimensional representation of a [ t ] ⊕ C on which a [ t ] acts by 0and acts as the identity, has a unique vertex algebra structure such that | i isthe image of 1 ⊗ V κ ( a ) and x ( z ) := ( x ( − | i )( z ) = X n ∈ Z ( xt n ) z − n − for all x ∈ a . We have X V κ ( a ) ∼ = a ∗ and, letting L κ ( a ) be the unique simple gradedquotient of V κ ( a ), X L κ ( a ) is a subvariety of a ∗ , which is Poisson and conic.For M ∈ O k on which L acts semisimply, we consider the multivariable charac-ter χ M of M , deﬁned by χ M ( τ, z, t ) = e π i kt tr M ( e π i z e π i τ ( L − c/ ) , ( τ, z, t ) ∈ H × h × C . We also write, in particular, χ L ( λ ) ( τ ) = χ L ( λ ) ( τ, , λ a closed form for χ L ( λ ) ( τ, z, t ) was given in [74]. It isconvenient to write v = 2 i π ( − τ D + z + tK ) ∈ e h ∗ as in [69], then the characterformula is χ L ( λ ) ( v ) = A λ + b ρ ( v ) A b ρ ( v ) , where A λ ( v ) = e − | λ | λ ( K ) ( δ,v ) X w ∈ c W ( λ ) ε ( w ) e h w ( λ ) ,v i . The complex number k is said to be admissible for b g if k Λ is admissible. Ifthis is the case, L k ( g ) is called a simple admissible aﬃne vertex algebra . By [76,Proposition 1.2], k is admissible if and only if k + h ∨ g = pq with p, q ∈ Z > , ( p, q ) = 1 , p > ( h ∨ g if ( r ∨ , q ) = 1 h g if ( r ∨ , q ) = r ∨ , (9)where r ∨ is the lacing number of g .If k is admissible with ( r ∨ , q ) = 1, we say that k is principal . If k is admissiblewith ( r ∨ , q ) = r ∨ , we say that k is co-principal . Theorem 3.1 ([10]) . Assume that the level k = − h ∨ g + p/q is admissible. Then X L k ( g ) = O k , where O k is a certain nilpotent orbit of g which only depends on thedenominator q . In particular, the associated variety of an admissible aﬃne vertex algebra iscontained in the nilpotent cone N g of g . We note that the converse is not true andthere are aﬃne vertex algebras at non-admissible levels whose associated variety iscontained in N g [18]. Theorem 3.2 ([8]) . Let k be admissible, λ ∈ b h ∗ k . Then L ( λ ) is a L k ( g ) -moduleif and only if λ is an admissible weight such that b ∆( λ ) = y ( b ∆( k Λ )) for some y ∈ f W . Moreover, any L k ( g ) -module that lies in O k is a direct sum of admissiblerepresentations L ( λ ) of b g of level k with b ∆( λ ) = y ( b ∆( k Λ )) for some y ∈ f W . For a principal admissible number, let Pr k be the set of admissible weights λ ∈ b h ∗ k such that b ∆( λ ) = y ( b ∆( k Λ )) for some y ∈ f W . Similarly, for a coprincipaladmissible number, let CoPr k be the set of admissible weights λ ∈ b h ∗ k such thatsuch that b ∆( λ ) = y ( b ∆( k Λ )) for some y ∈ f W . An element of Pr k (resp. CoPr k ) iscalled a principal admissible weight (resp. coprincipal admissible weight ) of level k .Occasionally we shall use Adm k to refer to the set Pr k or CoPr k , according as k isa principal or coprincipal admissible number.For λ ∈ b h ∗ let us denote by ¯ λ ∈ h ∗ the restriction of λ to h . For λ ∈ Pr k (resp. λ ∈ CoPr k ), the primitive ideal J ¯ λ is an maximal ideal of U ( g ), and J ¯ λ = J ¯ µ if andonly if µ ∈ W ◦ λ for λ, µ ∈ Pr k (resp. for λ, µ ∈ CoPr k ) ([12, Proposition 2.4]).Here and throughout ◦ denotes the ‘dot’ action w ◦ λ = w ( λ + ρ ) − ρ . Set[Pr k ] = Pr k / ∼ , [CoPr k ] = CoPr k / ∼ , (10)where λ ∼ µ ⇐⇒ µ ∈ W ◦ λ .We denote by Zhu( V ) the Ramond twisted

Zhu algebra of the Z -graded vertexalgebra V , brieﬂy recalling its construction from ([109, 36]). Bilinear products ∗ n : V ⊗ V ! V are deﬁned by a ∗ n b = X j ∈ Z + (cid:18) ∆( a ) j (cid:19) a ( n + j ) b, for a of conformal weight ∆( a ), b ∈ V arbitrary. We write in particular a ∗ b = a ∗ − b and a ◦ b = a ∗ − b . Then the Zhu algebra is a quotient of V by the subspace V ◦ V spanned by all elements of the form a ◦ b for a, b ∈ V . The operation a ⊗ b a ∗ b i.e., r ∨ = 1 for the types A, D, E , r ∨ = 2 for the types B, C, F , and r ∨ = 3 for the type G . OLLAPSING LEVELS OF W -ALGEBRAS 13 is well-deﬁned in the quotient and turns it into an associative unital algebra withunit [ | i ].The fundamental property satisﬁed by Zhu( V ) is the existence of a natural bi-jection between the set of irreducible Zhu( V )-modules and the set of irreduciblepositive energy Ramond twisted V -modules, sending a V -module M to its lowestgraded piece M low equipped with the Zhu( V )-action [ a ] · m = a m for a ∈ V and m ∈ M low .For the simple aﬃne vertex algebra L k ( g ) we haveZhu( L k ( g )) ∼ = U ( g ) /I k (11)for some two-sided ideal I k of U ( g ).The statement of Theorem 3.2 is strengthened by the following assertion. Theorem 3.3 ([13]) . Let k be admissible. We have Zhu( L k ( g )) ∼ = Y U ( g ) /J ¯ λ , where the product is taken over [ λ ] ∈ [Pr k ] (resp. [ λ ] ∈ [CoPr k ] ). Theorem 3.4.

Let k be admissible, V a conical self-dual conformal vertex algebra,and ϕ : V k ( g ) ! V a conformal vertex algebra homomorphism. Then ϕ factorsthrough an embedding L k ( g ) ֒ ! V . In particular, V is a direct sum of admissible b g -modules.Proof. It is known [73] that the maximal proper submodule e N of V k ( g ) is generatedby a singular vector v λ of weight, say, λ . Therefore N = ϕ ( e N ) is generated as a b g -module by ϕ ( v λ ) = 0, we shall show that ϕ ( v λ ) = 0.Note that since ϕ is conformal, for a graded V -module M , the restricted dual M ∗ = L d ∈ C Hom C ( M d , C ) ([51]) as a V -module is the same as the restricted dualof M as a V k ( g )-module. In particular, V is self-dual as a V k ( g )-module.Assume that ϕ ( v λ ) = 0. Then we have a non-splitting exact sequence 0 ! N ! V ! V /N ! b g -modules. By the self-duality of V , this gives a non-splittingexact sequence 0 ! ( V /N ) ∗ ! V ! N ∗ ! , (12)of V k ( g )-modules.Let w λ be a weight vector of V of weight λ that is mapped to the highest weightvector of N ∗ . Since V = C | i and (12) is non-splitting, we have | i ∈ U ( b g ) w λ ,and hence, ϕ ( V k ( g )) ⊂ U ( b g ) w λ . As w λ is not a singular vector because otherwise | i 6∈ U ( b g ) w λ , ϕ ( v λ ) and w λ are linearly independent vectors of U ( b g ) w λ , which areprimitive in the sense of [91, 2.6]. This implies that [ U ( b g ) w λ : L ( λ )] > P ( λ ) be the projective cover of L ( λ ) in O k , and consider the homomorphism g : P ( λ ) ։ U ( b g ) w λ ֒ ! V that sends the generator of P ( λ ) of weight λ to w λ . Since ϕ is conformal, the universal Casimir element C of e g acts as zero on V . It followsthat g factors through a homomorphism P ( λ ) /CP ( λ ) ! V . But then Lemma 3.5below says that the multiplicity of L ( λ ) in g ( P ( λ )) ∼ = U ( b g ) w λ is at most one. Sincethis is a contradiction, we get that ϕ ( v λ ) = 0.The last assertion follows from Theorem 3.2. (cid:3) The following assertion is a direct consequence of [15, Lemma 6.9].

Lemma 3.5.

Let µ ∈ b h ∗ k be dominant, suppose λ < µ and there is no ν such that λ < ν < µ . Then [ P ( λ ) /CP ( λ ) : L ( λ )] = 1 .Proof. We have [70] [ M ( µ ) : L ( λ )] = 1 and an exact sequence 0 ! M ( µ ) ! P ( λ ) ! M ( λ ) !

0, see the discussion just before [15, Lemma 6.9] for details.Hence [ P ( λ ) : L ( λ )] = 2. Let ˜ w λ be a generator of P ( λ ) of weight λ . By [15,Lemma 6.9], CP ( λ ) = 0, and hence C ˜ w λ = 0 because ˜ w λ is a generator. It followsthat [ CP ( λ ) : L ( λ )] = 1, and hence, [ P ( λ ) /CP ( λ ) : L ( λ )] = 1. (cid:3) If L ( λ ) is an ordinary module over an admissible aﬃne vertex algebra L k ( g ),then λ ∈ Pr k Z := { λ ∈ Pr k | h λ, α ∨ i ∈ Z for all α ∈ ∆ } (resp. λ ∈ CoPr k Z := { λ ∈ CoPr k | h λ, α ∨ i ∈ Z for all α ∈ ∆ } ) if k is principal (resp. if k is coprincipal). Wehave b ∆( λ ) = { α + nqδ | α ∈ ∆ , n ∈ Z } for λ ∈ Pr k Z , while b ∆( λ ) = { α + nqδ | α ∈ ∆ long , n ∈ Z } ⊔ { α + nqδ : α ∈ ∆ short , n ∈ Z } for λ ∈ CoPr k Z , where ∆ long is theset of long roots of ∆ and ∆ short is the set of short roots of ∆. The set of simpleroots of b ∆( λ ) for λ ∈ Pr k Z and λ ∈ CoPr k Z is given by S ( q ) = { γ , γ , . . . , γ } , where γ i = α i for i = 1 , . . . , ℓ and γ = ( − θ + qδ if ( r ∨ , q ) = 1 , − θ s + qδ/r ∨ if ( r ∨ , q ) = r ∨ , where θ s is the highest short root of ∆.Let φ : e h ∗ ! e h ∗ be the isometry deﬁned to act as the identity on the ﬁnite part h ∗ and to act by φ (Λ ) = (1 /q )Λ , φ ( δ ) = qδ. The adjoint φ ∗ : e h ! e h acts by φ ∗ ( K ) = (1 /q ) K, φ ( D ) = qD. We have [74] Pr k = [ y ∈ f Wy ( S ( q )) ⊂ b ∆re+ Pr ky , Pr ky = yφ ( b P p − h ∨ g ++ + b ρ ) − b ρ, CoPr k = [ y ∈ f Wy ( S ( q )) ⊂ b ∆re+ Pr ky , CoPr ky = yφ ( ◦ b P p − h g ++ + b ρ ) − b ρ. Here ◦ b P k ++ is the set of weights λ of level k such that h λ, α ∨ i i ∈ Z > for i = 1 , . . . , ℓ ,and h λ, − θ ∨ s + K i ∈ Z > , just as b P k ++ is the set of weights λ of level k such that h λ, α ∨ i i ∈ Z > for i = 1 , . . . , ℓ , and h λ, − θ ∨ + K i ∈ Z > . In particular,Pr k Z + b ρ = Pr k + b ρ = φ ( b P p − h ∨ g ++ + b ρ ) , CoPr k Z + b ρ = CoPr k + b ρ = φ ( ◦ b P p − h g ++ + b ρ ) . Proposition 3.6 ([74]) . Let k be principal admissible of the form (9) and let λ = yφ ( ν ) − b ρ ∈ Pr ky with y = t β y , β ∈ P ∨ , y ∈ W , and ν ∈ ◦ b P p − h ∨ + + b ρ . Let T ∈ R > and z ∈ h suchthat α ( z ) / ∈ Z for all α ∈ ∆ . Then as T ! + one has χ L ( λ ) ( iT, − iT z, ∼ b ( λ, z ) e π g / T , OLLAPSING LEVELS OF W -ALGEBRAS 15 where g = dim( g ) − | ρ ∨ | pq = (cid:18) − h ∨ g pq (cid:19) dim( g ) , and b ( λ, z ) = ε ( y ) | P/pqQ | − Y α ∈ ∆ + π ( α | ν ) p · sin π ( α | z − β ) q sin π ( α | z ) . The proof of the proposition below closely follows the arguments of [74], withadaptations to the co-principal case.

Proposition 3.7.

Let k be coprincipal admissible of the form (9) and let λ = yφ ( ν ) − b ρ ∈ CoPr ky , with y = t β y , β ∈ P ∨ , y ∈ W , and ν ∈ ◦ b P p − h ∨ + + b ρ . Let T ∈ R > and z ∈ h suchthat α ( z ) / ∈ Z for all α ∈ ∆ . Then as T ! + we have χ L ( λ ) ( iT, − iT z, ∼ b ( λ, z ) e π g / T , where g = dim( g ) − | ρ ∨ | pq = − h ∨ L g r ∨ pq ! dim( g ) , and b ( λ, z ) = ε ( y ) | P ∨ /pqQ | − Y α ∈ ∆ + π ( α ∨ | ν ) p · sin π ( α ∨ | z − β ) q sin π ( α | z ) . Proof.

The denominator of χ L ( λ ) ( τ, z, t ) = A λ + b ρ ( τ, z, t ) A b ρ ( τ, z, t )(13)is the standard Weyl denominator. Its asymptotic behaviour is well known [69,Proposition 13.13] to be A b ρ ( iT, − iT z, ∼ b ( ρ, z ) T − ℓ/ e − π dim( g ) / T , (14)where b ( ρ, z ) = Y α ∈ ∆ + π ( α | z ) . We analyse the numerator by writing it in terms of theta functions. Let m be apositive integer for which the lattice √ mQ is integral, and let b µ = m Λ + µ ∈ b h ∗ where µ ∈ P ∨ . Then we deﬁneΘ b µ,Q ( v ) = e − | µ | µ ( K ) h δ,v i X α ∈ Q e h t α µ,v i . The modular transformation of these theta functions is given byΘ b µ,Q ( iT, − iT z,

0) = T − ℓ/ | P ∨ /mQ | − / × X µ ′ ∈ P ∨ /mQ e − π i( µ ′ | µ ) /m Θ b µ ′ ,Q (cid:18) − iT , z, − iT ( z | z )2 (cid:19) , where in the sum b µ ′ = m Λ + µ ′ . In the sum deﬁning A λ + b ρ ( v ) the action of the Weyl group c W ( λ ) is intertwinedwith the action of the group c W ( S ( q ) ) = W ⋉ t qQ via the automorphism y . Thenumerator of (13) is thus expressed in terms of theta functions as A λ + b ρ ( v ) = e π i τ | λ + b ρ | k + h ∨ ) X u ∈ c W ( S ( q ) ) ε ( u ) e h yuφ ( ν ) ,v i = ε ( y ) X w ∈ W ε ( w )Θ qw ( b ν ) ,Q ((1 /q ) φ ∗ t ∗ β ( v )) . Now we compute the asymptotic behaviour of ε ( y ) X w ∈ W ε ( w )Θ qw ( b ν ) ,Q ( iT, − iT z, t ) . Since λ is co-principal the lattice pqQ is integral and we may use the modulartransformation formula to write X w ∈ W ε ( w )Θ qw ( b ν ) ,Q ( iT, − iT z, T − ℓ/ | P ∨ /pqQ | − / × X w ∈ W X b µ ∈ P ∨ / ( pq ) Q ε ( w ) e − π i( µ ′ | qw ( ν )) / ( pq ) Θ b µ,Q (cid:18) − iT , z, − iT ( z | z )2 (cid:19) = T − ℓ/ | P ∨ /pqQ | − / e π ( pq ) T ( z | z ) × X w ∈ W X b µ ∈ P ∨ / ( pq ) Q regular X γ ∈ µ/ ( pq )+ Q ε ( w ) e − π i p ( w ( ν ) | µ ) e − πT ( pq )( γ | γ )+2 π i( pq )( γ | z ) . (The summands corresponding to non regular b µ cancel out in the sum over W .) Inthe limit T ! + the dominant terms in the sum above come from the shortestregular µ ∈ P ∨ . Such µ consist precisely of the W -orbit of ρ ∨ . We therefore obtainthe asymptotic X w ∈ W ε ( w )Θ qw ( b ν ) ,Q ( iT, − iT z, ∼ T − ℓ/ | P ∨ / ( pq ) Q | − / e − πT · | ρ ∨| pq × X σ ∈ W " X w ∈ W ε ( w ) e − π i p ( σw ( ν ) | ρ ∨ ) e π i( ρ ∨ | σ ( z )) . The Weyl denominator formula asserts that X w ∈ W ε ( w ) e ( w ( ρ ∨ ) | λ ) = Y α ∈ ∆ + (cid:16) e ( α | λ ) / − e − ( α | λ ) / (cid:17) . (15)(There is a more standard version of the formula, from which (15) is obtained byapplying to the Langlands dual Lie algebra and noting that the duality exchanges ρ with ρ ∨ up to a factor of √ r ∨ .) Using (15) we reduce the double sum over W to Y α ∈ ∆ + π ( α ∨ | ν ) p sin π ( α ∨ | z ) . To deduce the asymptotic behaviour of A λ + b ρ we note that(1 /q ) φ ∗ t ∗ β (2 π i( − τ D + z + tK )) = 2 π i (cid:16) − τ D + ( z + τ ν − ( β )) /q + ( t + τ | β | ) K/q (cid:17) . OLLAPSING LEVELS OF W -ALGEBRAS 17 It then follows that A λ + b ρ ( iT, − iT z, ∼ ε ( y ) T − ℓ/ | P ∨ / ( pq ) Q | − / e − πT · | ρ ∨| pq × Y α ∈ ∆ + π ( α ∨ | ν ) p sin π ( α ∨ | z − β ) q , (16)and, combining with (14), ﬁnally χ L ( λ ) ( iT, − iT z, ∼ ε ( y ) | P ∨ / ( pq ) Q | − / e π T · (cid:18) dim( g ) − | ρ ∨| pq (cid:19) × Y α ∈ ∆ + π ( α ∨ | ν ) p · sin π ( α ∨ | z − β ) q sin π ( α | z ) . The second form for g given in the theorem statement follows from the Freudenthal-de Vries strange formula and the fact that Langlands duality exchanges ρ and ρ ∨ . (cid:3) Corollary 3.8.

Let k be an admissible number of the form (9) . The admissibleaﬃne vertex algebra L k ( g ) admits an asymptotic datum, and so does any simpleordinary representation L ( λ ) of L k ( g ) . (1) For λ ∈ Pr k Z , g L ( λ ) = (cid:18) − h ∨ g pq (cid:19) dim g , w L ( λ ) = 0 , A L ( λ ) = 1 q | ∆ + | | P/ ( pq ) Q ∨ | Y α ∈ ∆ + π ( λ + ρ | α ) p , qdim L ( k Λ ) L ( λ ) = Y α ∈ ∆ + ( λ + ρ | α ) t ( ρ | α ) t , where n t = ( t n − t − n ) / ( t − t − ) , t = e π i /p . (2) For λ ∈ CoPr k Z , g L ( λ ) = − r ∨ h ∨ L g pq ! dim g , w L ( λ ) = 0 , A L ( λ ) = ( r ∨ g ) | ∆ short+ | q | ∆ + | | P ∨ / ( pq ) Q | Y α ∈ ∆ + π ( λ + ρ | α ∨ ) p , qdim L ( k Λ ) L ( λ ) = Y α ∈ ∆ + ( λ + ρ | α ∨ ) t ( ρ | α ∨ ) t . Proof.

The assertion follows from Propositions 3.6 and 3.7 by taking the limit z ! β = 0 and ¯ y = 1. Let us explain the details for the coprincipal case. Thenormalised character χ L ( λ ) ( τ ) is just the specialization χ L ( λ ) ( τ, , β = 0, y = 1, ν = λ + ρ , andwe let z tend to 0 along a ray disjoint from the hyperplanes ( α | z ) = 0. Comparingthe deﬁnition of the asymptotic dimension A (Deﬁnition 2.1 above) with that of b ( λ, z ) yields A L ( λ ) = lim z ! b ( λ, z )= | P ∨ /pqQ | − / Y α ∈ ∆ + π ( α ∨ | λ + ρ ) p · | α ∨ | /q | α | = | P ∨ /pqQ | − / ( r ∨ ) | ∆ short+ | q | ∆ + | Y α ∈ ∆ + π ( α ∨ | ρ ) p . Here we used l’Hopital’s rule. (cid:3)

In the above we have A L ( λ ) > < ( λ + ρ, α ) = ( α | α )2 ( λ + ρ | α ∨ ) < p for λ ∈ Pr k , α ∈ ∆ + , and 0 < ( λ + ρ, α ∨ ) < p for λ ∈ CoPr k , α ∈ ∆ + .Note that | P/ ( pq ) Q ∨ | = ( pq ) ℓ | P/Q ∨ | and | P ∨ /pqQ ∨ | = (cid:0) pqr ∨ (cid:1) ℓ | P ∨ /r ∨ Q | if r ∨ | q .The values of | P/Q ∨ | and | P ∨ /r ∨ Q | , as well as other useful data, are collected inTable 1 for each simple Lie algebra. g A ℓ , ℓ > B ℓ , ℓ > C ℓ , ℓ > D ℓ , ℓ > E E E F G dim g ℓ ( ℓ + 2) ℓ (2 ℓ + 1) ℓ (2 ℓ + 1) ℓ (2 ℓ −

1) 78 133 248 52 14 h g ℓ + 1 2 ℓ ℓ ℓ − h ∨ g ℓ + 1 2 ℓ − ℓ + 1 2 ℓ − | P ∨ /Q ∨ | ℓ + 1 2 2 4 3 2 1 1 1 | P/Q ∨ | ℓ + 1 4 2 ℓ | P ∨ /r ∨ Q | ℓ + 1 2 ℓ | ∆ + | ℓ ( ℓ +1)2 ℓ ℓ ℓ ( ℓ −

1) 36 63 120 24 6 | ∆ short+ | ℓ ℓ ( ℓ −

1) 12 3

Table 1.

Some data for simple Lie algebras

Theorem 3.9.

Let k be admissible, and let V be a conical self-dual conformalvertex algebra equipped with a vertex algebra homomorphism ϕ : V k ( g ) ! V suchthat ϕ ( ω V k ( g ) ) ∈ V and ( ω V ) (2) ϕ ( ω V k ( g ) ) = 0 . Assume that V is a quotient of avertex algebra ˜ V that admits an asymptotic datum and χ ˜ V ( τ ) ∼ χ L k ( g ) ( τ ) , as τ , that is, A ˜ V = A L k ( g ) , w ˜ V = 0 , and g ˜ V = g L k ( g ) . Then ϕ factors through an isomorphism L k ( g ) ∼ = V .Proof. By Proposition 2.9, ϕ is conformal. Hence, by Theorem 3.4, V is a direct sumof admissible representations L ( λ ) with λ ∈ Pr k Z if k is principal (resp. λ ∈ CoPr k Z if k is coprincipal). So we can write V = M λ ∈ Pr k Z L ( λ ) ⊕ m λ , (resp. V = M λ ∈ CoPr k Z L ( λ ) ⊕ m λ )with m k Λ = 1 if k is principal (resp. if k is coprincipal). In particular, V admitsan asymptotic datum with g V = g L k ( g ) = g ˜ V , w V = 0, A V = P λ ∈ Pr k Z m λ A L ( λ )OLLAPSING LEVELS OF W -ALGEBRAS 19 (resp. A V = P λ ∈ CoPr k Z m λ A L ( λ ) ). Since A V A ˜ V = A L k ( g ) and A L ( λ ) > λ ∈ Pr k Z (resp. λ ∈ CoPr k Z ), we get that the assertion. (cid:3) For an admissible number k with denominator q , set λ o = ( ρ/q − ρ if ( q, r ∨ ) = 1 ,ρ ∨ /q − ρ if ( q, r ∨ ) = 1 . (17)Then by [14, Proposition 2.4], we have λ o,k := λ o + k Λ ∈ ( Pr k if ( q, r ∨ ) = 1 , CoPr k if ( q, r ∨ ) = 1 . (18) Theorem 3.10 ([14]) . For an admissible number k , we have Var( J λ o ) = O k .Remark . Let k be an admissible number and let λ ∈ h ∗ such that λ + k Λ ∈ Adm k . Since Var( I k ) = O k by [9, Theorem 9.5] (recall (11)) and L g ( λ ) is aZhu( L k ( g ))-module (Theorem 3.2), we haveVar( J λ ) ⊂ O k . Proposition 3.12.

For an admissible number k , L ( λ o,k ) has the minimal confor-mal dimension among the simple L k ( g ) -modules that belong to O k .Proof. By [8] a simple L k ( g )-module which belongs to O k is an irreducible highestweight module L ( b µ ) of highest weight b µ = µ + k Λ ∈ Adm k .First we consider the principal admissible case ( q, r ∨ ) = 1. We have, cf. Propo-sition 3.6 above and preceding remarks, b µ = yφ ( b ν ) − b ρ where b ν = ν + p Λ and ν ∈ P p, reg+ , where y = yt − η with η ∈ P ∨ ,q + .The conformal weight of L ( b µ ) is given by( µ | µ + 2 ρ )2( k + h ∨ g ) = | µ + ρ | − | ρ | k + h ∨ g ) . Now we consider | µ + ρ | = | yφ ( ν ) | = | ν − pq η | = 1 q | qν − pη | . We show that qν − pη ∈ P is a regular weight. Indeed suppose h qν − pη, α ∨ i i = 0for some i = 1 , . . . , ℓ and put m = h ν, α ∨ i i and n = h η, α ∨ i i . Then qm − pn = 0 andso p divides m , but this is a contradiction since 0 < m < p because ν is regular.The regular elements of P of minimal norm are ρ and its images under the ﬁniteWeyl group, so the claim is proved in the principal admissible case. The coprincipaladmissible case is similar. Since ( q, r ∨ ) = r ∨ we have qν ∈ qP ⊂ P ∨ and hence qν − pη ∈ P ∨ . The regular elements of P ∨ of minimal norm are ρ ∨ and its imagesunder the ﬁnite Weyl group, thus proving the claim. (cid:3) Remark . (1) The weight λ o,k is not the unique element of Adm k that gives the minimalconformal dimension unless k ∈ Z > . Indeed, for ˆ µ = µ + k Λ ∈ Adm k . L (ˆ µ ) has the minimal conformal dimension if and only of µ + ρ = w ( λ o + ρ )with w ∈ W such that w ( α ) ∈ ∆ + for α ∈ ∆ + with ( ρ, α ∨ ) ∈ q Z (resp. λ + ρ = w ( ρ ∨ ) /q with w ∈ W such that w ( α ) ∈ ∆ + for α ∈ ∆ + with( ρ ∨ , α ∨ ) ∈ q Z ). (2) The strange formula implies that the asymptotic growth coincides with theeﬀective central charge, that is, g L ( λ ) = c L k ( g ) − h min for all λ ∈ Adm k .4. Asymptotic data of W -algebras Let f be a nilpotent element of g . Recall [72] that a Z -grading g = M j ∈ Z g j Γ (19)is called good for f if f ∈ g − , ad f : g j Γ ! g j − is injective for j > and surjectivefor j . The grading is called even if g j Γ = 0 for j Z . By the Jacobson-MorosovTheorem, the nilpotent element f embeds into an sl -triple ( e, h, f ), and g therebyinherits a Z -grading induced by the eigenvalues of ad( h/ Dynkin grading . All Dynkin gradings are good, but not all good gradingsare Dynkin.Let x be the semisimple element of g deﬁning the grading (19), that is, g i Γ = { y ∈ g : [ x , y ] = iy } . We can assume that x is contained in the Cartan subalgebra h and that α ( x ) ∈ Z > for all α ∈ ∆ + . For j ∈ Z , we set∆ j Γ := { α ∈ ∆ : ( α | x ) = j } = { α ∈ ∆ : g α ⊂ g j Γ } , where g α is the α -root space.Since the bilinear form ( x, y ) ( f | [ x, y ]) is non-degenerate on g / × g / , theset ∆ / has even cardinality. We set ∆ , + = ∆ ∩ ∆ + . It is a set of positive rootsfor ∆ . Remark . Unless otherwise speciﬁed, we will always use the Dynkin gradingassociated with ( e, h, f ) as good grading Γ. In this case, x is h/

2, and we willbrieﬂy write x , g j , ∆ j , ∆ for x , g j Γ , ∆ j Γ , ∆ , + , respectively. However, in type A , even good grading always exist [43], and it will be convenient to opt for an evengood grading that is not necessarily the Dynkin grading.We denote by W k ( g , f ) the universal aﬃne W -algebra associated with g , f atlevel k and a good grading g Γ = L j ∈ Z g j Γ : W k ( g , f ) = H DS,f ( V k ( g )) , where H • DS,f (?) is the BRST cohomology functor of the quantized Drinfeld-Sokolovreduction associated with g , f ([46, 72]). The BRST complex of a b g -module M takesthe form C • ( M ) = M ⊗ F ch ⊗ F ne , where F ch denotes the Cliﬀord vertex algebra associated with g > ⊕ ( g > ) ∗ and itscanonical symmetric bilinear form, or in the terminology of [72] the charged freefermions vertex algebra associated with g > ⊕ ( g > ) ∗ made into a purely odd vectorsuperspace and canonical skew-supersymmetric form, and F ne denotes the neutral OLLAPSING LEVELS OF W -ALGEBRAS 21 free fermion vertex algebra associated with g / and bilinear form ( x, y ) ( f | [ x, y ]).We omit the detailed deﬁnition, referring the reader to ([72]).The W -algebra W k ( g , f ) is conformal provided that k = − h ∨ and its centralcharge c W k ( g ,f ) is given by c W k ( g ,f ) = c V k ( g ) − dim G.f −

32 dim g / + 24( ρ | x ) − k + h ∨ g ) | x | (20) = dim g −

12 dim g / − k + h ∨ g | ρ − ( k + h ∨ g ) x | (21)([72, Theorem 2.2]). Although the vertex algebra structure of W k ( g , f ) does notdepend on the choice of the good grading of g ([17, § W k ( g , f ) be the unique simple graded quotient of W k ( g , f ). Proposition 4.2.

The vertex algebra W k ( g , f ) is self-dual if the grading (19) isDynkin.Proof. By [13, Proposition 6.1 and Remark 6.2], it is suﬃcient to show that ( k + h ∨ )( h | v ) − tr g > (ad v ) = 0 for all v ∈ g f . Since ( h | v ) = ([ e, f ] | v ) = 0 for v ∈ g f ,it is enough to show that tr g > (ad v ) = 0. Note that g f is the centraliser g ♮ of the sl -triple { e, h, f } in g , which is a reductive Lie subalgebra of g . We clearly havetr g > (ad v ) = 0 for v ∈ [ g ♮ , g ♮ ] since g > is a ﬁnite-dimensional representation of thesemisimple Lie algebra [ g ♮ , g ♮ ]. On the other hand, Lemma 5.5 below shows thattr g > (ad v ) = 0 for an element v in the centre z ( g ♮ ) of g ♮ as well. (cid:3) The associated variety X W k ( g ,f ) is isomorphic to the Slodowy slice S f = f + g e , where g e is the centraliser of e in g ([33]). There is a natural C ∗ -action on S f thatcontracts to f ([100]), and X W k ( g ,f ) is a C ∗ -invariant subvariety of S f . Theorem 4.3 ([10]) . (1) We have H iDS,f ( M ) = 0 for any M ∈ KL k and i = 0 . Hence, the functor KL k ! W k ( g , f ) -Mod , M H DS,f ( M ) , is exact. (2) The W k ( g , f ) -module H DS,f ( M ) is ordinary for a ﬁnitely generated object M ∈ KL k . (3) For any quotient V of V k ( g ) , H DS,f ( V ) is a quotient of W k ( g , f ) providedthat it is nonzero, and we have X H DS,f ( V ) = X V ∩ S f , which is a C ∗ -invariant subvariety of S f ( [10] ). In particular, (a) H DS,f ( L k ( g )) = 0 if and only if f ∈ X L k ( g ) ; (b) If G.f ⊂ X L k ( g ) ⊂ N , then H DS,f ( V ) is quasi-lisse and so is W k ( g , f ) ; (c) If X L k ( g ) = G.f , then H DS,f ( V ) is lisse and so is W k ( g , f ) . Conjecture 4.4 ([72, 76]) . H DS,f ( L k ( g )) is either zero or isomorphic to W k ( g , f ) . Theorem 4.5 ([13]) . Conjecture 4.4 holds if k is an admissible level, f ∈ X L k ( g ) = O k , and f admits an even good grading. By Theorem 3.1 and Theorem 4.3, W k ( g , f ) is quasi-lisse if k is admissible and f ∈ O k ; W k ( g , f ) is lisse if k is admissible and f ∈ O k . Moreover, under the assumptionof Theorem 4.5, the associated variety of W k ( g , f ) is equal to the nilpotent Slodowyslice S O k ,f := S f ∩ O k . (22) Conjecture 4.6 ([50, 76, 10]) . W k ( g , f ) is rational if k is admissible and f ∈ O k . Conjecture 4.6 has been proved in the cases where f is principal [9], g is of type A [13], g is of type ADE and f is subregular [13], and g is of type B and f issubregular [45].By [36], Zhu( W k ( g , f )) is naturally isomorphic to the ﬁnite W -algebra [96] U ( g , f ) associated with ( g , f ). More generally, we have the following assertion. Theorem 4.7 ([9]) . We have the isomorphism

Zhu( H DS,f ( L k ( g )) ∼ = H DS,f (Zhu( L k ( g )) , where on the right-hand-side H DS,f (?) is the ﬁnite-dimensional analogue of theDrinfeld-Sokolov reduction functor that is denoted by M M † in [88] . For a two sided ideal I of U ( g ), let Var( I ) be the zero locus of gr I ⊂ gr U ( g ) = C [ g ∗ ] in g ∗ . We have H DS,f ( U ( g ) /I ) = 0 ⇐⇒ G.f ⊂ Var( I )(23)[56, 89], see [9, Section 2]. Therefore, the following assertion follows from Theorems3.3 and 4.7. Corollary 4.8.

Let k be an admissible number. We have Zhu( H DS,f ( L k ( g )) ∼ = Y H DS,f ( U ( g ) /J ¯ λ ) , where the product is taken over [ λ ] ∈ [Pr k ] (resp. [ λ ] ∈ [CoPr k ] ) such that G.f ⊂ Var( J ¯ λ ) . The image [ ω ] of the conformal vector of W k ( g , f ) in Zhu( H DS,f ( L k ( g )) acts on H DS,f ( U ( g ) /J ¯ λ ) as the constant multiplication by h λ = | λ + ρ | − | ρ | k + h ∨ g ) − k + h ∨ g | x | + ( x , ρ ) . (24)In particular, a simple module of H DS,f ( L k ( g )) corresponding to a simple moduleof H DS,f ( U ( g ) /J ¯ λ ) via Corollary 4.8 has the conformal dimension h λ .For a W k ( g , f )-module M , we set χ H • DS,f ( M ) ( τ, z ) = X i ∈ Z ( − i tr H iDS,f ( M ) ( e π i z q L − c/ ) , ( τ, z ) ∈ H × h f , when it is well-deﬁned, where now L = L Sug0 − ( x ) + L ch0 + L ne0 . Note that by Theorem 4.3, χ H • DS,f ( L ( λ )) ( τ, z ) is well-deﬁned for a ﬁnitely generatedobject M in KL k and we have χ H • DS,f ( M ) ( τ, z ) = tr H DS,f ( M ) ( e π i z q L − c/ ) . OLLAPSING LEVELS OF W -ALGEBRAS 23 Proposition 4.9.

Let M ∈ O k such that χ M ( τ, − τ z, ∼ A M ( z )( − i τ ) w M e π i τ g M , and suppose that χ H • DS,f ( M ) ( τ, z ) is well-deﬁned. Then χ H • DS,f ( M ) ( τ, − τ z ) ∼ A H • DS,f ( M ) ( z )( − i τ ) w M e π i τ ( g M − dim O f ) , where A H • DS,f ( M ) ( z ) := Q α ∈ ∆ + π ( α | z + x ) Q α ∈ ∆ , + ∪ ∆ / , + π ( α | z + x ) A M ( z + x ) . Proof.

Let us write χ M ( τ, z, t ) = ( A b ρ χ M )( τ, z, t ) A b ρ ( τ, z, t ) . The asymptotic behaviour of A b ρ ( τ, z, t ) is well known [69, Proposition 13.13] to be A b ρ ( τ, − τ z, ∼ b ( ρ, z )( − i τ ) − ℓ/ e − π i τ dim( g ) (25)where b ( ρ, z ) = Y α ∈ ∆ + π ( α | z ) . Hence χ M ( τ, − τ z, ∼ A M ( z )( − i τ ) w M e π i τ g M (26) ⇐⇒ ( A b ρ χ M )( τ, − τ z, ∼ b ( ρ, z ) A ( ρ, z )( − i τ ) w M − ℓ e π i τ ( g M − dim g ) . Now by Theorem 4.3 and the Euler-Poincar´e principle (see the discussion in [76,Section 2])ch H DS,f ( M ) ( τ, z ) = lim ε ! (I) · (II) , where (I) = tr M q L Sug0 − ( x ) e π i( z + εx ) and (II) = str F ch ⊗ F ne q L ch0 + L ne0 e π i(( z + εx ) ch0 +( z + εx ) ne0 ) . Here L Sug0 denotes the zero mode of the Sugawara conformal vector in V k ( g ) while L ch0 and L ne0 denote the conformal vectors in F ch and F ne given in [72, Section 2.2],or in [76, Section 2.1]. We thus have χ H DS,f ( M ) ( τ, z ) = lim ǫ ! ( A b ρ ch M )( τ, z − τ x + εx , ψ ( τ, z − τ x + εx , , (27)in which ψ ( τ, z, t ) = e π i h ∨ t η ( τ ) ℓ Y α ∈ ∆ , + ∪ ∆ / , + f ( τ, ( α | z )) , where f ( τ, s ) = e π i τ/ e π i s ∞ Y n =1 (cid:16) − e π i(( n − τ − s ) (cid:17) (cid:16) − e π i( nτ + s ) (cid:17) . We now describe the proof of (27) brieﬂy, referring the reader to [76, Section 2.2] fordetails. The numerator of (27) comes directly from the numerator of the expression(13) for χ L ( λ ) . The modiﬁcation z z − τ x corresponds to the shift of Virasoro operator L Sug0 L Sug0 − ( x ) . The Weyl denominator is η ( τ ) ℓ times a productof theta functions indexed by α ∈ ∆ + . At the level of characters the eﬀect of thetensor product with F ch is to cancel those factors associated with α ∈ ∆ j Γ for j > F ne reintroduces those factors associated with α ∈ ∆ / , + (whose cardinality is half that of ∆ / ). Ultimately this yields the denominator ψ ( τ, z − τ x , t ) of (27).The function ψ satisﬁes the asymptotic ψ ( τ, − τ z, ∼ Y α ∈ ∆ , + ∪ ∆ / , + π ( α | z )( − i τ ) − ℓ/ e i π dim( g f ) / τ (28)as τ

0. The asymptotic behaviour of the numerator A ν + b ρ ( τ, z − τ x , t ) wasestablished in the proof of Proposition 3.7.The required asymptotic follows from (26), (27) and (28), together with the factthat dim O f = dim g − dim g f . (cid:3) Proposition 4.10.

Let k = − h ∨ g + pq be an admissible level and let f ∈ X L k ( g ) .Then H DS,f ( L ( λ )) admits an asymptotic datum for all simple ordinary L k ( g ) -module L ( λ ) with w H DS,f ( L ( λ )) = 0 . Moreover: (1) ( [76] ) For λ ∈ Pr k Z , g H DS,f ( L ( λ )) = g L k ( g ) − dim G.f = dim g f − h ∨ g dim g pq , A H DS,f ( L ( λ )) = 12 | ∆1 / | q | ∆ , + | | P/ ( pq ) Q ∨ | × Y α ∈ ∆ + π ( λ + ρ | α ) p Y α ∈ ∆ + \ ∆ , + π ( x | α ) q . (2) For λ ∈ CoPr k Z , g H DS,f ( L ( λ )) = g L k ( g ) − dim G.f = dim g f − r ∨ h ∨ L g dim g pq , A H DS,f ( L ( λ )) = ( r ∨ g ) | ∆ short+ ∩ ∆ | | ∆1 / | q | ∆ , + | | P ∨ / ( pq ) Q | × Y α ∈ ∆ + π ( λ + ρ | α ∨ ) p Y α ∈ ∆ + \ ∆ , + π ( x | α ∨ ) q . In particular, qdim H DS,f ( L k ( g )) H DS,f ( L ( λ )) = qdim L k ( g ) L ( λ ) for any simple ordinary L k ( g ) -module L ( λ ) . OLLAPSING LEVELS OF W -ALGEBRAS 25 Proof.

Let k be coprincipal and let λ ∈ CoPr k . By Proposition 3.7 and Proposi-tion 4.9, χ H DS,f ( L ( λ )) ( iT, − iT z, ∼ ε ( y ) e − π g Lk ( g ) / T | P ∨ / ( pq ) Q | / · Q α ∈ ∆ + π ( α ∨ | ν ) p sin π ( α ∨ | z + x − β ) q Q α ∈ ∆ , + ∪ ∆ / , + π ( α | z + x ) , (29)where ν , ¯ y are as in Proposition 3.7.We are interested in the case λ ∈ CoPr k Z which corresponds to y = 1, β = 0 and ν = λ + b ρ . We now simplify the products appearing in the asymptotic above Y α ∈ ∆ + π ( α ∨ | λ + ρ ) p Y α ∈ ∆ / , +

12 sin π ( α | z + x ) Y α ∈ ∆ , + π ( α ∨ | z + x ) q π ( α | z + x ) Y α ∈ ∆ > / , + π ( α ∨ | z + x ) q . The second product becomes simply 2 −| ∆ / | / . In the third product the terms x are irrelevant since α ∈ ∆ , + . In the limit z ! Y α ∈ ∆ , + | α ∨ | /q | α | = ( r ∨ ) | ∆ , + ∩ ∆ short | q | ∆ , + | . In the fourth product the limit obtains by simply putting z = 0. Thus we haveobtained the stated formula. (cid:3) Theorem 4.11.

Let k be admissible, and let f ∈ g be a nilpotent element thatadmits an even good grading. (1) Let f ∈ O k , so that H DS,f ( L k ( g )) ∼ = W k ( g , f ) (see Theorem 4.5). Then h λ o is the minimal conformal dimension among simple positive energy rep-resentations of W k ( g , f ) (see (17) and (24) ), and we have g W k ( g ,f ) = c W k ( g ,f ) − h λ o . (2) Suppose further that f ∈ O k , so that W k ( g , f ) is lisse. Then there exists aunique simple W k ( g , f ) -module that has the minimal conformal dimension h λ o .Proof. (1) By Theorem 3.10, Var( J λ o ) = O k , which contains the orbit G.f bythe assumption. Hence, H DS,f ( U ( g ) /J λ o ) is nonzero by (23). It follows fromCorollary 4.8 that there exists a simple W k ( g , f )-module corresponding a simple H DS,f ( U ( g ) /J λ o )-module, which has the minimal conformal dimension h λ o amongsimple positive energy representations of W k ( g , f ) by Proposition 3.12 and (24).We have g W k ( g ,f ) = g L k ( g ) − dim G.f (Proposition 4.10)= c V k ( g ) − h λ o + k + h ∨ g | x | − ( x | ρ )) − dim G.f (Remark 3.13 and (24)) , where Γ is a good even grading. Hence the assertion follows from (21) and the factthat dim g / = 0 by the assumption. (2) By [13], each factor H DS,f ( U ( g ) /J ¯ λ ) in Corollary 4.8 is a simple algebra, which has a unique simple module. Hence theassertion follows from Remark 3.13. (cid:3)

In view of Proposition 2.4, Theorem 4.11 gives a supporting evidence for Con-jecture 4.6. 5.

Collapsing levels for W -algebras In general, if s is a semisimple Lie algebra, we write κ s for the Killing form of s .For now, assume that s a simple. We denote by h ∨ s its dual Coxeter number andby ( | ) s the normalised inner product 12 h ∨ s κ s so that ( θ | θ ) = 2 for θ the highestpositive root of s .As in Section 4, ﬁx an sl -triple ( e, h, f ) of g , with related notation. In particular,for j ∈ Z , recall that g j = { x ∈ g : [ h, x ] = 2 jx } , and g is the centraliser of h in g . The centraliser g ♮ of the sl -triple ( e, h, f ) in g is given by g ♮ = g ∩ g f = g e ∩ g f . (30)The Lie algebra g ♮ is a reductive subalgebra of g , and we can write g ♮ = g ♮ ⊕ [ g ♮ , g ♮ ],with g ♮ the centre of the reductive Lie algebra g ♮ . Denoting by g ♮ , . . . , g ♮s the simplefactors of [ g ♮ , g ♮ ], we get g ♮ = s M i =0 g ♮i . More generally, for g Γ = L j ∈ Z g j Γ a good grading for f , set g ♮ Γ := g ∩ g f . (31)We note that g ♮ Γ is no longer a reductive Lie algebra in general. By [43, Lemma1.2], g ♮ is a maximal reductive subalgebra of g f . On the other hand, since g ♮ isreductive and contained in L j > g j Γ , we have g ♮ ⊂ g . Hence, g ♮ is a maximalreductive subalgebra of g ♮ Γ .Let i ∈ { , . . . , s } . For any element x ∈ g ♮i ⊂ g ♮ Γ the adjoint action ad x restrictsto an endomorphism of g j Γ which we denote ρ jx , for any j . Setting κ g j Γ ( x, x ) =tr( ρ jx ◦ ρ jx ), one deﬁnes an invariant bilinear form on g ♮i by (see [75, Theorem 2.1]for the case where Γ is the Dynkin grading): φ ♮ Γ ,i ( x | x ) := k ( x | x ) g + 12 ( κ g ( x, x ) − κ g ( x, x ) − κ g / ( x, x )) . Thus, for i = 0, there exists a scalar k ♮ Γ ,i such that φ ♮ Γ ,i = k ♮ Γ ,i ( | ) i where ( | ) i := ( | ) g ♮i . (32)Note that V φ ♮ Γ , ( g ♮ ) ∼ = M (1) ⊗ rank φ ♮ Γ , , where M (1) is the Heisenberg vertexalgebra of central charge 1.Following [31], we say that two good gradings Γ , Γ ′ are adjacent if g = M i − j i + g i Γ ∩ g j Γ ′ , OLLAPSING LEVELS OF W -ALGEBRAS 27 where i − denotes the largest half-integer strictly smaller than i and i + denotes thesmallest half-integer strictly greater than i . Lemma 5.1.

Let Γ , Γ ′ be two good gradings for f . If Γ , Γ ′ are adjacent, then φ ♮ Γ ,i = φ ♮ Γ ′ ,i for i = 0 , . . . , s .Proof. According to [31, Theorem 2 and Lemma 26], there are Lagrangian subspaces l Γ and l Γ ′ of g / and g / ′ , respectively, with respect to the non-degenerate bilinearform ( f | [ · , · ]) such that l Γ ⊕ g > / = l Γ ′ ⊕ g > / ′ and l − Γ ⊕ g < − / = l − Γ ′ ⊕ g < − / ′ , where a − denotes the dual space to the subspace a ⊂ g with respect to the Killingform of g . Then for any x ∈ g ♮ , κ g ( x, x ) − κ g ( x, x ) − κ g / ( x, x ) = 2 κ l Γ ⊕ g > / ( x, x )= κ g ( x, x ) − κ g ′ ( x, x ) − κ g / ′ ( x, x ) . Indeed, from the decompositions g = ( l − Γ ⊕ g < − / ) ⊕ ( k − Γ ⊕ g ⊕ k Γ ) ⊕ ( l Γ ⊕ g > / ) , g = ( l − Γ ′ ⊕ g < − / ′ ) ⊕ ( k − Γ ′ ⊕ g ′ ⊕ k Γ ′ ) ⊕ ( l Γ ′ ⊕ g > / ′ ) , where k Γ (resp. k Γ ′ ) is a Lagrangian complement in g / (resp. g / ′ ) to l Γ (resp. l Γ ′ ),we deduce that κ g / ( x, x ) = κ k Γ ⊕ k − Γ ( x, x ) and κ g / ′ ( x, x ) = κ k Γ ′ ⊕ k − Γ ′ ( x, x ) . This completes the proof. (cid:3)

Denoting by Γ D the Dynkin grading, we set φ ♮i := φ ♮ Γ D ,i , and k ♮i := k ♮ Γ D ,i for i = 0 . Remark . In type A , according to [31, Lemma 26] and [31, Section 6], there isa chain Γ , . . . , Γ t of good gradings for f such that Γ = Γ, for any i ∈ { , . . . , t } ,Γ i − and Γ i are adjacent, and one of these good gradings is even. Hence, in type A , we are free to use an even good grading Γ to compute k ♮i , which is convenient(see the proof of Lemma 8.4).Set V k ♮ ( g ♮ ) := V φ ♮ ( g ♮ ) ⊗ s O i =1 V k ♮i ( g ♮i ) . (33)By [75, Theorem 2.1], there exists an embedding ι : V k ♮ ( g ♮ ) ֒ −! W k ( g , f )(34)of vertex algebras. We have ι ( ω V k♮ ( g ♮ ) ) ∈ W k ( g , f ) , ( ω W k ( g ,f ) ) (2) ι ( ω V k♮ ( g ♮ ) ) = 0 . (35)We denote by V k ( g ♮ ) the image in W k ( g , f ) of the embedding ι , and by V k ( g ♮ )the image of V k ( g ♮ ) by the canonical projection π : W k ( g , f ) ։ W k ( g , f ). Deﬁnition . If W k ( g , f ) ∼ = V k ( g ♮ ), we say that the level k is collapsing . Lemma 5.4.

The level k is collapsing if and only if W k ( g , f ) ∼ = L k ♮ ( g ♮ ) , where L k ♮ ( g ♮ ) stands for s O i =0 L k ♮i ( g ♮i ) . Equivalently, k is collapsing for W k ( g , f ) ifand only if there exists a surjective vertex algebra homomorphism W k ( g , f ) ։ L k ♮ ( g ♮ ) . For example, if W k ( g , f ) ∼ = C , then k is collapsing. Proof. If W k ( g , f ) ∼ = V k ( g ♮ ), then W k ( g , f ) is isomorphic to the quotient of N si =0 V k ♮i ( g i )by the kernel of the compound map π ◦ ι . Since W k ( g , f ) is simple we deduce thatthis quotient is isomorphic to N si =0 L k ♮i ( g i ) = L k ♮ ( g ♮ ). Conversely, if W k ( g , f ) ∼ = N si =0 L k ♮i ( g i ), then π ◦ ι factorises through N si =0 L k ♮i ( g i ), and so W k ( g , f ) is iso-morphic to the image of this induced map, so W k ( g , f ) ∼ = V k ( g ♮ ). (cid:3) Lemma 5.5.

The centre z ( g ♮ ) of the reductive Lie algebra g ♮ consists of semisim-ple elements of g . Moreover, for any x ∈ z ( g ♮ ) , we have tr g f (ad x ) = 0 and tr g > (ad x ) = 0 , where ad x stands for the endomorphism of g f (resp. g > ) inducedfrom the adjoint action of x acting on g f (resp. g > ).Proof. As previously mentioned, g ♮ is a maximal reductive subalgebra of g f ([43,Lemma 1.2]). Let t be a maximal torus of g = g h containing a maximal torus of g ♮ and set t ♮ := t ∩ g ♮ = t ∩ g f . We intend to show that z ( g ♮ ) is contained in t ♮ .We use the decomposition in t ♮ -weight spaces of g ♮ following [31, Section 2]. For α ∈ ( t ♮ ) ∗ and n >

0, let L ( α, n ) denote the irreducible t ♮ ⊕ s -modules of dimension( n + 1) on which t ♮ acts by the weight α , where s ∼ = sl is the Lie algebra generatedby e, h, f . Since each L ( α, n ) contains a nonzero vector annihilated by f , the set ofweights of t ♮ on g is also the set of weights of t ♮ on g f . Let Φ f ⊂ ( t ♮ ) ∗ be the set ofall nonzero weights of t ♮ on g f . We have the following decomposition g f = g t ♮ ∩ g f ⊕ M α ∈ Φ fi g f ( α, i ) , (36)where g t ♮ is the centraliser of t ♮ in g (it is a Levi subalgebra of g ) and g f ( α, i ) = { x ∈ g f : [ h, x ] = ix and [ t, x ] = α ( t ) x for all t ∈ t ♮ } . This decomposition is compatible with the decomposition g f = g ♮ ⊕ g f< , and wehave g ♮ = t ♮ ⊕ M α ∈ Φ ◦ f g f ( α, , where Φ ◦ f denotes the set of all nonzero element of Φ f .Let now x ∈ z ( g ♮ ) that we write x = x + P α ∈ Φ ◦ f x α relatively to the abovedecomposition, with x ∈ t ♮ and x α ∈ g f ( α,

0) for α ∈ Φ ◦ f . Let α ′ ∈ Φ ◦ f and pick OLLAPSING LEVELS OF W -ALGEBRAS 29 t ′ ∈ t ♮ such that α ′ ( t ′ ) = 0. From the equalities0 = [ y, x ] = [ y, x ] + X α ∈ Φ ◦ f [ y, x α ] = 0 + X α ∈ Φ ◦ fα = α ′ α ( t ) x α + α ′ ( t ′ ) x α ′ , we deduce that x α ′ = 0. Since this is true for each α ∈ Φ ◦ f , we get that x = x ∈ t ♮ .In particular, x is a semisimple elements of g .To show the second assertion, ﬁrst note that we have the following decomposition g = M α ∈ Φ f M n > m ( α, n ) L ( α, n ) , (37)where m ( α, n ) = 0 for all α ∈ Φ f . Since Φ f is a restricted root system in the senseof [31, Section 2], for α ∈ ( t ♮ ) ∗ , α ∈ Φ f implies that − α ∈ Φ f and the correspondingroot vector spaces have the same dimension. So the second assertion follows fromthe decompositions (36) and (37). (cid:3) The main strategy

We describe in this section our general strategy to ﬁnd new (admissible) collaps-ing levels using Theorem 3.9. 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 W k ( g , f ) and if Conjecture 4.4 istrue, then obviously X W k ( g ,f ) ∼ = X L k♮ ( g ♮ ) (38)as Poisson varieties. Lemma 6.1.

Suppose that k is collapsing. If W k ( g , f ) is quasi-lisse then φ ♮ isidentically zero on g ♮ . In particular, if k is admissible and f ∈ O k , then φ ♮ = 0 . Our convention is that φ ♮ = 0 when g ♮ = { } . Proof.

The associated variety X L k♮ ( g ♮ ) is a subvariety of ( g ♮ ) ∗ × ( g ♮ ) ∗ × · · · × ( g ♮s ) ∗ and the symplectic leaves of ( g ♮ ) ∗ × ( g ♮ ) ∗ × · · · × ( g ♮s ) ∗ are the coadjoint orbitsof G ♮ × · · · × G ♮s , where G ♮i is the adjoint group of g ♮i . Recall that V φ ♮ ( g ♮ ) isa Heisenberg vertex algebra of rank dim g ♮ . The associated variety of its simplequotient is C dim rank φ ♮ , provided that φ ♮ = 0. Hence, X L k♮ ( g ♮ ) has ﬁnitely manysymplectic leaves if and only if it is contained in N g × · · · × N g s , where N g i is thenilpotent cone of g ♮i . In particular, we must have X L φ♮ ( g ♮ ) = { } . This happens ifand only if φ ♮ = 0. (cid:3) If the centre g ♮ is one dimensional, then the Heisenberg vertex algebra V φ ( g ♮ )does not depend on the choice of a nonzero bilinear symmetric form φ on g ♮ . In thiscase, we set k ♮ := φ ♮ ( x, x ) for any nonzero x ∈ g ♮ and write V k ♮ ( g ♮ ) for V φ ♮ ( g ♮ ).Thus k ♮ is well-deﬁned only up to nonzero scalar. In most of cases, either g ♮ = 0 ordim g ♮ = 1. More speciﬁcally, for all nilpotent orbits G.f that we consider in theclassical cases we have dim g ♮ G.f in the exceptional cases except for two, we also have dim g ♮

1: see Tables 11–17. In the two exceptions (which occurs in types E and E ), g ♮ = g ♮ ∼ = C and, by abuse of notation, we will also use the symbol k ♮ to denote the value of φ ♮ ( x, x ) forsome nonzero x ∈ g ♮ . In view of Lemma 6.1, our purpose in only to check whether φ ♮ = 0 so that the abuse of notation is not a problem.With this notation, we call k ♮ admissible if k ♮ = 0 and k ♮i is admissible for g ♮i ,for each i = 1 , . . . , s . Remark . It may happen that k is admissible while k ♮ is not, even if g ♮ = 0.For example, for g = G , f in the nilpotent orbit labelled ˜ A (of dimension 8) and k = − / G , then k ♮ = k + 3 / − /

14 is not admissible for g ♮ ∼ = sl (see Table 11). However, we expect that k ♮ is admissible if k is admissible under additional conditions on f and k ; seeConjecture 6.4.Recall S O k ,f = S f ∩ O k , the nilpotent Slodowy slice. Assume that k and k ♮ are both admissible, f ∈ O k ,that the conjecture W k ( g , f ) = H DS,f ( L k ( g )) (Conjecture 4.4), which is veriﬁed if f admits an even good grading, is true, and that k is collapsing. Then accordingto Theorem 3.1 and Theorem 4.3, S O k ,f = O k ♮ × · · · × O k ♮s , where O k ♮ , . . . , O k ♮s are nilpotent orbits in g ♮ , . . . , g ♮s , respectively. This motivatesthe following deﬁnition: Deﬁnition . We say that a nilpotent Slodowy slice S O ,f is collapsing if S O ,f isisomorphic to a product of nilpotent orbit closures in g ♮ .Based on the above analysis, we consider pairs ( O , O ′ ) such that(1) O = O k is the associated variety of some simple admissible aﬃne vertexalgebra L k ( g ),(2) O ′ ⊂ O ,(3) for f ∈ O ′ , S O ,f is a collapsing nilpotent Slodowy slice which is the associ-ated variety X of some simple admissible aﬃne vertex algebra L k ♮ ( g ♮ ). (Inparticular, X is a product of nilpotent orbit closures in g ♮ .)The nilpotent orbit O k , for admissible k = − h ∨ g + p/q , is given by [10, Tables 2–10]and depends only on the denominator q : it is described in term of the correspondingpartition of n for g simple of classical type, that is, g = sl n , sp n or so n , and in theBala-Carter classiﬁcation for g simple of exceptional type.Motivating by the determination of generic singularities of the nilpotent closures O in simple Lie algebras g of classical types, Kraft and Procesi [82, 84] described thesmooth equivalences of singularities between Slodowy slices S O ,f and S O ,f , where S O ,f is obtained from S O ,f by the row/column removal rule (see Lemmas 8.2and 9.2). It turns out that these smooth equivalences actually yield isomorphismsof varieties, see [87, Proposition 7.3.2] .Thus, exploiting the row/column removal rule and [10] we can ﬁnd many exam-ples of pairs ( O , O ′ ) satisfying the above conditions in the classical types. In ourwork, we consider all such pairs. This is also mentioned without detail in [52, § OLLAPSING LEVELS OF W -ALGEBRAS 31 For the simple Lie algebras of exceptional types, the authors of [52] determine theisomorphism type of most Slodowy slices S O ,f when G.f is a minimal degenerationof O , that is, G.f is a maximal orbit in O \ O : for the remaining cases, they obtaina weaker information. They also determine the isomorphism type of some Slodowyslices S O ,f where G.f is not a minimal degeneration of O . From their work togetherwith [10], one ﬁnd a number of pairs ( O , O ′ ) satisfying the above conditions in theexceptional types. In fact, our work is more exhaustive and we do not consideronly these cases. To be more precise, in order to study collapsing levels in thesimple Lie algebras of exceptional types, we directly exploit the description of the k ♮i ’s as in Tables 11–16 and the central charge (see § S O ,f and a product of nilpotent orbit closures in g ♮ observed in [52],and some of them seem to be new.Furthermore, notice that our results ensure that these kind of isomorphisms areisomorphisms of Poisson varieties (for the classical types as well). Conjecture 6.4. If k is admissible and if f ∈ O k is such that S O k ,f is collapsing,then k ♮ is admissible, provided that k ♮ = 0 . We will verify the conjecture in the classical cases in the cases where S O k ,f iscollapsing and the isomorphism between S O k ,f and a product of nilpotent orbitclosures in g ♮ is obtained from the row/column removal rule of Kraft-Procesi (seeLemma 8.4 and Lemma 9.5). We feel that these cases exhaust all possible casesof collapsing nilpotent Slodowy slices so that, together with Lemma 6.5 below, itwould complete the proof of the conjecture. Lemma 6.5.

Conjecture 6.4 is true if g is simple of exceptional type.Proof. Assume that k is admissible. It is known that S O k ,f is equidimensional ofdimension dim O k − dim G.f ([56, Corollary 1.3.8]). Moreover, by the main resultof [20], S O k ,f is irreducible since k is admissible.If S O k ,f is collapsing then in particular it is contained in the nilpotent cone N g ♮ of g ♮ , and so dim S O ,f dim N g ♮ = dim g ♮ − rk g ♮ .The semisimple type of g ♮ and the values of the k ♮i ’s are computed in Tables11–17. Hence, ﬁxing f , we ﬁrst consider nilpotent orbits O containing G.f suchthat dim O − dim G.f dim g ♮ − rk g ♮ and such that O = O k for some admissiblelevel k for g . This heavily restricts the possibilities for the denominator q of k .Then we can ask whether for such q , the corresponding level k ♮ is admissible for g ♮ .Let us illustrate with an example.Assume that g = F and that f belongs to the nilpotent orbit labelled B in theBala-Carter classiﬁcation (dimension 42). According to Table 12, g ♮ ∼ = A . Hencedim S O ,f dim A − rk A = 2, so 42 dim O k

44 from which we see that O k must be B , C or F ( a ).Recall that k = − p/q with ( p, q ) = 1 and p > q is odd, and p >

12 if q is even. By [10], if O k = B then q = 8, if O k = F ( a ) then q = 7 or q = 10, andﬁnally the orbit C cannot occur as O k . Now we examine admissibility of k ♮ = 8 k + 60 = − p − q ) q in each case. ∗ If q = 7, then 2(4 p − q ) > × − ×

7) = 2, so k ♮ is admissible. ∗ If q = 8, then 2(4 p − q ) /q = (4 p − × / p − q > × − × k ♮ is admissible. ∗ If q = 10, then 2(4 p − q ) /q = (4 p − × / p − q > × − ×

10 = 2so k ♮ is admissible.In most cases considerations of orbit dimension, like those above, suﬃce to reachthe desired conclusion. In some cases, however, a more detailed analysis of theHasse diagrams of g and g ♮ is necessary (we use the diagrams of [52]). This is thecase, for example, for the minimal nilpotent orbits of F and E . We explain theveriﬁcation for these two cases.Let us ﬁrst consider the minimal nilpotent orbit of F (dimension 16). In thiscase g ♮ ∼ = C . Since the nilpotent cone of C has dimension 18 we have 16 dim O k

34. These inequalities are satisﬁed by 6 nilpotent orbits, but of theseonly the orbits A (the minimal orbit itself) and A + e A can occur as O k .If O k = A + ˜ A then q = 4 (here k = − p/q as above), and if O k = A then q = 2. Now we examine admissibility of the level k ♮ = k + 5 / − p − q q , in each case. ∗ If q = 2, then (2 p − q ) / (2 q ) = ( p − / p − > − > h C so k ♮ is admissible. ∗ If q = 4, then (2 p − q ) / (2 q ) = ( p − / p >

13 gives us only p − > p − > O A + ˜ A − dim O A = 18. Soif S O A

2+ ˜ A ,A were collapsing, then necessarily S O A

2+ ˜ A ,A would be isomorphic tothe nilpotent cone N C of C since it is irreducible. In particular, it should containa nilpotent G ♮ -orbit of dimension 16 (corresponding to the subregular nilpotentorbit of C ∼ = sp ). Here G ♮ denotes the centraliser in F of the sl -triple associatedwith f in the minimal nilpotent orbit A of F whose Lie algebra is g ♮ . But fromthe Hasse diagram of the F we see that this is not possible. Indeed, the Hassediagram of the G ♮ -action on S O A

2+ ˜ A ,A is just the interval between the orbitslabelled A + ˜ A and A in the Hasse diagram of F , and the dimension of thecorresponding G ♮ -orbits would be 18 , , , ,

0. Since it does not coincide withthe Hasse diagram of N C , this case is ruled out.Consider now the minimal nilpotent orbit A of E (dimension 34). In this case g ♮ ∼ = D . Since the nilpotent cone of D has dimension 60 we have 34 dim O k

94. Only the orbits 4 A and 2 A + A can satisfy this conditions and occur as O k .If O k = 4 A then q = 2 (here k = −

18 + p/q ), and if O k = 2 A + A then q = 3.Now we examine admissibility of the level k ♮ = k + 4 = −

10 + ( p − q ) /q, in each case. ∗ If q = 2, then ( p − q ) /q = ( p − / p − > − >

19 = h ∨ D so k ♮ is admissible. ∗ If q = 3, then ( p − q ) /q = ( p − / p >

19 gives us only p − > p − >

10 as required for admissibility.

OLLAPSING LEVELS OF W -ALGEBRAS 33 To exclude the second case, we observe instead that dim O A + A − dim O A = 56.In N D there are exactly two nilpotent orbits of dimension 56: there are associatedwith the partition (7 ,

5) and (9 , ). So if S O A

2+ ˜ A ,A were collapsing, then neces-sarily S O A

2+ ˜ A ,A would be isomorphic to one of the nilpotent orbit closures of D of dimension 56 since it is irreducible. Both of them contain the nilpotent G ♮ -orbitof dimension 54, associated with the partition (7 , , ) while, looking at the Hassediagram of the G ♮ -action on S O A A ,A in E we see that this is not possible.The rest of the veriﬁcations are left to the reader. (cid:3) Central charge.

Let k be an admissible level for g . We recall (1) that thenilpotent orbit X L k ( g ) ⊂ N g completely determines the denominator q of the level k = − h ∨ g + p/q (cf. Theorem 3.1), and (2) that the k ♮i ’s, deﬁned by (32), are allpolynomials of degree one in k or, equivalently, in p .Assume that both k and k ♮ are admissible. Then both L k ♮ ( g ♮ ) and W k ( g , f ) areconformal vertex algebra since admissible levels are never critical. Denoting by c V the central charge of a conformal vertex algebra V , we recall that c L k ( g ) = k dim g k + h ∨ g . If the centre g ♮ vanishes then obviously c L k♮ ( g ♮ ) = s X i =1 c L k♮i ( g ♮i ) . (39)In this case the possible values of the numerator p of k + h ∨ g for admissible k arenow determined as solutions of c H DS,f ( L k ( g )) = c L k♮ ( g ♮ ) , (40)considered as an equation in an unknown p . Recall that c H DS,f ( L k ( g )) is given inequation (21). If there are no solutions in p then k is not collapsing, while if thereare solutions (with p > h or p > h ∨ so that k = − h ∨ g + p/q is admissible) then weproceed to the next step ( § g ♮ is nonzero then, by Lemma 6.1, necessarily φ ♮ = 0. In particular c L k♮ ( g ♮ ) continues to be given by (39). Now the condition φ ♮ = 0 entirely determines k ,and the equation c H DS,f ( L k ( g )) = c L k♮ ( g ♮ ) is now either true or false. If true thenwe proceed to the next step ( § k is not collapsing.The data needed to compute the levels k ♮i in term of k , for i = 0 , . . . , s , are col-lected in Tables 2–4 for the classical types, and in Tables 11–17 for the exceptionaltypes. Notice that the data for the exceptional types have been obtained using thesoftware GAP4 .6.3.

Asymptotic growth and asymptotic dimension.

The ﬁrst and secondsteps ( § § k and k ♮ .As this point, we can attempt to apply Theorem 3.9. In fact, as a consequence ofTheorem 3.9, we have the following powerful result. Proposition 6.6.

Let k and k ♮ be admissible, f ∈ O k . Suppose that g H DS,f ( L k ( g )) = g L k♮ ( g ♮ ) = s X i =1 g L ki ( g i ) , A H DS,f ( L k ( g )) = A L k♮ ( g ♮ ) = s Y i =1 A L ki ( g i ) . Then k is collapsing.Proof. The assertion follows immediately by applying Theorem 3.9 for V = W k ( g , f )and ˜ V = H DS,f ( L k ( g )). (cid:3) This proposition ensures that it is enough to compare the asymptotic growthsand the asymptotic dimension of the vertex algebras H DS,f ( L k ( g )) and L k ♮ ( g ♮ ) thatwe compute using Corollary 3.8. This is the goal of next sections. In the classicalcases, we sometimes directly use the asymptotic growths (when f admits an evengood grading) to detect possible values of k instead of the central charge argument,because the equation given by the central charge is often diﬃcult to solve. Remark . In Proposition 6.6 suppose further that f ∈ O k . Then we get that W k ( g , f ) ∼ = H DS,f ( L k ( g ))without the assumption that f admits an even good grading. Indeed, f ∈ O k implies that H DS,f ( L k ( g )) is lisse by Theorem 4.3. Hence L k ♮ ( g ♮ ) must be inte-grable and the homomorphism τ must factors through the embedding L k ♮ ( g ♮ ) ֒ ! H DS,f ( L k ( g )) ([40]). In particular, H DS,f ( L k ( g )) is a direct sum of integrable rep-resentations of the aﬃne Kac-Moody algebra associated with g ♮ . It follows that theproof of Theorem 3.9 goes through to obtain that H DS,f ( L k ( g )) ∼ = L k ♮ ( g ♮ ).If the isomorphism W k ( g , f ) ∼ = H DS,f ( L k ( g )) holds, which is the case when f admits an even good grading, Proposition 6.6 gives a necessary and suﬃcientcondition for admissible k to be collapsing. Unfortunately, in general, it gives onlya suﬃcient condition.7. Some useful product formulas

We recall the well-known identity n − Y j =1 jπn = n (41)and its immediate consequence n Y j =1 j − / πn = 2 . (42)In this section we record some further identities similar to these which will be veryhelpful when we come to apply Corollary 3.8.Firstly from (41) we deduce n − Y j =1 (cid:18) jπn (cid:19) n − j = ⌊ n − ⌋ Y j =1 jπn n = n − Y j =1 jπn n = n n . (43) OLLAPSING LEVELS OF W -ALGEBRAS 35 Next, we have the following identities.

Lemma 7.1 ([71, 69]) . (1) We have Y α ∈ ∆ + π ( ρ | α ) h ∨ g = | P/Q ∨ | (cid:0) h ∨ g (cid:1) ℓ . (2) For g simple, not of type C ℓ , G , F , we have Y α ∈ ∆ + π ( ρ | α ) h ∨ g + 1 = (cid:0) h ∨ g + 1 (cid:1) ℓ . (3) We have Y α ∈ ∆ + π ( ρ | α ∨ ) h g = Y α ∈ ∆ + π ( ρ ∨ | α ) h g = | P ∨ /Q ∨ | ( h g ) ℓ . (4) We have Y α ∈ ∆ + π ( ρ | α ∨ ) h g + 1 = Y α ∈ ∆ + π ( ρ ∨ | α ) h g + 1 = ( h g + 1) ℓ . The lemma is probably known. We provide a proof for the convenience of thereader, and to clear up an ambiguity from [71] (see Remark 7.2).

Proof.

As a rule, in this proof, we write ρ g (resp. ρ ∨ g ) for the half-sum of positiveroots (resp. coroots) of g . For p ∈ Z > , set Π g ( p ) := Y α ∈ ∆ + π ( ρ g | α ) p , Π ∨ g ( p ) := Y α ∈ ∆ + π ( ρ g | α ∨ ) p . Since { ht( α ) : α ∈ ∆ + } = { ht( α ∨ ) : α ∈ ∆ + } , note that for any p > h g , Π ∨ g ( p ) = Y α ∈ ∆ + π ( ρ ∨ g | α ) p . We use the data of Table 1.(1) The identity is established in [69, Chapter 13, (13.8.1)] using modular invari-ance properties. It can also be checked using a case-by-case argument exploitingidentities (41) and (43).(2) We check the identity using a case-by-case argument. ∗ Type A ℓ . We have h ∨ A ℓ + 1 = ℓ + 2. Since ( ρ A ℓ | α ) = ht( α ), we easily that Π A ℓ ( ℓ + 2) = Π A ℓ +1 ( ℓ + 2) ℓ +1 Y j =1 jπℓ + 2 − . So by (1) applied to A ℓ +1 , we obtain the expected statement, Π A ℓ ( ℓ +2) = ( ℓ +2) ℓ/ , using the identity (41). ∗ Type B ℓ and D ℓ . We have h ∨ B ℓ = 2 ℓ − h ∨ D ℓ + 1 = 2 ℓ −

1. We ﬁrst showthe statement for D ℓ . We have to show that Π D ℓ (2 ℓ −

1) = (2 ℓ − ℓ/ . By (1) applied to B ℓ , we have Π B ℓ (2 ℓ −

1) = 2(2 ℓ − ℓ/ . So it suﬃces to show that the ratio Π B ℓ (2 ℓ − Π D ℓ (2 ℓ −

1) equals 2. Observing that ( ρ D ℓ | α ) =ht( α ), we get that Π D ℓ (2 ℓ −

1) = ℓ − Y j =1 (cid:18) jπ ℓ − (cid:19) ℓ − j × ℓ − Y j =1 jπ ℓ − × ℓ − Y i =1 ℓ +( i − Y j =2 i +1 jπ ℓ − . On the other hand, observing that ( ρ B ℓ | α ) = ht( α ) if α is long and ( ρ B ℓ | α ) = ht( α )if α is short, we get that Π B ℓ (2 ℓ −

1) = ℓ Y j =1 j − π ℓ − × ℓ − Y j =1 (cid:18) jπ ℓ − (cid:19) ℓ − j × ℓ − Y i =1 ℓ +( i − Y j =2 i jπ ℓ − . Using the identity (41), we show that ℓ Y j =1 j − π ℓ −

1) = 2 , ℓ − Y j =1 jπ ℓ − ℓ − Y j =1 jπ ℓ − ℓ − / . From this, we obtain that Π B ℓ (2 ℓ − Π D ℓ (2 ℓ −

1) = 2 as desired.We now turn to the statement for B ℓ . We have to show that Π B ℓ (2 ℓ ) = (2 ℓ ) ℓ/ . By (1) applied to D ℓ +1 , we have Π D ℓ +1 (2 ℓ ) = 2(2 ℓ ) ( ℓ +1) / . So it suﬃces to show that the ratio Π D ℓ +1 (2 ℓ ) Π B ℓ (2 ℓ ) equals 2(2 ℓ ) / . As before, comput-ing the heights of roots, we obtain that Π D ℓ +1 (2 ℓ ) Π B ℓ (2 ℓ ) = ℓ Y j =1 jπ ℓ ℓ Y j =1 j − π ℓ − ℓ Y j =1 j − π ℓ . Using (41) we show that ℓ Y j =1 jπ ℓ = 2 ℓ / , ℓ Y j =1 j − π ℓ = 2 / , ℓ Y j =1 j − π ℓ = 2 , whence Π D ℓ +1 (2 ℓ ) Π B ℓ (2 ℓ ) = 2(2 ℓ ) / , as desired. ∗ Types E , E , E . By direct calculations, we easily obtain that Π E (13) = Y k =1 (cid:18) kπ (cid:19) = Y k =1 kπ ! / = 13 . Similarly, we get Π E (19) = 19 / , Π E (31) = 31 . (3) and (4). We prove both identities together. OLLAPSING LEVELS OF W -ALGEBRAS 37 By (2), it suﬃces to check the statement for the non simply-laced cases. Weeasily check that Π ∨ G (6) = 6 , Π ∨ G (7) = 7 , Π ∨ F (12) = 12 , Π ∨ F (13) = 13 . It remains to consider the cases where g has type B ℓ or C ℓ . ∗ Type B ℓ . Let us ﬁrst prove the identity (4). By (1) applied to B ℓ +1 , we have Π B ℓ +1 (2 ℓ + 1) = 2(2 ℓ + 1) ( ℓ +1) / . Hence it suﬃces to show that Π B ℓ +1 (2 ℓ + 1) Π ∨ B ℓ (2 ℓ + 1) = 2(2 ℓ + 1) / . Using the computations of (2) and the identity (41), we easily obtain the expectedequality.Let us now prove the identity (3) for B ℓ . By (2) applied to B ℓ , we have Π B ℓ (2 ℓ ) = (2 ℓ ) / . Hence it suﬃces to show that Π ∨ B ℓ (2 ℓ ) Π B ℓ (2 ℓ ) = 2 / since | P ∨ /Q ∨ | = 2 for the type B ℓ . Using (41) and the computations of (2), weeasily obtain the expected equality. ∗ Type C ℓ . Notice that h C ℓ = h B ℓ = 2 ℓ . Hence it suﬃces to show that Π ∨ B ℓ (2 ℓ ) Π ∨ C ℓ (2 ℓ ) = 1 and Π ∨ B ℓ (2 ℓ + 1) Π ∨ C ℓ (2 ℓ + 1) = 1 . since | P ∨ /Q ∨ | = 2 for the types B ℓ and C ℓ . Again using (41), we easily obtain theexpected equalities. This concludes the proof of the lemma. (cid:3) Remark . The identities of the lemma are also stated in [71, Proposition 4.30].However, contrary to what should follow from (4.30.2) of [71], identity (2) does nothold for the types C ℓ , ℓ > G and F . For these types, it seems that there is nopleasant formula for Y α ∈ ∆ + ( C ℓ ) π ( ρ | α ) h ∨ g + 1 .In the next two sections we study collapsing levels in the classical cases, imple-menting the strategy described in Section 6.8. Collapsing levels for type sl n Let n ∈ Z > . In this section, it is assumed that g is the simple Lie algebra sl n ,the Lie algebra of traceless n × n matrices with coeﬃcients in C . The Killing formof g = sl n is given by κ g ( x, y ) = 2 n tr( xy ) and ( x | y ) g = tr( xy ).Denote by P ( n ) the set of partitions of n . As a rule, unless otherwise speciﬁed,we write an element λ of P ( n ) as a decreasing sequence λ = ( λ , . . . , λ r ) omittingzeroes. Thus, λ > · · · > λ r > λ + · · · + λ r = n. Let us denote by > the partial order on P ( n ) relative to dominance. Moreprecisely, given λ = ( λ , · · · , λ r ) , µ = ( µ , . . . , µ s ) ∈ P ( n ), we have λ > µ if P ki =1 λ i > P ki =1 µ i for 1 k min( r, s ). By [34, Theorem 5.1.1], nilpotent orbits of sl n are parametrised by P ( n ). For λ ∈ P ( n ), we shall denote by O λ the corresponding nilpotent orbit of sl n . If λ , µ ∈ P ( n ), then O µ ⊂ O λ if and only if µ λ . Deﬁnition . Let λ ∈ P ( n ). A degeneration of λ is an element µ ∈ P ( n ) suchthat O µ ( O λ , that is, µ < λ . A degeneration µ of λ is said to be minimal if O µ is open in O λ \ O λ .Fix λ ∈ P ( n ). As proved in [43] the set of good gradings for f ∈ O λ arein bijection with the set of pyramids of shape λ . We refer to [31] for the preciseconstruction of pyramids associated with good gradings.For our purpose, let us just recall that a pyramid is a diagram consisting of n boxes each of size 2 units by 2 units drawn in the upper half of the xy -place, withmidpoints having integer coordinates. By the coordinates of the box i , we meanthe coordinates of its midpoint. We will also speak of the row number of a box,by which we mean its y -coordinate, and the column number of a box, meaning its x -coordinate. A pyramid of shape λ consists of r rows, with the i th row consistingof λ i horizontally consecutive boxes. The rows are positioned so that the boxes ofthe ﬁrst row are centred on the y -axis and have y -coordinate 1, the boxes of thesecond row have y -coordinate 3, etc. In addition, no box of row i is permitted tohave x -coordinate smaller than the minimal x -coordinate of the boxes in row i − x -coordinate be greater than the maximal x -coordinates of the boxes inrow i −

1. We obtain the Dynkin pyramid (corresponding to the Dynkin grading)when the boxes of any row are centred around the y -axis. Note that a good gradingis even if and only if the x -coordinates of all boxes have the same parity. This isalways the case for the left-adjusted and the right-adjusted pyramids of shape λ .For the Dynkin pyramid, this happens if and only if all parts λ i have the sameparity.We number the boxes of the pyramid from top right to bottom left and so thatcol(1) > col(2) > . . . > col( n ). Then one can choose a representative f in O λ asfollows. Set f = P i,j e i,j , where the sum is over all i, j ∈ { , . . . , n } such thatcol( j ) = col( i ) + 2, and set x = εI n + ˜ x , where ˜ x = P ni =1 12 col( i ) e i,i and ε is chosen so that tr( x ) = 0. Here e i,j standsfor the ( i, j )-matrix unit. Then f ∈ O λ and g i Γ := { x ∈ g : [ x , x ] = ix } yields a good grading g = M j ∈ Z g j Γ for f (see [43] or [31, § λ = (3 , • • • Figure 1.

Left-adjusted, Dynkin and right-adjusted pyramids ofshape λ = (3 , OLLAPSING LEVELS OF W -ALGEBRAS 39 Lemma 8.2 (row/column removal rule in sl n ) . Let λ ∈ P ( n ) and µ a degenerationof λ . Assume that the ﬁrst l rows and the ﬁrst m columns of λ and µ coincide.Denote by λ ′ and µ ′ the partitions obtained by erasing these l common rows and m common columns. Then S O λ ,f ∼ = S O λ ′ ,f ′ , as algebraic varieties, with f ∈ O µ and f ′ ∈ O µ ′ . In particular, if f ′ = 0 , then S O λ ,f ∼ = O λ ′ . By [10] if k is an admissible level for sl n , we have O k = O λ , where λ = ( q e m , e s ) , e s e q − . (44) Lemma 8.3.

Let µ be a partition of n such that O µ ⊂ O λ . Let λ ′ and µ ′ be thepartitions obtained from λ and µ by erasing all common rows and columns of λ and µ . Then, µ ′ corresponds to the zero nilpotent orbit of sl | µ ′ | , that is, µ ′ = (1 | µ ′ | ) ifand only if µ is of one of the following types: (a) µ = ( q e m , e s ) = λ , (b) µ = ( q m , s ) with m e m and s > , (c) µ = ( q e m − , ( q − ) and e s = q − .Here | µ ′ | stands for the sum of the parts of µ ′ .Proof. Since O µ ⊂ O λ = O k , one can write µ = ( q m , ν ), with 0 m e m and ν = ( ν , . . . , ν t ) with ν < q . Assume that µ = λ , the case µ = λ being obvious.Let λ ′′ and µ ′′ be the partitions obtained from ( q e m , e s ) and µ by erasing allcommon rows of λ = ( q e m , e s ) and µ . Then λ ′′ = ( q e m − m , e s ) and µ ′′ = ν . Thepartition µ ′′ corresponds to the zero nilpotent orbit of sl | µ ′′ | if and only if ν = (1 | ν | ).This leads to the partitions of type (b). We illustrate in Figures 2 the row removalrule in the case where λ = (3 ,

1) and µ = (3 , ) is of type (b). O λ ∩ S f , f ∈ O µ ∼ = O (3 , ∩ S f ′ , f ′ = 0 Figure 2.

Row removal rule for λ = (3 ,

1) and µ = (3 , ) Consider now the common columns of λ and µ . Observe that λ = ( q e m , e s ) and µ = ( q m , ν , . . . , ν t ) have at least one common column if only if e m + 1 = m + t ,that is, t = e m − m + 1 . Assume that this condition holds.We illustrate in Figures 3 and 4 the row/column removal rule in the case where λ = (5 , µ = (5 , , λ = (5 , µ = (5 , ). O λ ∩ S f , f ∈ O µ ∼ = O (3) ∩ S f ′ , f ′ ∈ O (2 , Figure 3.

Row/column removal rule for λ = (5 ,

2) and µ = (5 , , O λ ∩ S f , f ∈ O µ ∼ = O (2) ∩ S f ′ , f ′ = 0 Figure 4.

Row/column removal rule for λ = (5 ,

3) and µ = (5 , ) We obtain that λ ′ = (( q − e s ) e m − m ) and µ ′ = ( ν − e s, . . . , ν t − e s ). Furthermore, µ ′ corresponds to the zero nilpotent orbit of sl | µ ′ | if and only if ν − e s = · · · = ν t − e s = 1.If so, then necessarily ( e m − m )( q − e s ) = t since λ ′ and µ ′ are partitions of the sameinteger, whence ( e m − m )( q − e s −

1) = 1 . (45)using t = e m − m + 1. But condition (45) holds if and only if q − e s − e m − m = 1. This leads to the partitions of type (c).Conversely, it is easy to verify that if µ is of type (a), (b) or (c) then µ ′ corre-sponds to the zero nilpotent orbit of sl | µ ′ | . (cid:3) In view of Lemma 8.3, we describe the centraliser g ♮ and the values of the k ♮i ’sfor particular sl -triples ( e, h, f ) of sl n . Lemma 8.4.

Let f be a nilpotent element of sl n associated with µ ∈ P ( n ) , with µ as in the ﬁrst column of Table 2. Then the centraliser g ♮ of an sl -triple ( e, h, f ) and the values of the k ♮i ’s are given by Table 2. Moreover, if k is admissible, thenso is k ♮ , provided that k ♮ = 0 . In Table 2, the numbering of the levels k ♮i ’s follows the order in which the simplefactors of g ♮ appears. Proof.

By Lemma 5.1 and Remark 5.2, in order to describe g ♮ and compute the k ♮i ’s,one can use the left-adjusted pyramid of shape µ . This pyramid always correspondsto an even good grading for f .(1) Consider ﬁrst the case µ = ( q m , s ), with s possibly zero. The pyramidconsists of a q × m rectangle surmounted by a vertical strip of dimension 1 × s onthe left as in Figure 5. For example, for µ = (5 , ) we get the pyramid as inFigure 6. OLLAPSING LEVELS OF W -ALGEBRAS 41 • ... m ... n Figure 5.

Pyramidfor ( q m , s ) • Figure 6.

Pyramidfor (5 , ) From the pyramid, we easily see that g = (cid:8) diag( x , . . . , x q − , y ) ∈ sl n : x i ∈ gl m , y ∈ gl m + s (cid:9) ∼ = C q − × ( sl m ) q − × sl m + s , and g ♮ = { diag( x, . . . , x, y ) ∈ sl n : x ∈ gl m , y = diag( x, x ′ ) , x ′ ∈ gl s } ⊂ g ∼ = ( C × sl m × sl s if s = 0 , sl m if s = 0 . First, pick t = diag( x, . . . , x, y ) ∈ g ♮ ∼ = sl m , with x = diag(1 , − , , . . . , ∈ gl m and y = diag( x, x ′ ) ∈ gl m + s with x ′ = 0. We have k ( t | t ) g + ( κ g ( t, t ) − κ g ( t, t )) / qk + 2( qn − qm − s ) , and ( t | t ) ♮ = 2, whence k ♮ = qk + qn − n. This terminates the case s = 0.Assume now s = 0. Pick t = diag( x, . . . , x, y ) ∈ g ♮ , with x = diag( − s, . . . , − s ) ∈ gl m and y = diag( x, x ′ ) ∈ gl m + s with x ′ = diag ( qm, . . . , qm ). The projection of t onto the semisimple part of g is diag(0 , . . . , , z ) with z = diag (cid:18) − snm + s , . . . , − snm + s , mnm + s , . . . , mnm + s (cid:19) ∈ sl m + s . From this, we get k ( t | t ) g + ( κ g ( t, t ) − κ g ( t, t )) / smn ( qk + qn − n ) . Choose ( | ) so that ( t | t ) = smn , whence k ♮ = qk + qn − n .Pick ﬁnally t = diag( x, . . . , x, y ) ∈ g ♮ ∼ = sl s , where x = 0 ∈ gl m and y =diag( x, x ′ ) ∈ gl m + s with x ′ = diag(1 , − , , . . . , k ( t | t ) g + ( κ g ( t, t ) − κ g ( t, t )) / k + (4 n − m + s )) / t | t ) ♮ = 2, whence k ♮ = k + qm − m. (2) Assume now that µ = ( q m , ( q − ). Here, we easily see that the embedding g ♮ ֒ ! g is as follows. We have g = (cid:8) diag( x, y , . . . , y q − ) ∈ sl n : x ∈ gl m , y i ∈ gl m +2 (cid:9) ∼ = C q − ∼ = ( sl m ) × ( sl m +2 ) q − , g ♮ = (cid:8) diag( x, y, . . . , y ) ∈ sl n : x ∈ gl m , y = diag( z, x ) ∈ gl m +2 , z ∈ gl ∈ (cid:9) ∼ = C × sl m × sl . µ g ♮ = L i g ♮i k ♮i conditions( q m , s ) C × sl m k ♮ = qk + qn − n s q − k ♮ = qk + qn − n ( q m ) sl m k ♮ = qk + qn − n ( q m , s ) C × sl m × sl s k ♮ = qk + qn − n s > k ♮ = qk + qn − nk ♮ = k + qm − m ( q m , ( q − ) C × sl m × sl k ♮ = qk + qn − nk ♮ = qk + qn − nk ♮ = ( q − k + n − m − Table 2.

Centralisers of some sl -triples ( e, h, f ) in sl n , with f ∈ O µ We compute k ♮ , k ♮ , k ♮ similarly as the previous case. So we omit the details.(3) Finally, assume that µ = ( q m , s ), with m > < s q −

1. Here, weeasily see that the embedding g ♮ ֒ ! g is as follows. We have g = (cid:8) diag( x , . . . , x s , y , . . . , y q − s ) ∈ sl n : x i ∈ gl m +1 , y j ∈ gl m (cid:9) ∼ = ( gl m +1 ) s × ( gl m ) q − s and g ♮ Γ = (cid:8) diag( x, . . . , x, y, . . . , y ) ∈ sl n : y ∈ gl m , x = diag( λ, y ) ∈ gl m +1 (cid:9) ∼ = C × sl m . We compute k ♮ , k ♮ as in the ﬁrst case. So we omit the details.The last assertion of the lemma is then easy to verify. Indeed, the conditions k ♮ = 0 (when k ♮ appears) implies p = n . Then k ♮ = 0 and k ♮ = − s + s/q (when k ♮ appears), which is admissible for sl s . If g ♮ = 0, then µ = ( q m ) and k ♮ = p − n = − m + p − m ( q −

1) which is admissible for sl m since p − m ( q − > m is equivalent to the condition p > n . (cid:3) We now brieﬂy discuss the dependence of the asymptotic dimension A H DS,f ( L k ( g )) on the auxiliary choice of a good grading made in the construction of the functor H DS,f (?). While it is known that H DS,f ( L k ( g )) is independent of this choice as avertex algebra, its conformal structure and consequently its asymptotic datum candepend on the choice of grading. At a collapsing level the W -algebra is isomorphicas a conformal vertex algebra to a simple aﬃne vertex algebra and so ex hypothesi its asymptotic dimension is independent of the choice of good grading.For the pairs ( k, f ) that arise in our search for collapsing levels for sl n therelationship between the orbits O k and G.f is such that the independence of A onthe choice of good grading is clear via combinatorial arguments. Namely, we havethe following lemma. Lemma 8.5.

Assume that µ is a partition of type (a), (b), (c) as in Lemma 8.3,and pick f ∈ O µ ⊂ O k . Then the expression given for A H DS,f ( L k ( g )) in Proposi-tion 4.10 is independent of the choice of good grading Γ . OLLAPSING LEVELS OF W -ALGEBRAS 43 Proof.

The expression in question is the product of a term manifestly independentof Γ with A ′ = 12 | ∆1 / | q | ∆ , + | Y α ∈ ∆ + \ ∆ , + π ( x | α ) q . (46)Now we consider k = − n + p/q an admissible level for g = sl n and the Hamil-tonian reduction H DS,f ( L k ( g )) for some f ∈ O k .We ﬁrst consider the partitions ( q m , s ) of n .The pyramids associated with λ = ( q m , s ) all consist of a q × m rectangle sur-mounted by a horizontal strip of dimension s ×

1. In Figures 7 and 8 we illustratetwo pyramids associated with n = 16, q = 5. In general there are 2( q − s ) + 1distinct pyramids associated with λ . i Figure 7.

An evengood grading i Figure 8.

An oddgood grading

We rewrite (46) as A ′ = Y i

Assume that k = − h ∨ g + p/q = − n + p/q is admissible for g = sl n .Pick a nilpotent element f ∈ O k , so that W k ( g , f ) is rational. Then k is collapsingif and only if n ≡ , ± q ) and p = ( h ∨ g if n ≡ ± q ) h ∨ g + 1 if n ≡ q ) . Furthermore (1) If n ≡ ± q ) , then W − n + p/q ( sl n , f ) ∼ = C . (2) If n ≡ q ) , W − n +( n +1) /q ( sl n , f ) ∼ = L ( sl e m ) , where e m is deﬁned by (44) .Proof. Fix a nilpotent element f ∈ O k = O ( q m ,s ) , with m := e m and s := e s in thenotation of (44).(a) Case s = 0.According to Table 2, we have g ♮ ∼ = C × sl m , and k ♮ = k ♮ = qk + qn − n = p − n .If k is collapsing, then necessarily k ♮ = 0, whence p = n . Assume from now that p = 0, whence k ♮ = k ♮ = 0. By (47), if k is collapsing, then W k ( sl n , f ) ∼ = C andthe asymptotic growth g W k ( sl n ,f ) = g H DS,f ( L k ( sl n ) must be 0. By Corollary 3.8, wehave g L k♮ ( g ♮ )) = 0 while by Proposition 4.10, g H DS,f ( L k ( sl n )) = dim g f − h ∨ g dim sl n pq = ( s − (cid:18) q − ( s + 1) q (cid:19) since dim g f = m ( q − s ) + ( m + 1) s −

1. Therefore, g H DS,f ( L k ( sl n )) = 0 if and onlyif s = 1 or s = q −

1. The case s = 1 will be dealt with in Theorem 8.7 with s = 1.We consider here only the case s = q − W − n + n/q ( sl n , f ) ∼ = C , for f corresponding to the parti-tion ( q m , q − H DS,f ( L k ( sl n )) is 1.Let g = L i g i Γ be the even good grading for f corresponding to the left-adjustedpyramid of shape ( q m , s ) as in the proof of Lemma 8.4. By Proposition 4.10 andLemma 7.1 (1), we get A H DS,f ( L k ( sl n )) = 1 q | ∆ +Γ , | q n − Y α ∈ ∆ + \ ∆ , + π ( x | α ) q , (48)with n = qm + q − | ∆ +Γ , | = m ( n − f . For i ∈ { , . . . , n − } , OLLAPSING LEVELS OF W -ALGEBRAS 45 let x i be the x -coordinate of the box labelled i in the pyramid. Note that for j ∈ Z ,we have { α ∈ ∆ + : ( x | α ) = j } = { ( i, l ) ∈ { , . . . , n − } : i l, | x i − x l | / j } . (49)In this way, using (49) and the identity (43), we obtain that Y α ∈ ∆ + \ ∆ , + π ( x | α ) q = q − Y j =1 (cid:18) jπq (cid:19) ( m +1)( n − j ( m +1)) = q q ( m +1)22 − ( m +1) . Combining this with (48), we verify that A H DS,f ( L k ( sl n ) = 1, as desired. Thisconcludes this case.(b) Case s = 0.According to Table 2, we have g ♮ ∼ = sl m and k ♮ = qk + qn − n . If k is collapsing,then by (47), we must have g L k♮ ( sl m ) = g H DS,f ( L k ( sl n )) . We have g L k♮ ( sl m ) = ( m − (cid:18) − mp − m ( q − (cid:19) , g H DS,f ( L k ( sl n )) = qm − − n ( n − pq since dim g f = m q −

1. Solving the equation( m − (cid:18) − mp − m ( q − (cid:19) = qm − − n ( n − pq with unknown p we obtain that p must be either equal to n + 1 or n −

1. Only thecase p = n + 1 is greater than h ∨ g = n .From now, it is assumed that p = n + 1, whence k ♮ = 1 = − m + ( m + 1) /

1. Weapply Proposition 6.6 to prove that k is collapsing. It is enough to show that L ( sl m )and H DS,f ( L k ( sl n )) share the same asymptotic dimension. By Corollary 3.8 andLemma 7.1 (2), A L ( sl m ) = 1 √ m (50)On the other hand, by Proposition 4.10 and Lemma 7.1 (2), A H DS,f ( L − n +( n +1) /q ( sl n )) = 1 q | ∆ , + | q n − n Y α ∈ ∆ + \ ∆ , + π ( x | α ) q , (51)with n = qm and | ∆ , + | = qm ( m − /

2. Moreover, computing the cardinality ofthe sets { α ∈ ∆ + : ( x | α ) = j } as in the previous case, we obtain by (43) that Y α ∈ ∆ + \ ∆ , + π ( x | α ) q = q qm / . Combining this with (51), we get that A H DS,f ( L − n +( n +1) /q ( sl n )) = √ m as expected.This completes the proof. (cid:3) We now consider the partitions µ ∈ P ( n ) as in Lemma 9.3 of type (b) and (c).This leads us to the following results. Theorem 8.7.

Assume that k = − n + p/q is admissible for g = sl n . (1) Pick a nilpotent element f ∈ O k corresponding to the partition ( q m , s ) ,with m > and s > . Then k is collapsing if and only if p = n = h ∨ sl n .Moreover, W − n + n/q ( sl n , f ) ∼ = L − s + s/q ( sl s ) . (2) Assume that e s = q − and pick a nilpotent element f ∈ O k correspondingto the partition ( q m , ( q − ) , with m = e m − in the notation of (44) .Then k is collapsing if and only if p = n = h ∨ g . Moreover, W − n + n/q ( sl n , f ) ∼ = L − /q ( sl ) . Remark . For s = 1 in (1) the formula has to be understood as W − n + n/q ( sl n , f ) ∼ = C . Since for s = 1, g L − s + s/q ( sl s ) = ( q − s − q = 0 and A L − s + s/q ( sl s ) = q − ( s − / = 1(see the below proof), the formulas make sense and are compatible with Theo-rem 8.6. Proof.

As in the proof of Theorem 8.6, we let g = L i g i Γ be the even good gradingfor f corresponding to the left-adjusted pyramid associated with the partition of f .(1) Fix a nilpotent element f ∈ O k corresponding to the partition ( q m , s ).According to Table 2, we have g ♮ ∼ = C × sl m × sl s , k ♮ = k ♮ = p − n and k ♮ = k + qm − m . If k is collapsing, then necessarily k ♮ = 0, that is, p = n . We assumefrom now on that p = n . Hence k ♮ = k + qm − m = − s + s/q , which is an admissiblelevel for sl s . By Proposition 4.10, we get g H DS,f ( L − n + n/q ( sl n )) = (cid:18) − q (cid:19) ( s −

1) = g L − s + s/q ( sl s ) since dim g f = ( m + s ) + ( q − m −

1. By Corollary 3.8 and Lemma 7.1 (1), wehave A L − s + s/q ( sl s ) = 1 q s ( s − / q ( s − / = 1 q ( s − / . On the other hand, by Proposition 4.10, we have A H DS,f ( L − n + n/q ( sl n )) = 1 q | ∆ , + | q ( n − / Y α ∈ ∆ + \ ∆ , + (cid:18) π ( x | α ) q (cid:19) , (52)with | ∆ , + | = ( m + s )( m + s − + m ( q − m − . Indeed, by Lemma 8.5, the asymptoticdimension does not depend on the good grading for such an f . Using the left-adjusted pyramid of shape ( q m , s ), we easily see that Y α ∈ ∆ + \ ∆ , + (cid:18) π ( x | α ) q (cid:19) = q − Y j =1 (cid:18) jπq (cid:19) m ( q − j )+ sm = q qm / sm (53)using (43). Combining (52) and (53), we conclude that A H DS,f ( L − n + n/q ( sl n )) = 1 q ( s − / = A L − s + s/q ( sl s ) , as desired. By Proposition 6.6 it follows that k is collapsing.(2) Fix a nilpotent element f ∈ O k corresponding to the partition ( q m , ( q − ).According to Table 2, we have g ♮ ∼ = C × sl m × sl , k ♮ = k ♮ = p − n and k ♮ = OLLAPSING LEVELS OF W -ALGEBRAS 47 ( q − k + n − m − k is collapsing, then necessarily k ♮ = 0, that is, p = n . Weassume from now on that p = n . Hence k ♮ = ( q − n/q − m −

2) = − /q , which isan admissible level for sl . Note that q is odd since ( q, n ) = 1 and n = qm +2( q − g H DS,f ( L − n + n/q ( sl n )) = 3 (cid:18) − q (cid:19) = g L − /q ( sl ) since dim g f = m + ( m + 2) ( q − −

1. Moreover, from (1) we know that A L − /q ( sl ) = 1 q / . On the other hand, by Proposition 4.10 and Lemma 8.5, we have A H DS,f ( L − n + n/q ( sl n )) = 1 q | ∆ , + | q ( n − / Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) , with | ∆ , + | = m ( m − + ( m +1)( m +2)( q − . Using the left-adjusted pyramid of shape( q m , ( q − ), we easily see that Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) = q − Y j =1 (cid:18) jπq (cid:19) m ( m +2)+( m +2) ( q − j − . As in the previous cases, we conclude using (41) and (43) that A H DS,f ( L − n + n/q ( sl n )) = A L − /q ( sl ) , as desired. By Proposition 6.6 it follows that k is collapsing. (cid:3) Remark . As has been observed in the above proof, if k is collapsing for f ∈ O ( q m ) , with n = qm , then necessarily k = − n + ( n + 1) /q or k = − n + ( n − /q .Only the ﬁrst case leads to an admissible level. However, one may wonder whetherthe following holds: W − n +( n − /q ( sl n , f ) ∼ = L − ( sl m ) . (The two above vertex algebras have the same central charge.)We propose the following conjectural extension of Theorem 8.7. Conjecture 8.10.

Let f ∈ sl n be a nilpotent element associated with a partition ( q m , ν ) , where < m e m in the notation of (44) , and ν = ( ν , . . . , ν t ) is apartition of s := n − qm such that ν < q . Then W − n + n/q ( sl n , f ) ∼ = W − s + s/q ( sl s , f ′ ) , where f ′ is a nilpotent element in sl s associated with the partition ν . Note that Conjecture 8.10 has been proven in the special case where n = 7, q = 3 and s = 4 by Francesco Allegra [5]. This case is in fact a particular case ofTheorem 8.7 (2). It seems that this conjecture has been stated in [106].The associated variety of W − n + n/q ( sl n , f ) is S O ( q f m, e s ) ,f while the associated va-riety of W − s + s/q ( sl s , f ′ ) is S O ( q f m − m, e s ) ,f ′ . These two nilpotent Slodowy slices areisomorphic by Lemma 8.2. The following proposition gives further evidence forConjecture 8.10. Proposition 8.11.

In the above notations, the vertex algebras W − n + n/q ( sl n , f ) and W − s + s/q ( sl s , f ′ ) have the same asymptotic growth.Proof. Write µ = ( µ , . . . , µ r ) the partition ( q m , ν ) corresponding to f . No-tice that g W − n + n/q ( sl n ,f ) = dim g f − nnq dim sl n = P qi =1 ( µ ∗ i ) − − ( n − q and g W − s + s/q ( sl s ,f ′ ) = dim g f − ssq dim sl s = P qi =1 ( µ ∗ i − m ) − − ( s − q , where ( µ ∗ , . . . , µ ∗ q )is the dual partition to µ . But we readily verify from the values of the µ i ’s that thefollowing identity holds: P qi =1 ( µ ∗ i ) − ( n − q = P qi =1 ( µ ∗ i − m ) − ( s − q , whencethe lemma. (cid:3) Collapsing levels for types sp n and so n Let n ∈ Z > . We study in this section collapsing levels for sp n and so n . Notations for sp n . We realise g = sp n as the set of n -size square matrices x suchthat x t J n + J n x = 0 where J n is the anti-diagonal matrix given by J n := (cid:18) U n/ − U n/ (cid:19) , where for m ∈ Z > , U m stands for the m -size square matrix with unit on the anti-diagonal. For a m -size square matrix x , we write b x for the matrix U m x t U m , where x t is the transpose matrix of x . Thus, sp n = (cid:26)(cid:18) a bc − ˆ a (cid:19) : a, b, c ∈ gl n/ , b = ˆ b, c = ˆ c (cid:27) . Writing e i,j for the i, j -matrix unit as in the sl n case, the following matrices give aChevalley basis for g : { e i,j − e − j, − i } i,j n/ ∪ { e i, − j + e j, − i , e − i,j + e − j,i } i

2. Set P ( n ) := { λ ∈ P ( n ) : number of parts of each even number is even } . By [34, Theorems 5.1.2 and 5.1.4], nilpotent orbits of so n are parametrised by P ( n ), with the exception that each very even partition λ ∈ P ( n ) (i.e., λ hasonly even parts) corresponds to two nilpotent orbits. For λ ∈ P ( n ), not very even,we shall denote by O λ , or simply by O λ when there is no possible confusion, thecorresponding nilpotent orbit of so n . For very even λ ∈ P ( n ), we shall denote by O I λ and O II λ the two corresponding nilpotent orbits of so n . In fact, their unionforms a single O ( n )-orbit. Thus nilpotent orbits of o n are parametrised by P ( n ).If λ , µ ∈ P ( n ), then O • µ ( O • λ if and only if µ < λ , where O • λ is either O λ , O I λ or O II λ according to whether λ is very even or not.Given λ ∈ P ( n ), there exists a unique λ + ∈ P ( n ) such that λ + λ , and if µ ∈ P ( n ) veriﬁes µ λ , then µ λ + . Deﬁnition . Assume that λ ∈ P ε ( n ), for ε ∈ {± } . An ε -degeneration of λ isan element µ ∈ P ε ( n ) such that O ε ; µ ( O ε ; λ , that is, µ < λ . A ε -degeneration µ of λ is said to be minimal if O ε ; µ is open in O ε ; λ \ O ε ; λ .9.1. Symplectic and orthogonal pyramids.

As in the sl n case, there is a bi-jection between the set of good gradings of sp n or so n compatible with a givennilpotent element and the set of some pyramids of shape the corresponding parti-tion of P ε ( n ), with ε ∈ {± } . Such pyramids are called symplectic pyramids for sp n ( ε = −

1) and orthogonal pyramids for so n ( ε = 1). For sp n and so n , we will beonly using symplectic or orthogonal Dynkin pyramids. This is a diagram consistingof n boxes each of size 2 units by 2 units drawn in the xy -plane. As in the sl n case, the coordinates of a box are the coordinates of its midpoint, and the row andcolumn numbers of a box mean its y - and x -coordinate, respectively, but there area few diﬀerences.Let us ﬁrst explain what are the symplectic Dynkin pyramids of shape λ ∈ P − ( n ). The parts of λ indicate the number of boxes in each row, and the rowsare added to the diagram so that we get a symmetric pyramid with respect to thepoint (0 , λ i of λ has odd multiplicity, then the ﬁrst time a row of this length is added to the diagram it is split into two halves, the right half is added to the next free row in the upperhalf plane in columns 1 , , . . . , λ i − skew rows . The missing boxes in skew rows are drawn as a box witha cross through it. We number the boxes of the symplectic Dynkin pyramid withlabels 1 , . . . , n/ , − n/ , . . . , − i and − i appear in centrallysymmetric boxes, for i = 1 , . . . , n/

2. As a rule, we will number the ﬁrst n/ λ = (5 , ) and λ = (5 , , • Figure 9.

SymplecticDynkin pyramid of shape(5 , ) • Figure 10.

Symplec-tic Dynkin pyramid ofshape (5 , , Let us now explain what are the orthogonal Dynkin pyramids of shape λ ∈ P ( n ). Assume to start with that n is even. Then the orthogonal Dynkin pyramidis constructed as in the symplectic case, adding rows of lengths determined by theparts of λ working outwards from the x -axis starting with the largest part, in acentrally symmetric way. The only diﬃculty is if some (necessarily odd) part of λ appears with odd multiplicity. As n is even, the number of distinct parts havingodd multiplicity is even. Choose i < j < · · · < i r < j r such that λ i > λ j > · · · > λ i r > λ j r are representatives for all the distinct odd parts of λ having oddmultiplicity. Then the ﬁrst time the part λ i s is needs to be added to the diagram,the part λ j s is also added at the same time, so that the parts λ i s and λ j s of λ contribute two centrally symmetric rows to the diagram, one row in the upper halfplane with boxes in columns 1 − λ j s , − λ j s , . . . , λ i s − − λ i s , − λ i s , . . . , λ j s −

1. We will referto the exceptional rows arising in this way as skew rows . We number the boxesexactly as in the symplectic case.For example, we represent in Figure 11 and Figure 12 the numbered orthogonalDynkin pyramids of shape λ = (4 , ) and λ = (3 , ), respectively. • Figure 11.

Orthogo-nal Dynkin pyramid ofshape (4 , ) • Figure 12.

Orthogo-nal Dynkin pyramid ofshape (3 , ) OLLAPSING LEVELS OF W -ALGEBRAS 51 If n is odd, there is one additional consideration. There must be some odd partappearing with odd multiplicity. Let λ i be the largest such part, and put λ i boxesinto the zeroth row in columns 1 − λ i , − λ i , . . . , λ i −

1; we also treat this zerothrow as a skew row. Now remove the part λ i from λ , to obtain a partition of an evennumber. The remaining parts are then added to the diagram exactly as in the case n even. We number the boxes exactly as in the symplectic Dynkin pyramid withlabels 1 , . . . , n/ , , − n/ , . . . , − i and − i appear in centrallysymmetric boxes, for i = 1 , . . . , n/

2, except that there is here a box numbered 0.We represent in Figure 13 and Figure 14 two more examples: the numberedorthogonal Dynkin pyramids of shape λ = (5 , ) and λ = (4 , Figure 13.

Orthogo-nal Dynkin pyramid ofshape (5 , ) Figure 14.

Orthogo-nal Dynkin pyramid ofshape (4 , We can now ﬁx a choice of an sl -triple ( e, h, f ) such that f ∈ O − λ by setting f = P i,j σ i,j e i,j for all 1 i, j n , where the sum of over all pairs i, j of boxessuch in the Dynkin pyramid such thateither col( j ) = col( i ) + 2 and row( i ) = row( j ) , or col( j ) = 1 , col( i ) = − i ) = − row( j )is a skew-row in the upper half plane , and h = P ni =1 col( i ) e i,i (so x = P ni =1 12 col( i ) e i,i ).The following lemma is a reﬁnement of [84, Theorem 12.3]. We refer to [87,Propositions 8.5.1 and 8.5.2] for a proof. Lemma 9.2 (row/column removal rule for sp n and so n ) . Let λ ∈ P ε ( n ) , for ε ∈ {± } , and µ a degeneration of λ . Assume that the ﬁrst l rows and the ﬁrst m columns of λ and µ coincide and denote by λ ′ and µ ′ the partitions obtained byerasing these l common rows and common m columns. Then S O ε ; λ ,f ∼ = S O ε ; λ ′ ,f ′ , as algebraic varieties, with f ∈ O ε ; µ and f ′ ∈ O ε ; µ ′ . In particular, if f ′ = 0 , then S O ε ; λ ,f ∼ = O ε ; λ ′ . By [10, Tables 2 and 3], the nilpotent orbit O k , for k admissible for g = sp n or g = so n , is described as follows: • If k is a principal admissible level for sp n (that is, q is odd), then O k = O − e λ − , where e λ = ( q e m , e s ) ∈ P ( n ), with 0 e s q − • If k is a co-principal admissible level for sp n (that is, q is even), then O k = O − e λ − where e λ = ( q + 1 , ( q ) e m , e s ) ∈ P ( n ), with 0 e s q − • If k is a principal admissible level for so n (that is, either n is even, or both n and q are odd), then O k = O e λ + , where e λ = ( q + 1 , q e m , e s ) ∈ P ( n ), with0 e s q − • If k is a co-principal admissible level for so n (that is, n is odd and q iseven), then O k = O e λ + , where e λ = ( q e m , e s ) ∈ P ( n ), with 0 e s q − sp n . Lemma 9.3 (Case sp n ) . Let λ ∈ P − ( n ) be such that O k = O − , λ . Fix a partition µ ∈ P − ( n ) of n such that O µ ⊂ O λ . Let λ ′ and µ ′ be the partitions obtainedfrom λ and µ by erasing all common rows and columns of λ and µ . Then, µ ′ corresponds to the zero nilpotent orbit of sp | µ ′ | , that is, µ ′ = (1 | µ ′ | ) if and only if µ is of one of the following types: (a) µ = λ , (b) µ = ( q m , s ) , with q odd, m even, s > , (c) µ = ( q e m − , q − , e s +1 ) , with q odd, (d) µ = ( q e m − , ( q − ) and e s = q − , with q odd, (e) µ = (( q ) m , s ) , with q even, q , s even, m odd or even. (f) µ = (( q ) e m , ( q − ) and e s = q − , with q even, q even, (g) µ = ( q + 1 , ( q ) m , s ) , with q even, q odd, m, s even, (h) µ = (( q ) e m +2 , e s ) , with q even, q odd. (i) µ = ( q + 1 , ( q ) m , q − , s ) , with q even, q odd, m, s even, (j) µ = ( q + 1 , ( q ) e m − , ( q − ) , and e s = q − , with q even, q odd, (k) µ = (( q − ) , e m = 0 and e s = q − , with q even, q odd.Here | µ ′ | stands for the sum of the parts of µ ′ .Proof. We argue as in the proof of Lemma 8.3. One can assume that µ = λ , thecase where µ = λ being obvious. According to the above description of O k , ﬁvetypes of partitions for λ can be distinguished. We consider the diﬀerent types.(1) λ = ( q e m , e s ), with q odd, e m even and 0 e s q −

1. This case is very similarto that of sl n . This leads to the partitions of type (b), the case that µ is of type( q e m − , ( q − ) being excluded since q and e m − λ = ( q e m − , q − , e s +1), with q odd, e m − e s +1 q −

1. Assume ﬁrstthat µ = ( q e m − , q − , ν ), with ν = ( ν t , . . . , ν t ) and ν q −

1. Then λ ′ = ( e s + 1)and µ ′ = ν . So the only possibility for that µ ′ corresponds to the zero orbit is that ν i = 1 for any i , that is, µ is of the form (c).Assume now that µ = ( q m , ν ), with 0 m e m − ν = ( ν t , . . . , ν t ) and ν q −

1. Let λ ′′ and µ ′′ be the partitions obtained by erasing all common rowsof λ and µ . Then λ ′′ = ( q e m − − m , q − , e s + 1) and µ ′′ = ν . If ν i = 1 for all i ,then we get a partition of type (b). Otherwise, observe that λ ′′ and µ ′ have atleast one common column if and only if t = e m − m + 1. If so, then ν t > e s + 1 and λ ′ = (( q − e s − e m − − m , q − − e s ), µ ′ = ( ν − e s − , . . . , ν t − e s − µ ′ corresponds to the zero orbit if and only if ν i − e s − i . Then e m − m + 1 = ( e m − − m )( q − e s −

1) + q − − e s, that is, 2 = ( e m − m )( q − e s − . OLLAPSING LEVELS OF W -ALGEBRAS 53 Since e m − m is odd, the only possibility is that e m − m = 1, that is, m = e m − q − e s − e s = q −

4. Hence t = 2 and ν = ν = q −

2. So µ of the form(d).(3) λ = (( q ) e m +1 , e s +1), with q even, q even, e m odd or even and 0 e s +1 q − sl n one. So we conclude similarly. This leads to thepartitions of type (e) or (f).(4) λ = ( q + 1 , ( q ) e m , e s ), with q even, q odd, e m even and 1 e s q −

1. Assumeﬁrst that µ = ( q + 1 , ( q ) m , ν ), with 0 m e m , ν = ( ν t , . . . , ν t ) and ν q − λ ′′ and µ ′′ be the partitions obtained by erasing all common rows of λ and µ .Then λ ′′ = ( q e m − m , e s ) and µ ′′ = ν . So we are led to the case (3), but the partition µ = ( q + 1 , ( q ) e m − , ( q − ) is excluded since both q and e m − µ = (( q ) m , ν ), with 0 m e m + 2, ν = ( ν t , . . . , ν t ) and ν q −

1. Then λ and µ have no common rows. Moreover, λ and µ have at leastone common column if and only if m + t = e m + 2. If so, then ν t > e s and we have λ ′ = ( q + 1 − e s, ( q − e s ) e m ), µ ′ = (( q − e s ) m , ν − e s, . . . , ν t − e s ). The later correspondsto the zero orbit if and only if either q − e s = 1 and ν i − e s = 1 for all i , whence m + t = e m + 2 and e s = q − µ has type (h) with e s = 0. Or m = 0and ν i − e s = 1 for all i , whence e m + 2 = e m (cid:16) q − e s (cid:17) + q − − e s, that is, 2 = ( e m + 1) (cid:16) q − − e s (cid:17) . Hence for parity reasons we get that e m = 0, λ = ( q + 1 , q −

3) and µ = (( q − ).This yields to the partition of type (k).(5) λ = ( q + 1 , ( q ) e m − , q − , e s + 1), with q even, q odd, e m − e s + 1 q − µ if q + 1. By erasing the ﬁrst (common) rowof λ and µ we go to the situation (3) with q is place of q . So we get partitions oftypes (g), (i) or (j).Assume now that µ = (( q ) m , ν ), with 0 m e m + 1, ν = ( ν t , . . . , ν t ) and ν q −

1. Then λ and µ have no common rows. Moreover, λ and µ have at leastone common column only if m + t = e m +2. If so, either ν t = e s +1 and then necessarily µ is of the type (h), or ν t > e s + 1 and we have λ ′ = ( q − e s, ( q − e s − e m − , q − − e s )and µ ′ = (( q − e s − m , ν − e s − , . . . , ν t − e s − q − e s − ν i − e s − i ,whence m + t = e m + 1 and e s + 1 = q −

1, and µ has type (h) with e s = q −

2, or m = 0 and ν i − e s − i , whence e m + 1 = ( e m + 1) q e s + 1 , that is, 0 = ( e m + 1) (cid:16) q − (cid:17) + e s + 1 , whence, q = 1 and e s + 1 = 0 which is impossible.Conversely, we easily that all the partitions of type (a)–(k) verify the desiredconditions. (cid:3) We now state the analog of Lemma 8.3 for so n . Lemma 9.4 (Case so n ) . Let λ ∈ P ( n ) be such that O k = O λ . Fix a partition µ of n such that O µ ⊂ O λ . Let λ ′ and µ ′ be the partitions obtained from λ and µ by erasing all common rows and columns of λ and µ . Then, µ ′ corresponds to thezero nilpotent orbit of sp | µ ′ | , that is, µ ′ = (1 | µ ′ | ) if and only if µ ∈ P ( n ) is of oneof the following types: (a) µ = λ , (b) µ = ( q m , s ) , with q odd or even, m odd or even, s > , (c) µ = ( q e m , ( q − ) , e s = q − and e m odd, with q odd, (d) µ = (3 e m − , ) , q = 3 , e s = 1 and e m even, (e) µ = ( q e m +1 , ) , e s = 3 and e m + 1 odd, with q odd, (f) µ = ( q e m , ( q − , , e s = q − and e m even, with q odd, (g) µ = ( q e m − , q − , s ) , with q even, s > , (h) µ = ( q e m − , q − , ) and e s = 3 , with q even, (i) µ = ( q e m − , ( q − , and e s = q − , with q even, (j) µ = ( q + 1 , q m , s ) , with q, m even and s > , (k) µ = ( q + 1 , q m , q − , s ) , with q, m even and s > , (l) µ = ( q + 1 , q e m − , q − , ) and e s = 3 , with q even, (m) µ = ( q + 1 , q e m − , ( q − , and e s = q − , with q even, (n) µ = ( q e m +1 , q − , and e s = q − , with q even.Here | µ ′ | stands for the sum of the parts of µ ′ .Proof. The proof is very similar to that of Lemma 9.3. The details are left to thereader. (cid:3)

In particular, by Lemma 9.2, if µ is a partition of n as in Lemma 9.3 (resp.Lemma 9.4), the nilpotent Slodowy slice S O k ,f is isomorphic to a nilpotent orbitclosure in sp s (resp. so s ) for f ∈ O ε ; µ with ε = − ε = 1) and s := | µ ′ | . If µ is of type (a), note that S O k ,f ∼ = { f } .In view of Lemmas 9.3 and 9.4, we describe the centraliser g ♮ and the values ofthe k ♮i ’s for particular sl -triples ( e, h, f ) of sp n and so n . Lemma 9.5.

Let ε = − (resp. ε = 1 ), and f ∈ O ε ; µ a nilpotent element of sp n (resp. so n ) with µ as in the ﬁrst column of Table 3 (resp. Table 4). Then the cen-traliser g ♮ of an sl -triple ( e, h, f ) and the k ♮i ’s are given by Table 3 (resp. Table 4).Moreover, if k is admissible, then so is k ♮ , provided that k ♮ = 0 . In Tables 3 and Table 4, the numbering of the levels k ♮i ’s follows the order inwhich the simple factors of g ♮ appears. Remark . In Tables 3 and Table 4, the conclusions for small s or m remainvalid, up to possible changes of numbering for the factors g ♮i . More speciﬁcally, ifthe factor g ♮i for some i is isomorphic to so m with m = 2, then g ♮i ∼ = C , so i mustbe replaced by 0 and the value of k ♮i remains valid.If the factor g ♮i for some i , let’s say i = 1, is isomorphic to so m with m = 4, then g ♮ is not simple and has to replaced by g ♮ × g ♮ , with g ♮ ∼ = sl , g ♮ ∼ = sl . Moreover,we have to read k ♮ = k ♮ = · · · instead of k ♮ = · · · , and the value of k ♮ remainsvalid.If the factor g ♮i for some i is isomorphic to so m or sl m with m = 1, then thisfactor does not appear (we use the convention that sl = so = 0), and we justforget about k ♮i . OLLAPSING LEVELS OF W -ALGEBRAS 55 Finally, if the factor g ♮i for some i is isomorphic to so ∼ = sl , so ∼ = sl or sp ∼ = sl then the conclusions remain valid without any change. Proof.

We proceed as in Lemma 8.4. In order to describe g ♮ we use the symplectic(resp. orthogonal) Dynkin pyramid of shape µ . Indeed, as explained in § sl -triple ( e, h, f ) with f ∈ O µ . One easily cancompute g = g h from the pyramid and, hence, g ♮ = g ∩ g f . Case sp n . We detail the proof for µ = ( q m , s ) ∈ P − ( n ). The other cases aredealt similarly, and the veriﬁcations are left to the reader. There are two cases.(1) µ = ( q m , s ) ∈ P − ( n ), with q odd, m even, s even. In this case, the orbit O − µ is even, and the symplectic Dynkin pyramid associated with the partition µ is as in Figure 9 if q = 5, m = s = 2. From the pyramid, we obtain that g = (cid:8) diag( x , . . . , x ( q − / , y, − \ x ( q − / , . . . , c x : x i ∈ gl m , y ∈ sp m + s (cid:9) ∼ = ( gl m ) ( q − / × sp m + s , g ♮ = { diag( x, . . . , x, y, − b x, . . . , − b x ) : x = (cid:18) a bc − ˆ a (cid:19) , y = a b z c − ˆ a , b = ˆ b, c = ˆ c, z ∈ sp s ⊂ g ∼ = sp m × sp s . To compute k ♮ , pick t ∈ g ♮ with a = diag(1 , . . . , b = c = 0 and z = 0 in theabove description of g ♮ . We obtain that ( t | t ) ♮ = 2, ( t | t ) g = 2 q , κ g ( t, t ) = 4 q (cid:0) n + 1 (cid:1) , κ g ( t, t ) = 2 m ( q −

1) + 4 (cid:0) m + s + 1 (cid:1) . Therefore, k ♮ = qk + q (cid:0) n + 1 (cid:1) − (cid:0) n + 1 (cid:1) . Tocompute k ♮ , pick t ∈ g ♮ with z = diag(1 , , . . . , , −

1) and x = 0 in the abovedescription of g ♮ . We obtain that ( t | t ) ♮ = 2, ( t | t ) g = 2, κ g ( t, t ) = 4 (cid:0) n + 1 (cid:1) , κ g ( t, t ) = 4 (cid:0) m + s + 1 (cid:1) . Therefore, k ♮ = k + (cid:0) n + 1 (cid:1) − (cid:0) m + s + 1 (cid:1) .(2) µ = ( q m , s ), with q even, m odd or even, s even. The orbit O − µ is odd,and we have g = (cid:8) diag( x , . . . , x q/ , y, − d x q/ , . . . , c x : x i ∈ gl m , y ∈ sp s (cid:9) ∼ = ( gl m ) q/ × sp s , g ♮ = { diag( x, . . . , x, y, − b x, . . . , b x : x ∈ so m , y ∈ sp s } ⊂ g ∼ = so m × sp s . If m = 2 or m >

4, we compute k ♮ by picking t ∈ g ♮ with x = diag(1 , , . . . , , − y = 0 in the above description of g ♮ . We obtain that ( t | t ) ♮ = 1, ( t | t ) g = 2 q , κ g ( t, t ) = 4 q (cid:0) n + 1 (cid:1) , κ g ( t, t ) = 2 mq , κ / ( t, t ) = 2 s . Therefore, k ♮ = 2 qk +2 q (cid:0) n + 1 (cid:1) − n, whence the expected result. To compute k ♮ , pick t ∈ g ♮ with x = 0 and y = diag(1 , , . . . , , −

1) in the above description of g ♮ . We obtain that( t | t ) ♮ = 2, ( t | t ) g = 2, κ g ( t, t ) = 4 (cid:0) n + 1 (cid:1) , κ g ( t, t ) = 4 (cid:16) s (cid:17) , κ / ( t, t ) = 2 m .Therefore, k ♮ = k + (cid:0) n + 1 (cid:1) − (cid:16) m m (cid:17) , whence the expected result.Assume m = 4. The isomorphism so ∼ = sl × sl (54) is obtained through the assignments t = diag(1 , , − , − h , t = diag(1 , − , , − h ,e , − e − , − e , e , − − e , − e , e , − e − , − f , e − , − e − , f , with span( h i , e i , f i ) ∼ = sl . To compute k ♮ , choose t ∈ g ♮ such that x = diag(1 , , − , − y = 0 in the above description of g ♮ . We obtain that k ♮ = 2 qk + 2 q (cid:0) n + 1 (cid:1) − n .To compute k ♮ , choose t ∈ g ♮ such that x = diag(1 , − , , −

1) and y = 0 in theabove description of g ♮ . In the same way we obtain that k ♮ = 2 qk + 2 q (cid:0) n + 1 (cid:1) − n .The computation of k ♮ in this case works as in the case m > k ♮ . This termi-nates this case following Remark 9.6. Case so n . As for the sp n , we detail the proof only for µ = ( q m , s ) ∈ P ( n ). Theother cases are dealt similarly, and the veriﬁcations are left to the reader. Thereare two cases.(1) µ = ( q m , s ) with q odd.(a) Assume ﬁrst that m is even. Then the orbit O µ is even, and from thepyramid, we easily obtain that g = (cid:8) diag( x , . . . , x ( q − / , y, − \ x ( q − / , . . . , − c x ) : x i ∈ gl m , y ∈ so m + s (cid:9) ∼ = ( gl m ) ( q − / × so m + s , g ♮ = (cid:26) diag( x, . . . , x, y, − b x, . . . , − b x ) : x = (cid:18) a bc − b a (cid:19) , b = − b b, c = − b c,y = a b z c − b a z ∈ so s ⊂ g ∼ = so m × so s . Assume m = 2 or m >

4. In order to compute k ♮ , pick t ∈ g ♮ with a =diag(1 , , . . . ,

0) and b = c = 0, z = 0 in the above description of g ♮ . Then ( t | t ) ♮ = 1,( t | t ) g = q , κ g ( t, t ) = 2 q ( n −

2) and κ g ( t, t ) = 2 m ( q −

1) + 2( m + s − k ♮ = qk + q ( n − − ( n − . In order to compute k ♮ , choose t ∈ g ♮ such that z = diag(1 , . . . , −

1) and a = b = c = 0 in the above description of g ♮ . Then( t | t ) ♮ = 1, ( t | t ) g = 1, κ g ( t, t ) = 2( n −

2) and κ g ( t, t ) = 2( m + s − k ♮ = k + n − m − s. If m = 4, using the isomorphism (54) as in the sp n case, weeasily verify that k ♮ = k ♮ = qk + q ( n − − ( n −

2) and k ♮ = k + n − m − s . So thesame conclusions hold following Remark 9.6.(b) Assume now that m is odd. Two cases have to be distinguished. ∗ If s is even, then the orbit O µ is even and the Dynkin pyramid is as inFigure 15 if q = 7, m = 3 and s = 2. OLLAPSING LEVELS OF W -ALGEBRAS 57 Figure 15. orthogonal Dynkin Pyramid for (7 , )

369 215487-11111012-8-7-9-5-2 -4-6-1-3 -12-10

Figure 16. orthogonal Dynkin Pyramid for (7 , ) From the pyramid, we obtain that g = (cid:8) diag( x , . . . , x ( q − / , y, − \ x ( q − / , . . . , − c x ) : x i ∈ gl m , y ∈ so m + s (cid:9) ∼ = ( gl m ) ( q − / × so m + s , g ♮ = diag( x, . . . , x, y, − b x, . . . , − b x ) : x = a u bv − b uc − b v − b a ∈ so m , a, b, c ∈ gl ( m − / ,b = − b b, c = − b c, y = a u b a ′ b ′ v − b u c ′ − b a ′ c − b v − b a ∈ so m + s , b ′ = − b b ′ , c ′ = − b c ′ ∼ = so m × so s . We compute the k ♮i ’s as in the case where m is even. ∗ If s is odd, then the orbit O µ is even and the Dynkin pyramid is as Figure 16if q = 7, m = 3 and s = 3.From the pyramid we obtain that g = (cid:8) diag( x , . . . , x ( q − / , y, − \ x ( q − / , . . . , − c x ) : x i ∈ gl m , y ∈ so m + s (cid:9) ∼ = ( gl m ) ( q − / × so m + s , and g ♮ = diag( x, . . . , x, y, − b x, . . . , − b x ) : x = a u b v − b uc − b v − b a , a, b, c ∈ gl ( m − / , b = − b b, c = − b c,y = w z − z a u u t − v − b u b z − t − v − b u − b z b v b v b t − b t b w , w ∈ so s − ∼ = so m × so s . The ﬁrst statements then easily follows. We compute the k ♮i ’s as in the case where m is even.(2) µ = ( q m , s ) with q even, m even and s > O µ is odd, and we have g = (cid:8) diag( x , . . . , x q/ , y, − d x q/ , . . . , − c x ) : x i ∈ gl m , y ∈ so s } ∼ = ( gl m ) q/ × so s , g ♮ = { diag( x, , . . . , , x, y, − b x, , . . . , , − b x ) : x = (cid:18) a bc − ˆ a (cid:19) , b = ˆ b, c = ˆ c, y ∈ so s (cid:27) ∼ = sp m × so s . Let g ♮ be the simple component of g ♮ isomorphic to sp m . To compute k ♮ , pick t ∈ g ♮ with a = diag(1 , . . . , b = c = 0, y = 0 in the above description of g ♮ .Then ( t | t ) g ♮ = 2, ( t | t ) g = q , κ g ( t, t ) = 2 q ( n − κ g ( x, x ) = 2 mq , κ g / ( t, t ) = 2 s .Therefore 2 k ♮ = qk + q ( n − − n, whence the expected result.Assume s = 3 or s >

4. Let g ♮ be the simple component of g ♮ isomorphic to so s . To compute k ♮ , pick y = diag(1 , . . . , −

1) and x = 0 in the above descriptionof g ♮ . Then ( t | t ) g ♮ = 1, ( t | t ) g = 1, κ g ( t, t ) = 2( n − κ g ( x, x ) = 2( s − κ g / ( t, t ) = 2 m . Therefore k ♮ = k + n − m − s, whence the expected result.We compute similarly k ♮ for s = 2 in which case the so s component is the centre g ♮ of g ♮ . Also, using the isomorphism so ∼ = sl × sl as in (1), m = 4 case, weeasily obtain that k ♮ = k ♮ = k + n − m − s. It remains to check the last assertion. We do it for g = sp m and µ = ( q m , s ),the other cases can be checked easily as well. In this case, g ♮ ∼ = sp m × sp s , k ♮ = qk + q ( n + 1) − ( n + 1) and k ♮ = k + n − m − s . Therefore k ♮ = p − ( n + 1)which is a nonnegative integer. In particular it is admissible for sp m . On the otherhand, k ♮ = − ( s + 1) + p − q m q , with ( p − q m , q ) = 1 and p − q m > s + 1. So k ♮ isadmissible for sp s . (cid:3) We are now in a position to state our results on collapsing levels for sp n . First,we consider the case where W k ( g , f ) is lisse, that is, f ∈ O k . Theorem 9.7.

Assume that k = − h ∨ g + p/q = − (cid:0) n + 1 (cid:1) + p/q is an admissiblelevel for g = sp n . Pick a nilpotent element f ∈ O k so that W k ( g , f ) is lisse. OLLAPSING LEVELS OF W -ALGEBRAS 59 µ g ♮ = L i g ♮i k ♮i conditions( q m , s ) sp m k ♮ = qk + q ( n + 1) − ( n + 1) q odd, m even, 0 s q − q m , s ) sp m × sp s k ♮ = qk + q ( n + 1) − ( n + 1) q odd, m even, s > k ♮ = k + n − m − s ( q m , q − , s ) sp m k ♮ = qk + q ( n + 1) − ( n + 1) q odd, m even, 0 s q − q m , q − , s ) sp m × sp s k ♮ = qk + q ( n + 1) − ( n + 1) q odd, m even, s > k ♮ = k + n − m − s − ( q m , ( q − ) sp m × sl k ♮ = qk + q ( n + 1) − ( n + 1) q odd, m even k ♮ = ( q − k + n − m ) − q + 1 , ( q ) m , s ) sp m k ♮ = q k + q ( n + 1) − n +12 q even, q odd, m even, 0 s q − q + 1 , ( q ) m , s ) sp m × sp s k ♮ = q k + q ( n + 1) − n +12 q even, q odd, m even, s > k ♮ = k + n − m − s − ( q + 1 , ( q ) m , q − , s ) sp m k ♮ = q k + q ( n + 1) − n +12 q even, q odd, m even, 2 s q − q + 1 , ( q ) m , q − , s ) sp m × sp s k ♮ = q k + q ( n + 1) − n +12 q even, q odd, m even, s > k ♮ = k + n − m − s − q + 1 , ( q ) m , ( q − ) sp m × sl k ♮ = q k + q ( n + 1) − n +12 q even, q odd, m even k ♮ = ( q − k + n − m − ) − q ) m , s ) so m k ♮ = qk + q ( n + 1) − n q even, q even, m odd or even, 0 s q − q ) m , s ) so m × sp s k ♮ = qk + q ( n + 1) − n q even, q even, s > k ♮ = k + n − m − s (( q ) m , ( q − ) so m × sl k ♮ = qk + q ( n + 1) − n q even, q even, m odd or even k ♮ = ( q − k + n − m ) − Table 3.

Centralisers of some sl -triples ( e, h, f ) in sp n , with f ∈ O − µ (1) Assume that k is principal. If p = h ∨ sp n = n + 1 , then for generic q , k iscollapsing if and only if n ≡ , − q . If n ≡ , − q , then forgeneric q , k is collapsing if and only if p = h ∨ sp n . Moreover, if n ≡ , − q , then W − h ∨ sp n + h ∨ sp n /q ( sp n , f ) ∼ = H DS,f ( L k ( g )) ∼ = C . (2) Assume that k is a co-principal admissible level for g = sp n . If p = h sp n +1 = n + 1 , then for generic q , k is collapsing if and only if n ≡ , q . If n ≡ , q , then for generic q , k is collapsing if and onlyif p = h sp n + 1 Moreover, if n ≡ , q with q odd, then W − h ∨ sp n +( h sp n +1) /q ( sp n , f ) ∼ = H DS,f ( L k ( g )) ∼ = C , and if n ≡ , q with q even, then W − h ∨ sp n +( h sp n +1) /q ( sp n , f ) ∼ = H DS,f ( L k ( g )) ∼ = L ( so m ) . In the above theorem, the isomorphism W k ( sp n , f ) ∼ = H DS,f ( L k ( g )) holds thanksto Remark 6.7. Proof.

Let λ be the partition corresponding to f ∈ O k . In the below proof, wealways exploit the symplectic Dynkin pyramid of shape λ as described in § I := { , . . . , n , − n , . . . , − } the set of labels, we notice that for j ∈ Z > , { α ∈ ∆ + : ( x | α ) = j } = { ( i, l ) ∈ I : 0 < i | l | , | col( i ) − col( l ) | / j } . (55) µ g ♮ = L i g ♮i k ♮i conditions( q m , s ) so m k ♮ = qk + q ( n − − ( n − q odd, m odd or even, 1 s q − q m , s ) so m × so s k ♮ = qk + q ( n − − ( n − q odd, m odd or even, s > k ♮ = k + n − m − s ( q m , ( q − ) so m × sl k ♮ = qk + q ( n − − ( n − q odd, m odd or even k ♮ = q − ( k + n − m − q m , ( q − , so m × sl k ♮ = qk + q ( n − − ( n − q odd, m even k ♮ = q − ( k + n − m − q m , s, so m k ♮ = qk + q ( n − − ( n − q odd, m odd or even, 3 s q − q m , ) so m × sl k ♮ = qk + q ( n − − ( n − q odd, m odd or even k ♮ = k + n − m − m , ) so m × sp k ♮ = 3 k + 3( n − − ( n − m odd k ♮ = k + 2 m + 2( q + 1 , q m , s ) sp m k ♮ = qk + q ( n − − ( n − q even, m even, 1 s q − q + 1 , q m , s ) sp m × so s k ♮ = qk + q ( n − − ( n − q even, m even, s > k ♮ = k + n − m − s − q + 1 , q m , q − , s, sp m k ♮ = qk + q ( n − − ( n − q even, m even, 3 s q − q + 1 , q m , q − , s ) sp m × so s k ♮ = qk + q ( n − − ( n − q even, m even, s > k ♮ = k + n − m − s − q + 1 , q m , q − , ) sp m × sl k ♮ = qk + q ( n − − ( n − q even, m even k ♮ = k + n − m − q + 1 , q m , ( q − , sp m × sl k ♮ = qk + q ( n − − ( n − q even, m even k ♮ = q − ( k + n − m − − ( q m , s ) sp m k ♮ = qk + q ( n − − n q even, m even, 1 s q − q m , s ) sp m × so s k ♮ = qk + q ( n − − n q even, m even, s > k ♮ = k + n − m − s ( q m , q − , s, sp m k ♮ = qk + q ( n − − n q even, m even, 3 s q − q m , q − , s ) sp m × so s k ♮ = qk + q ( n − − n q even, m even, s > k ♮ = k + n − m − s − q m , q − , ) sp m × sl k ♮ = qk + q ( n − − n q even, m even k ♮ = k + n − m − q m , ( q − , sp m × sl k ♮ = qk + q ( n − − n q even, m even k ♮ = q − ( k + n − m − q m , ( q − , sp m × sl k ♮ = qk + q ( n − − n q even, m even k ♮ = q − ( k + n − m − − Table 4.

Centralisers of some sl -triples ( e, h, f ) in so n , with f ∈ O µ In this way, the pyramid allows us to compute the central charge and the asymptoticdimension of H DS,f ( L k ( g )). Indeed, the central charge of H DS,f ( L k ( g )) is given by c H DS,f ( L k ( g )) = dim g −

12 dim g / − (cid:18) qp | ρ | − ρ | x ) + pq | x | (cid:19) , with | x | = 1 h ∨ g X α ∈ ∆ + ( x | α ) , ( ρ | x ) = 12 X α ∈ ∆ + ( x | α ) , while | ρ | is computed using the strange formula: | ρ | = h ∨ g dim g

12 = n ( n + 1)( n + 2)48 . OLLAPSING LEVELS OF W -ALGEBRAS 61 On the other hand, (55) enables to compute the term Q α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) inthe asymptotic dimension of H DS,f ( L k ( g )) in the principal case.For the co-principal case, note that Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ∨ ) q (cid:19) = Y α ∈ ∆ long+ \ ∆ (cid:18) π ( x | α ) q (cid:19) Y α ∈ ∆ short+ \ ∆ (cid:18) π ( x | α ) q (cid:19) , and that for j ∈ Z > , { α ∈ ∆ short+ : ( x | α ) = j } = { ( i, l ) ∈ I : 0 < i < | l | , | col( i ) − col( l ) | / j } , (56) { α ∈ ∆ long+ : ( x | α ) = j } = { ( i, l ) ∈ I : i > , l = − i, | col( i ) − col( l ) | / j } , We now follow the strategy of Section 6.(1) In this part, q is assumed to be odd. From the description of O k , either O k = O ( q f m , e s ) or O k = O ( q f m − ,q − , e s +1) . We consider the two cases separately.(a) Assume that O k = O ( q f m , e s ) , with e m, e s even, and set m := e m , s := e s forsimplicity. According to Table 3, we have g ♮ ∼ = sp m , and k ♮ = p − ( n + 1). Usingthe symplectic pyramid of shape ( q m , s ), we establish for j = 1 , . . . , ( q − / { α ∈ ∆ + : ( x | α ) = 2 j − } = (cid:18) q − j + 12 (cid:19) × m + ( s/ − j + 1 if j s/

20 else. , { α ∈ ∆ + : ( x | α ) = 2 j } = (cid:18) q − j − (cid:19) × m + m ( m + 1)2 + ( s/ − j if j s/

20 else. . The cardinality of roots α ∈ ∆ + such that ( x | α ) is a half-integer can be com-puted as follows. ∗ If s − q − − s + 1, then { α ∈ ∆ + : ( x | α ) = (2 j − / } = sm if j = 1 , . . . , s − , (cid:0) s + (cid:4) j (cid:5)(cid:1) m if j = s, . . . , q − − s + 1 , (cid:0) s − i + (cid:4) j (cid:5)(cid:1) m if j = q − − s + 1 + i with i = 1 , . . . , s − . ∗ If s − > q − − s + 1, then { α ∈ ∆ + : ( x | α ) = (2 j − / } = sm if j = 1 , . . . , q − − s + 1 , ( s − i ) m if j = q − − s + 1 + i with i = 1 , . . . , s − − q − , (cid:0) s − i + (cid:4) j (cid:5)(cid:1) m if j = q − − s + 1 + i with i = s − − q − , . . . , s − . From this, we get an expression of c H DS,f ( L k ( g )) depending on q, m, s . On theother hand c L k♮ ( g ♮ ) = m ( m + 1)(2 − p + mq + s )2( − p + m ( q −

1) + s ) . Fix p = h ∨ g = n +1. For generic q , the only solutions of the equation c H DS,f ( L k ( g )) = c L k♮ ( g ♮ ) with unknown s are s = 0 and s = q −

1. Now if we ﬁx s = 0, for generic q , the only solutions of the equation c H DS,f ( L k ( g )) = c L k♮ ( g ♮ ) with unknown p are, p = n p = n + 12 . Only p = n + 1 is greater than h ∨ g . If we ﬁx s = q −

1, for generic q , the onlysolution of the equation c H DS,f ( L k ( g )) = c L k♮ ( g ♮ ) with unknown p is p = n + 1. ∗ Assume p = n + 1 and s = 0. In this case, k ♮ = 0 and g H DS,f ( L k ( sp n )) = 0. Weaim to apply Proposition 6.6. By Proposition 4.10 and Lemma 7.1 (1), we have A H DS,f ( L − ( n n /q ( sp n )) = 1 q | ∆ | q n Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) (57)with n = qm and | ∆ | = ( q − m ( m − + m , since the orbit O − λ is even. By (55)and the previous computations, we show that Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) = q qm + m , (58)using the identities (41) and (43). Combining (57) and (58) we conclude that A H DS,f ( L − ( n n /q ( sp n )) = 1 as desired. ∗ Assume p = n + 1 and s = q −

1. Then k ♮ = 0 and g H DS,f ( L k ( sp n )) = 0. Asin the previous case, we aim to apply Proposition 6.6. By Proposition 4.10 andLemma 7.1 (1), we have A H DS,f ( L − ( n n /q ( sp n )) = 12 | ∆1 / | q | ∆ | q n Y α ∈ ∆ + \ ∆ sin (cid:18) π ( x | α ) q (cid:19) (59)with n = qm + q − | ∆ | = ( q − m ( m − + m and | ∆ / | = ( q − m . Using theidentities (41) and (43) and the previous computations, we show that Y α ∈ ∆ + \ ∆ sin (cid:18) π ( x | α ) q (cid:19) = (cid:18) (cid:19) ( q − m (cid:16) q q − (cid:17) q ( m m − . (60)Combining (59) and (60), we conclude that A H DS,f ( L − ( n n /q ( sp n )) = 1 asdesired.(b) Assume now that O k = O ( q m ,q − ,s ) , with m := e m − s := e s + 1. We argueas in (a). According to Table 3, we have g ♮ ∼ = sp m , and k ♮ = p − ( n + 1). Wecompute the central charges of H DS,f ( L k ( g )) and L k ♮ ( g ♮ ) similarly as in (a). Fix p = h ∨ g = n + 1. Solving the equation c H DS,f ( L k ( g )) = c L k♮ ( g ♮ ) with unknown s ,we get that for generic q , the only solutions are 0 or q + 1, the second one beingexcluded. Fix s = 0. Solving the equation c H DS,f ( L k ( g )) = c L k♮ ( g ♮ ) with unknown p , we get that for generic q , the only solutions are p = n p = n . Only p = n + 1 is greater than h ∨ g . But the case p = n + 1 and s = 0 has beenalready dealt with in part (a). OLLAPSING LEVELS OF W -ALGEBRAS 63 (2) In this part, q is assumed to be even. It follows from the description of O k .that either O k = O (( q ) f m +1 , e s +1) with even q , or O k = O ( q +1 , ( q ) f m , e s ) with odd q , or O k = O ( q +1 , ( q ) f m − , q − , e s +1) with odd q .(a) Assume that O k = O (( q ) f m +1 , e s +1) , with q even, and set m := e m + 1, s := e s + 1for simplicity. According to Table 3, we have g ♮ ∼ = so m , and k ♮ = p − n . The orbit O k is even. Hence one can directly use the asymptotic growth. We have g H DS,f ( L k ( sp n )) = 12 ( m (cid:16) q − s (cid:17) + ( m + 1) s ) − ( n − n ( n + 2) pq , with n = qm + s , while g L k♮ ( g ♮ ) = m ( m − (cid:18) − m − p − qm − s + m − (cid:19) . Fixing s = 0, we ﬁnd that, for generic q , the only solution of the equation g H DS,f ( L k ( sp n )) = g L k♮ ( g ♮ ) with unknown p is p = n + 1. Fixing p = n + 1, the only solutions of theequation g H DS,f ( L k ( sp n )) = g L k♮ ( g ♮ ) with unknown s are s = 0 or s = q + 1, thesecond case being excluded.From now on it is assumed that s = 0 and p = n + 1. Thus k ♮ = 1 and g H DS,f ( L k ( sp n )) = m . By Corollary 3.8, A L ( sp m ) = 12 , and by Proposition 4.10, A H DS,f ( L − ( n n +1) /q ( sp n )) = 2 | ∆ short+ ∩ ∆ | q | ∆ | (cid:0) q (cid:1) n √ Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ∨ ) q (cid:19) with | ∆ | = | ∆ short+ ∩ ∆ | = q e m ( e m − since O k is even. Using the symplecticpyramid of shape (( q ) m ), we establish for j = 1 , . . . , q/ { α ∈ ∆ short+ : ( x | α ) = 2 j − } = m (cid:16) q − j (cid:17) + m ( m + 1)2 , { α ∈ ∆ long+ : ( x | α ) = 2 j − } = e m, { α ∈ ∆ short+ : ( x | α ) = 2 j } = m (cid:16) q − j (cid:17) . From this, we deduce as in previous computations that A H DS,f ( L − ( n n +1) /q ( sp n )) = 12whence the statement by Proposition 6.6.(b) Assume that O k = O ( q +1 , ( q ) f m , e s ) with q odd, m := e m even, s := e s even.According to Table 3, we have g ♮ ∼ = sp m , and k ♮ = p − n +12 . Using the symplecticpyramid of shape ( q + 1 , ( q ) m , s ), we compute { α ∈ ∆ short+ : ( x | α ) = j } and { α ∈ ∆ long+ : ( x | α ) = j } for each j . For s = 0, we establish that for j = 1 , . . . , q/ { α ∈ ∆ short+ : ( x | α ) = (2 j − / } = m (cid:0) q − ( j − (cid:1) , { α ∈ ∆ short+ : ( x | α ) = 2 j − } = q +24 − j + (cid:0) q +24 − j (cid:1) m , { α ∈ ∆ long+ : ( x | α ) = 2 j − } = 1, { α ∈ ∆ short+ : ( x | α ) = 2 j } = q +24 − j + (cid:0) q +24 − j (cid:1) m + m ( m − ,and that for j = 1 , . . . , ( q + 2) / − { α ∈ ∆ long+ : ( x | α ) = 2 j } = m . For s = 0, we establish that { α ∈ ∆ short+ : ( x | α ) = (2 j − / } = m (cid:16) q − ( j − (cid:17) + ms if j q +2 − s ,m ( s − q +2 − s + j ) if q +2 − s + 1 j q +2 − s + s − , q +2 − s + s j q , { α ∈ ∆ short+ : ( x | α ) = 2 j − } = (cid:18) q + 24 − j (cid:19) m + q +2 − s − (2 j −

1) + 3 + 4 (cid:0) s − j (cid:1) if j q +2 − s , q +2 − s + 1 j q +2 − s + s − , q +2 − s + s j q , { α ∈ ∆ long+ : ( x | α ) = 2 j − } = ( j − s − , { α ∈ ∆ short+ : ( x | α ) = 2 j } = (cid:18) q + 24 − j (cid:19) m + m ( m − q +2 − s − j + 2 + 4 (cid:0) s − j (cid:1) if j q +2 − s , q +2 − s + 1 j q +2 − s + s − , q +2 − s + s j q , { α ∈ ∆ long+ : ( x | α ) = 2 j } = m. Fixing p = n + 1, the only solutions of the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown s are s = 0 and s = q −

1. Fixing s = 0 or s = q −

1, we ﬁnd thatfor generic q the only admissible solution of the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown p is p = n + 1. ∗ Assume that p = n + 1 and s = 0. We have k ♮ = 0, g H DS,f ( L k ( sp n )) = 0 and A L ( sp m ) = 1. Here we have | ∆ | = ( q − m ( m − + ( q − (cid:0) m (cid:1) , | ∆ short+ ∩ ∆ | = qm ( m − + ( q − + m ( m − , | ∆ / | = m + m ( q − A H DS,f ( L − ( n n +1) /q ( sp n )) = 1, whence the statement byProposition 6.6. ∗ Assume that p = n + 1 and s = q −

1. We have k ♮ = 0, g H DS,f ( L k ( sp n )) = 0and A L ( sp m ) = 1. Here we have | ∆ | = ( q − m ( m − + (cid:0) m (cid:1) , | ∆ short+ ∩ ∆ | = qm ( m − + m ( m − , | ∆ / | = qm and using the above computations we deduce that A H DS,f ( L − ( n n +1) /q ( sp n )) = 1, whence the statement by Proposition 6.6.(c) Assume that O k = O ( q +1 , ( q ) m , q − ,s ) with q odd, m := e m − s := e s + 1even. According to Table 3, we have g ♮ ∼ = sp m , and k ♮ = p − n +12 . As in the previouscases, we compute the central charge c H DS,f ( L k ( sp n )) using the symplectic pyramidof shape ( q + 1 , ( q ) m , q − , s ). We omit here the details. Fixing p = n + 1, theonly solutions of the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown s are s = 0or s = q + 1, the second case being excluded. Fixing s = 0, we ﬁnd that forgeneric q the only admissible solution of the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown p is p = n + 1. But the case where p = n + 1 and s = 0 has beenalready dealt with in part (2) (b). OLLAPSING LEVELS OF W -ALGEBRAS 65 This terminates the proof of part (2). (cid:3)

Remark . Since C and L ( so m ) are rational, Theorem 9.7 conﬁrms Conjec-ture 4.6 for g = sp n , k + h ∨ g = h ∨ g /q with n ≡ , − q ), q odd, and k + h ∨ g = ( h ∨ g + 1) /q with n ≡ , q/ q even. Remark . Although our main goal is to identify collapsing levels, i.e., to proveisomorphisms between certain W -algebras and aﬃne vertex algebras, it is worthremarking that our methods can be used to prove other isomorphisms too, as wenow illustrate.We consider the coprincipal admissible level k = − h ∨ g + p/q , for g = sp n , where q is twice an odd integer and p = h + 1. Then O k = O λ where λ = ( q + 1 , ( q ) m , s )for some m, s even. Let f ∈ O k . In Theorem 9.7 above we showed that if s = 0then H DS,f ( L k ( g )) is isomorphic to the trivial vertex algebra C .If instead s = 2 then the central charge of H DS,f ( L k ( g )), which in general isgiven by the formula c = − s ( q − s − (cid:0) qs + q − s − s + 4 (cid:1) q becomes c = 13 − q − q , which is the central charge of the Virasoro minimal model Vir ,q/ . The values of( x | α ) for α ∈ ∆ + can be read oﬀ from the symplectic pyramid of λ as was donein the proof of Theorem 9.7, to obtain a formula for the asymptotic dimension A H DS,f ( L k ( g )) . On the other hand the asymptotic dimension of Vir ,q/ is given byformula (6) and the two expressions can be shown to coincide by an elementary,though very tedious, calculation. It follows that W k ( g , f ) ∼ = H DS,f ( L k ( g )) ∼ = Vir ,q/ . This gives a yet another evidence for Conjecture 4.4 and Conjecture 4.6.We now turn to collapsing levels for sp n in the non lisse cases, that is, we considerthe partitions µ ∈ P − ( n ) identiﬁed in Lemma 9.3 cases (b)–(k). Our study leadsus to the following results. Theorem 9.10.

Assume that k = − h ∨ g + p/q = − (cid:0) n + 1 (cid:1) + p/q is admissible for g = sp n . (1) Assume that k is principal, that is, q is odd. (a) Pick a nilpotent element f ∈ O k corresponding to the partition ( q m , s ) ,with m, s even. For generic q , k is collapsing if and only if p = h ∨ sp n ,and W − h ∨ sp n + h ∨ sp n /q ( sp n , f ) ∼ = L − h ∨ sp s + h ∨ sp s /q ( sp s ) . (b) Pick a nilpotent element f ∈ O k corresponding to the partition ( q m , q − , s ) , with s even. For generic q , k is collapsing only if p = h ∨ sp n andwe have the following ﬁnite extension: W − h ∨ sp n + h ∨ sp n /q ( sp n , f ) ∼ = L − h ∨ sp s +( h sp s +1) / (2 q ) ( sp s ) ⊕ L − h ∨ sp s +( h sp s +1) / (2 q ) ( sp s ; ̟ ) . (2) Assume that k is co-principal, that is, q is even. (a) Pick a nilpotent element f ∈ O k corresponding to the partition ( q +1 , ( q ) m , s ) , with q odd, m even, s even. If p = h sp n + 1 , then k iscollapsing if and only if s = 0 or s = 2 . If s = 2 , then for generic q , k is collapsing if and only if p = h sp n + 1 . Moreover, if s = 2 , then W − h ∨ sp n +( h sp n +1) /q ( sp n , f ) ∼ = L − h ∨ sp s + h ∨ sp s / ( q/ ( sp s ) . (b) Pick a nilpotent element f ∈ O k corresponding to the partition ( q +1 , ( q ) m , q − , s ) , with q odd and s even. For generic q , k is collapsingonly if p = h sp n + 1 and we have the following ﬁnite extension: W − h ∨ sp n +( h sp n +1) /q ( sp n , f ) ∼ = L − h ∨ sp s +( h sp s +1) /q ( sp s ) ⊕ L − h ∨ sp s +( h sp s +1) /q ( sp s ; ̟ ) . (c) Pick a nilpotent element f ∈ O k corresponding to the partition (( q ) m , s ) ,with q even, m odd or even, and s even. For generic q , k is collapsingonly if p = h sp n + 1 and we have the following ﬁnite extension: W − h ∨ sp n +( h sp n +1) /q ( sp n , f ) ∼ = L ( so m ) ⊗ ( L − h ∨ sp s +( h sp s +1) /q ( sp s )) ⊕ ( L ( so m ; ̟ ) ⊗ ( L − h ∨ sp s +( h sp s +1) /q ( sp s ; ̟ ))) . Remark . Similarly to Remark 8.8, notice that (3) and (4) are compatible with(1) for s = 0, and (5), (6), (7) is compatible with (2) with s = 0. Proof.

We argue as in the proof of Theorem 9.7. So we omit some details when thecomputations are very similar to those considered in Theorem 9.7.(1) In this part, q is odd.(a) Fix a nilpotent element f ∈ O k corresponding to the partition ( q m , s ).According to Table 3, we have g ♮ ∼ = sp m × sp s , k ♮ = p − ( n + 1) and k ♮ = − ( s +1) + ( p − qm ) /q . Since f is even, W k ( g , f ) ∼ = H DS,f ( L k ( sp n )), and Proposition 6.6gives a necessary and suﬃcient condition for that k is collapsing.By Proposition 4.10, we have g H DS,f ( L k ( sp n )) = 12 (( m + s ) + m ( q −

1) + m + s ) − n ( n + 1)( n + 2)4 pq , with n = qm + s , while by Corollary 3.8, g L k♮ ( g ♮ ) = g L − ( m m − n p ) / ( sp m ) + g L − ( s p − qm /q ( sp s ) = m ( m + 1)2 (cid:18) − m + 2 m − n + 2 p (cid:19) + s ( s + 1)2 (cid:18) − s + 2(2 p − qm ) q (cid:19) . By solving the equation g H DS,f ( L k ( sp n )) = g L k♮ ( g ♮ ) with unknown p , we obtain that,for generic q , the only solutions which are nonnegative integers are p = n + 12 and p = n . Only the solution p = n + 1 is greater than h ∨ g . From now on it is assumed that p = n +1. Then k ♮ = 0 and k ♮ = − ( s +1)+( s +1) /q . We now apply Proposition 6.6to prove that k is collapsing. By Corollary 3.8 and Lemma 7.1 (1), we have A L − ( s s /q ( sp s ) = 1 q s / q s/ . OLLAPSING LEVELS OF W -ALGEBRAS 67 On the other hand, by Proposition 4.10 and Lemma 7.1 (1), we have A H DS,f ( L − ( n n /q ( sp n )) = 1 q | ∆ | q n Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) (61)with n = qm + s and | ∆ | = ( q − m ( m − + ( m + s ) , since the orbit G.f is even.Using the pyramid associated with the partition ( q m , s ) we establish that for j =1 , . . . , ( q − / { α ∈ ∆ + : ( x | α ) = 2 j − } = (cid:0) q − j − (cid:1) × m + m ( m + s ) , { α ∈ ∆ + : ( x | α ) = 2 j } = (cid:0) q − j − (cid:1) × m + m ( m + s ) + m ( m +1)2 , { α ∈ ∆ + : ( x | α ) = q − j } = jm , { α ∈ ∆ + : ( x | α ) = q − j + 1 } = ( j − m + m ( m +1)2 , We deduce that Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) = q qm + ms + m (62)by (41) and (43). Combining (61) and (62), we conclude that A H DS,f ( L − ( n n /q ( sp n )) = A L − ( s s /q ( sp s ) , as desired.(b) Fix a nilpotent element f ∈ O k corresponding to the partition ( q m , q − , s ).According to Table 3, we have g ♮ ∼ = sp m × sp s , k ♮ = p − ( n + 1) and k ♮ = − ( s + 1) + p − mq − q q , with (2 p − mq − q, q ) = 1. Using the pyramid of shape ( q m , q − , s ),we establish that for j = 1 , . . . , ( q − / { α ∈ ∆ + : ( x | α ) = j − } = (cid:0) q − (cid:1) m + (cid:16) q − (2 j − (cid:17) m + s , { α ∈ ∆ + : ( x | α ) = q +2 j − } = (cid:16) q − (2 j − (cid:17) m , { α ∈ ∆ + : ( x | α ) = 2 j − } = (cid:16) q − (2 j − (cid:17) m + (cid:16) q − (2 j − (cid:17) + ( ms if 2 j − q − , , { α ∈ ∆ + : ( x | α ) = 2 j } = (cid:0) q − j − (cid:1) m + (cid:0) q − j − (cid:1) + m ( m +1)2 + ( ms if 2 j − q − H DS,f ( L k ( sp n )). By solving the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown p , we obtain that for generic q the only admissible solutionis p = n . Assume that p = n + 1. Then k ♮ = 0 and k ♮ = − ( s + 1)+ s +12 q which isco-principal admissible for sp s . We easily verify that g H DS,f ( L − ( n n /q ( sp n )) = g L − ( s s +12 q ( sp s ) . By Corollary 3.8 and Lemma 7.1 (4), we have A L − ( s s +1) / (2 q ) ( sp s ) = 12 s +1 q s ( s +1) . On the other hand, by Proposition 4.10 and Lemma 7.1 (1), A H DS,f ( L − ( n n /q ( sp n )) =12 | ∆ / | / q | ∆ | q n Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) with n = qm + q − s , | ∆ | = ( q − m ( m − + ( m + s ) and | ∆ / | = ( q − m + s .From the above computations, we get that A H DS,f ( L − ( n n /q ( sp n )) = 2 A L − ( s s +12 q ( sp s ) . Since g H DS,f ( L − ( n n /q ( sp n )) = g L − ( s s +12 q ( sp s ) , it follows from Theorem 3.4that H DS,f ( L − ( n +1)+( n +1) /q ( sp n )) is a direct sum of admissible representations of L − ( s +1)+ s +12 q ( sp s ). Denoting by L k ♮ ( sp s ; λ ) the highest irreducible representationof L k ♮ ( sp s ) of admissible weight λ , we get that H DS,f ( L − ( n +1)+( n +1) /q ( sp n )) = M ∆ λ ∈ Z L − ( s +1)+ s +12 q ( sp s ; λ ) ⊗ m λ , where ∆ λ is the lowest L -eigenvalue of L − ( s +1)+ s +12 q ( sp s ; λ ). By [10], the abovesum is ﬁnite since there are only ﬁnitely many simple objects in the category O − ( n +1)+( n +1) /q .We have p − h sp s = ( s + 1) − s = 1, where k ♮ + h ∨ sp s = p /q . Recall that∆ λ ( sp s ) = ( λ | λ + 2 ρ sp s )2( k ♮ + h ∨ sp s ) . For generic q , we observe that λ = ̟ is the only fundamental weight for which∆ λ ∈ Z . One the other hand, we easily verify using Proposition 4.10 thatqdim( L − ( s +1)+( s +1) / (2 q ) ( sp s ; ̟ )) = 1 , that is, A L − ( s s +1) / (2 q ) ( sp s ; ̟ ) = A L − ( s s +1) / (2 q ) ( sp s ) . As a result, m A L − ( s s /q ( sp s ;0) + m ̟ A L − ( s s /q ( sp s ; ̟ ) = 2 A L − ( s s /q ( sp s ;0) . But m must be at most 1. We conclude that H DS,f ( L − ( n +1)+( n +1) /q ( sp n )) ∼ = L − ( s +1)+ s +12 q ( sp s ) ⊕ L − ( s +1)+ s +12 q ( sp s ; ̟ ) , whence the expected statement.(2) In this part, q is even.(a) Pick a nilpotent element f ∈ O k corresponding to the partition ( q +1 , ( q ) m , s ),with odd q . According to Table 3, we have g ♮ ∼ = sp m × sp s , k ♮ = p − n +12 and k ♮ = − ( s + 1) + p − mq − q q .Using the pyramid of shape ( q +1 , ( q ) m , s ), we establish that for j = 1 , . . . , q/ { α ∈ ∆ short+ : ( x | α ) = 2 j − } = (cid:18) q + 2 − j (cid:19) m + ( s if 2 j − q , OLLAPSING LEVELS OF W -ALGEBRAS 69 that for j = 1 , . . . , ( q + 2) / { α ∈ ∆ short+ : ( x | α ) = 2 j − } = (cid:18) q + 2 − j (cid:19) ( m + 1) + ( ms if 2 j q − { α ∈ ∆ long+ : ( x | α ) = 2 j − } = 1 , and that for j = 1 , . . . , ( q − / { α ∈ ∆ short+ : ( x | α ) = 2 j } = (cid:18) q + 2 − j (cid:19) ( m + 1) + m ( m − ( ms if 2 j q − { α ∈ ∆ long+ : ( x | α ) = 2 j } = m. The above cardinalities allows us to compute the central charge of H DS,f ( L k ( sp n )).By solving the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown p , we obtainthat, for generic q , the only admissible solution is p = n + 1. Fixing p = n + 1, weobtain that the only nonnegative integer solutions the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown s are s = 0 and s = 2. The case where p = n + 1 and s = 0 has been already dealt with in the proof of Theorem 9.7. We now assumethat p = n + 1 and s = 2. Then k ♮ , k ♮ = ( s + 1) / ( q/

2) and we easily verify that g H DS,f ( L k ( sp n )) = g L k♮ ( g ♮ ) . Let us compare the asymptotic dimensions. We obtainhere that A L − ( s s/ / ( q/ ( sp s ) = 1( q/ s + s , and that A H DS,f ( L − ( n n +1) /q ( sp n )) =2 | ∆ short ∩ ∆ | | ∆ / | / q | ∆ | ( q/ n × √ Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) with n = qm/ q/ | ∆ | = ( q − m ( m − + ( m +2) , | ∆ short ∩ ∆ | = | ∆ |− m +22 and | ∆ / | = qm/ A H DS,f ( L − ( n n +1) /q ( sp n )) and we get that A H DS,f ( L − ( n n +1) /q ( sp n )) equals A L − ( s s/ / ( q/ ( sp s ) , whence the expected result.(b) Pick a nilpotent element f ∈ O k corresponding to the partition ( q +1 , ( q ) m , q − , s ), with odd q . According to Table 3, we have g ♮ ∼ = sp m × sp s , k ♮ = p − n +12 and k ♮ = − ( s + 1) + p − qm − qq . Using the pyramid of shape ( q + 1 , ( q ) m , q − , s )we establish that for j = 1 , . . . , q/ { α ∈ ∆ short+ : ( x | α ) = (2 j − / } = m + m ( q − j ) + s if j ( q − / ,s if j = ( q + 2) / , j = 1 , . . . , ( q − / { α ∈ ∆ short+ : ( x | α ) = 2 j − } = 2 + (cid:0) q +2 − j (cid:1) ( m + 4) + 1+ ( ms if 2 j − ( q − / , { α ∈ ∆ long+ : ( x | α ) = 2 j − } = 2 , { α ∈ ∆ long+ : ( x | α ) = q/ } = 1 , { α ∈ ∆ short+ : ( x | α ) = 2 j } = 2 + (cid:0) q − − j (cid:1) ( m + 4) + m ( m − + ( ms if 2 j ( q − /

40 otherwise, , { α ∈ ∆ long+ : ( x | α ) = 2 j } = m, By solving the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown p , we obtainthat, for generic q , the only solution is p = n + 1. From now on we assume that p = n +1. Then k ♮ = 0 and k ♮ = − ( s +1)+ s +1 q and we have g W − ( n n +1 q ( sp n ,f ) = g L − ( s s +1 q ( sp s ) . Moreover, A L − ( s s +1 q ( sp s ) = 2 s ( s − q s ( q/ s √ , and, as in the previous case, we compute A H DS,f ( L − ( n n +1 q ( sp n )) . Here we getthat A H DS,f ( L − ( n n +1 q ( sp n )) = 2 A L − ( s s +1 q ( sp s ) . We now argue and conclude exactly as in (1) (b). So we omit the details.(c) Fix a nilpotent element f ∈ O k corresponding to the partition (( q ) m , s ),with even q . According to Table 3, we have g ♮ ∼ = so m × sp s , k ♮ = p − n and k ♮ = − ( s + 1) + ( p − qm ) /q . Using the pyramid associated with the partition(( q ) m , s ) we establish that for j = 1 , . . . , q/ { α ∈ ∆ short+ : ( x | α ) = (2 j − / } = ms, { α ∈ ∆ short+ : ( x | α ) = 2 j − } = m (cid:0) q − (cid:1) + m ( m − , { α ∈ ∆ long+ : ( x | α ) = 2 j − } = m, and that for j = 1 , . . . , q/ − { α ∈ ∆ short+ : ( x | α ) = 2 j } = m (cid:0) q − (cid:1) . By solving the equation c H DS,f ( L k ( sp n )) = c L k♮ ( g ♮ ) with unknown p , we obtainthat, for generic q , the only solutions are p = n and p = n + 1 . Only the solution p = n + 1 leads to an admissible level. From now on it isassumed that p = n + 1. Then k ♮ = 1 and k ♮ = − ( s + 1) + s +1 q and we have g W − ( n n +1 q ( sp n ,f ) = g L ( so m ) + g L − ( s s +1 q ( sp s ) = m + s (2+ q − s + qs )2 q .By Corollary 3.8 and Lemma 7.1 (1), A L ( so m ) ⊗ L − ( s s +12 ( sp s ) = 12 × s ( s − − q s ( s +1) . On the other hand, by Proposition 4.10 and Lemma 7.1 (1), A H DS,f ( L − ( n n +1 q ( sp n )) = 2 | ∆ short+ ∩ ∆ | | ∆ / | / q | ∆ | q n Y α ∈ ∆ + \ ∆ (cid:18) π ( x | α ) q (cid:19) OLLAPSING LEVELS OF W -ALGEBRAS 71 with n = qm + s , | ∆ short+ ∩ ∆ | = m ( m − q + s ( s − , | ∆ | = m ( m − q + (cid:0) s (cid:1) and | ∆ / | = ms . Using the above computations, we get that A H DS,f ( L − ( n n +12 ( sp n )) = 2 A L ( so m ) ⊗ L − ( s s +12 ( sp s ) . We now argue as in (1) (b). For the sp s factor, it is exactly as in (1) (b) withdenominator q (which is even) instead of 2 q . So we omit the details. For the so m factor, we ﬁrst observe that k ♮ = 1 = − ( m −

2) + m − m − − h ∨ so m = 1.Moreover, for generic q , λ = ̟ is the only fundamental weight for which ∆ λ ∈ Z .In addition, we easily verify using Proposition 4.10 that A L ( so m ; ̟ ) = A L ( so m ) . Then we conclude exactly as in (1) (b). (cid:3)

Remark . As it has been observed in the proof of Theorem 9.7 (1), if k iscollapsing for f ∈ O ( q m , s ) , then necessarily p = n + 1 or p = n +12 . Only the ﬁrstcase veriﬁes that p > h ∨ g . However, one may wonder whether the following holds: W − ( n +1)+ n +12 q ( sp n , f ) ∼ = L − / ( sp m ) ⊗ L − ( s +1)+ s +12 q ( sp s ) . (The two above vertex algebras have the same central charge.)We now state our main results on collapsing levels for so n . Here also we startwith the lisse case, that is, f ∈ O k . Theorem 9.13.

Assume that k = − h ∨ g + p/q = − ( n −

2) + p/q is admissible for g = so n . Pick a nilpotent element f ∈ O k so that W k ( g , f ) is lisse. (1) Assume that q is odd. If p = h ∨ so n = n − , then for generic q , k is collapsingif and only if n ≡ , q . If n ≡ , q , then for generic q , k iscollapsing if and only if p = h ∨ so n . Moreover, if n ≡ , q , then W − h ∨ so n + h ∨ so n /q ( so n , f ) ∼ = H DS,f ( L k ( g )) ∼ = C . (2) Assume that n and q are even. If p = h so n + 1 = n − , then for generic q , k is collapsing if and only if n ≡ , q . If n ≡ , q , then forgeneric q , k is collapsing if and only if p = h so n + 1 . Moreover, if n ≡ , q , then W − h ∨ so n +( h so n +1) /q ( so n , f ) ∼ = H DS,f ( L k ( g )) ∼ = C . (3) Assume that n is odd and that q is even. If p = h so n + 1 = n , then forgeneric q , k is collapsing if and only if n ≡ − , q . If n ≡ − , q , then for generic q , k is collapsing if and only if p = h so n + 1 .Moreover, if n ≡ − , q , then W − h ∨ so n +( h so n +1) /q ( so n , f ) ∼ = H DS,f ( L k ( g )) ∼ = C . In the above theorem, the isomorphism W k ( so n , f ) ∼ = H DS,f ( L k ( g )) holds thanksto Remark 6.7. Proof.

We argue as in the proof of Theorem 9.7. We exploit here the orthog-onal Dynkin pyramid of shape λ corresponding to f ∈ O k . Here, we set I = { , . . . , n , − n , . . . , − } if n is even, and I = { , . . . , n , , − n , . . . , − } if n is odd.Moreover, for j ∈ Z > , { α ∈ ∆ long+ : ( x | α ) = j } = { ( i, l ) ∈ I : 0 < i | l | , | col( i ) − col( l ) | / j } , { α ∈ ∆ short+ : ( x | α ) = j } = { ( i, l ) ∈ I : i > , l = 0 , | col( i ) − col( l ) | / j } , and { α ∈ ∆ + : ( x | α ) = j } = { α ∈ ∆ long+ : ( x | α ) = j } ∪ { α ∈ ∆ short+ : ( x | α ) = j } , with { α ∈ ∆ short+ : ( x | α ) = j } = ∅ if n is even. From this, we compute the centralcharge and the asymptotic dimension of H DS,f ( L k ( g )) using the pyramid exactlyas in the case where g = sp n . Since the proof is very similar to that of Theorem 9.7,we omit the details. (cid:3) Remark . As in Remark 9.9 we now comment on a few isomorphisms between W -algebras obtained with similar methods as in the proof of Theorem 9.13 above.Let q > n be odd, and k = − h ∨ g + p/q acoprincipal admissible level for g = so n , where p = h g + 1. Then O k = O λ where λ = ( q m , s ). If we choose n so that s = 3 and take f ∈ O k then we obtain anisomorphism W k ( g , f ) ∼ = H DS,f ( L k ( g )) ∼ = Vir ,q/ , proved by comparison of asymptotic growth and asymptotic dimension.If q and m are odd so that n = mq + 3 is even, and k = − h ∨ g + p/q is the principaladmissible level for g = so n where p = h ∨ g , then O k = O λ where λ = ( q m , f ∈ O k then we obtain an isomorphism W k ( g , f ) ∼ = H DS,f ( L k ( g )) ∼ = Vir ,q , again proved by comparison of asymptotic growth and asymptotic dimension. Theorem 9.15.

Assume that k = − h ∨ g + p/q = − ( n −

2) + p/q is admissible for g = so n . (1) Assume that q is odd so that k is principal. (a) Pick a nilpotent element f ∈ O k corresponding to the partition ( q m , s ) with s > . For generic q , k is collapsing only if p = h ∨ so n or p = h ∨ so n + 1 . Moreover, W − h ∨ so n + h ∨ so n /q ( so n , f ) ∼ = L − h ∨ so s + h ∨ so s /q ( so s ) , and (for m > ) W − h ∨ so n +( h ∨ so n +1) /q ( so n , f ) ∼ = L ( so m ) ⊗ L − h ∨ so s +( h ∨ so s +1) /q ( so s ) ⊕ L ( so m ; ̟ ) ⊗ L − h ∨ so s +( h ∨ so s +1) /q ( so s ; ̟ ) . (b) Pick a nilpotent element f ∈ O k corresponding to the partition ( q m , ( q − ) . For generic q , k is collapsing if and only if p = h ∨ so n and, W − h ∨ so n + h ∨ so n /q ( so n , f ) ∼ = L − /q ( sl ) , (2) Assume that q and n are even so that k is principal. (a) Pick a nilpotent element f ∈ O k corresponding to the partition ( q +1 , q m , s ) , with even m , odd s . Then k is collapsing if and only if p = h so n + 1 and, W − h ∨ so n +( h so n +1) /q ( so n , f ) ∼ = L − h ∨ so s +( h so s +1) /q ( so s ) . OLLAPSING LEVELS OF W -ALGEBRAS 73 (b) Pick a nilpotent element f ∈ O k corresponding to the partition ( q +1 , q m , q − , s ) , with even m, s , s > . Then for generic q , k iscollapsing only if p = h so n + 1 . Moreover, for generic q , W − h ∨ so n +( h so n +1) /q ( so n , f ) ∼ = L − h ∨ so s +( h so s +1) /q ( so s ) ⊕ L − h ∨ so s +( h so s +1) /q ( so s ; ̟ ) . (3) Assume that n is odd and q is even so that k is co-principal. (a) Pick a nilpotent element f ∈ O k corresponding to the partition ( q m , s ) ,with even m and odd s . Then for generic q , k is collapsing only if p = h so n + 1 . Moreover, W − h ∨ so n +( h so n +1) /q ( so n , f ) ∼ = L − h ∨ so s +( h so s +1) /q ( so s ) . (b) Pick a nilpotent element f ∈ O k corresponding to the partition ( q m , q − , s ) , with even m, s , s > . Then for generic q , k is collapsing onlyif p = n . Moreover, for generic q , W − h ∨ so n +( h so n +1) /q ( so n , f ) ∼ = L − h ∨ so s +( h so s +1) /q ( so s ) ⊕ L − h ∨ so s +( h so s +1) /q ( so s ; ̟ ) . Proof.

This proof is very similar to that of Theorem 9.10, except that, obviously,we use here orthogonal Dynkin pyramids. We omit the details. (cid:3)

Remark . Similarly to Remark 8.8, notice that (4) with s = 0 or s = 1 iscompatible with (1), (5) with s = 1 is compatible with (2), (7) with s = 1 iscompatible with (3), and (8) with s = 0 is compatible with (3). Remark . It might be also interesting to consider the case where ( n − , q ) = 1.For example, consider the Lie algebra so (type B ) and f ∈ O (5 , ) . We ﬁndthat k = −

25+ 255 = −

20 is not admissible, and k ♮ = −

5+ 55 = − W − ( so , f ) ∼ = L − ( so ) . Other examples are W − ( so , f ) ∼ = L − ( so ) . W − ( so , f ) ∼ = L − ( so ) . The last example is interesting because one knows [20, Theorem 7.1] that, for ℓ > X L − ( so ℓ +1 ) = O short . Conjecture 9.18. (1)

The cases covered by Theorems 9.7 and 9.10 gives theexhaustive list of pairs ( f, k ) where f is a nilpotent element of sp n and k isan admissible collapsing levels for sp n . (2) The cases covered by Theorems 9.13 and 9.15 gives the exhaustive list ofpairs ( f, k ) where f is a nilpotent element of so n and k is an admissiblecollapsing levels for so n . Collapsing levels in the exceptional types

In this section we state our main results and conjectures concerning collapsinglevels in the exceptional types. As in the preceding sections, our proofs followthe strategy described in Section 6; considering pairs ( O k , G.f ) such that S O k ,f iscollapsing. Data on nilpotent orbits, sl -triples and centralisers, which we will usethroughout this section, is recorded in Tables 11, 12, 13, 14-15, 16-17.The results of this section are organised by type, the isomorphisms in type E , E , E , G and F presented in Theorems 10.1, 10.4, 10.7, 10.10 and 10.12 respec-tively. In place of f we write the label of G.f in the Bala-Carter classiﬁcation, and g and g ♮ are denoted by their types. The results are summarized in Tables 5, 6, 8,9, and 10. We present a complete proof only for Theorem 10.1, the others beingvery similar.In the tables, we indicate for each triple ( O k , G.f, O ♮ ∼ = S O k ,f ), the values of p/q = k + h ♮ g , k ♮i + h ∨ g ♮i , for i = 1 , . . . , s (if i >

1, we write in the ﬁrst column k ♮ + h ∨ g ♮ ,and in second column k ♮ + h ∨ g ♮ , etc.), the central charge c V , the asymptotic growth g V and the asymptotic dimension A V with V being either H DS,f ( L k ( g )) or L k ♮ ( g ♮ ).Then we write in the last column the symbol if all invariants match, we write“ﬁn. ext.” if we expect that H DS,f ( L k ( g )) is a ﬁnite extension of L k ♮ ( g ♮ ) (usually,this happens when all invariant match except the asymptotic dimension, and moredetails are furnished in the corresponding theorem). As a rule, when one theinvariants does not coincide, we write in ﬁrst position the invariant correspondingto H DS,f ( L k ( g )).Nilpotent orbits are given by their the Bala-Carter classiﬁcation in the excep-tional types. Nilpotent orbits in classical types are given in term of partitions.Note that the associated variety of L k ( g ) determines the possible denominators q of k + h ∨ g since k is admissible (see Theorem 3.1). So the values of q are alwaysamong these possible denominators.We also indicate in the table the isomorphism type of S O k ,f (in the third column)when it is known (if so, it is a product of nilpotent orbits in g ♮ ). When the iso-morphism O ♮ ∼ = S O k ,f comes from a minimal degeneration, then we always obtaina minimal nilpotent orbit closure [52]. Following Kraft and Procesi [82, 84], werefer to the minimal nilpotent orbit O min of a simple Lie algebra by the lower caseletters for the ambient simple Lie algebra: a k , b k , c k , d k ( k > g , f , e , e , e .Similarly, we refer to the minimal special nilpotent orbit for the ambient simpleLie algebra as a spk , b spk , c spk , d spk ( k > g sp , f sp , e sp , e sp , e sp . The nilpotent coneof a Lie algebra of type X will denoted by N X . In Table 10, the letter m refers toa non-normal type of singularity which is neither a simple surface singularity nora minimal singularity (see [52, § a +2 appearing alsoin Table 10 it refers to the singularity a together with the action of a subgroup K ⊂ Aut( sl ) which lifts a subgroup of the Dynkin diagram of sl = A (see [52, § S O k ,f is not known (to the best of our knowledge)we write “unknown”. Sometimes, when the type is not known, we observe however There is an order-reversing map d on the set of nilpotent orbits in g that becomes an involutionwhen restricted to its image [101]. Orbits in the image of d are called special . There is a uniqueminimal special nilpotent orbit, which is of dimension 2 h g −

2. Note that the minimal nilpotentorbit of g has dimension 2 h ∨ g − OLLAPSING LEVELS OF W -ALGEBRAS 75 that S O k ,f is birationally equivalent to a certain product of nilpotent orbits in g ♮ .If so, we use the symbol ≈ to indicate this (See Remarks 10.2, 10.5, 10.8, 10.13 and10.11). Theorem 10.1.

The following isomorphisms hold, providing collapsing levels for g = E . W − / ( E , E ) ∼ = C , W − / ( E , E ) ∼ = C , W − / ( E , E ( a )) ∼ = C , W − / ( E , D ) ∼ = L − / ( A ) , W − / ( E , A ) ∼ = L − / ( A ) , W − / ( E , D ) ∼ = L − / ( A ) , W − / ( E , A ) ∼ = L − / ( A ) , W − / ( E , A + A ) ∼ = C , W − / ( E , A ) ∼ = L − / ( G ) , W − / ( E , A ) ∼ = L ( A ) . Moreover, the following isomorphisms of W -algebras with ﬁnite extensions of ad-missible aﬃne vertex algebras hold. W − / ( E , A ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) , W − / ( E , A ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) . Proof.

We detail below only a few cases. The chief tool to prove the isomorphismsis Theorem 3.9, using data summarised in Table 5. • Assume q >

12. Let p >

12 and ( p, q ) = 1. Computing the central charge, weobserve that c W − p/q ( E ,E ) = 0 if and only if ( p, q ) = (12 ,

13) or ( p, q ) = (13 , g W − p/q ( E ,E ) = 0 and A W − p/q ( E ,E ) = 1. ByTheorem 3.9 the ﬁrst two isomorphisms follow. • Assume that q = 6 or 7, and pick f ∈ A . According to Table 13, we have g ♮ ∼ = sl and k ♮ = k + 17 /

2. Computing the central charge, we easily verify that if k is collapsing then necessarily ( p, q ) = (13 ,

6) or ( p, q ) = (12 , ∗ Assume ﬁrst that ( p, q ) = (13 , k ♮ = − /

3. Since k = −

12 + 13 / k ♮ = − / E and A , respectively, it suﬃces to applyTheorem 3.9 to prove that W − / ( E , A ) and L − / ( A ) are isomorphic.We easily check using Corollary 3.8 and Proposition 4.10 that L − / ( A ) and H DS,A ( L − / ( E )) share the same asymptotic growth of 2. Let us comparetheir asymptotic dimensions. By Corollary 3.8 and Lemma 7.1 (1), we have A L − / ( A ) = 13 √ , while by Proposition 4.10 and Lemma 7.1 (2), A H DS,A ( L − / ( E )) = 12 | ∆ / | / | ∆ | √ Y α ∈ ∆ + \ ∆ π ( x | α )6 , with | ∆ / | = 6 and | ∆ | = 1. Computing the cardinality of { α ∈ ∆ + : ( x | α ) = j } for j >

0, we verify that Y α ∈ ∆ + \ ∆ π ( x | α )6 = 2 , whence A H DS,A ( L − / ( E )) = √ , as required. ∗ Assume now that ( p, q ) = (12 , g H DS,A ( L − / ( E )) = g L − / ( A )6 TOMOYUKI ARAKAWA, JETHRO VAN EKEREN, AND ANNE MOREAU and that A H DS,A ( L − / ( E )) = 2 A L − / ( A ) . Moreover, we can verify that the central charge of H DS,A ( L − / ( E )) and L − / ( A ) are both equal to −

25. We argue as in the proof of Theorem 9.10 (1)(b). By Theorem 3.4, there is a full embedding, L / ( A ) ֒ ! H DS,A ( L − / ( E )) . Denoting by L / ( A ; λ ) the highest irreducible representation of L / ( A )of admissible weight λ , we get that H DS,A ( L − / ( E )) = M ∆ λ ∈ Z L / ( A ; λ ) , where ∆ λ is the lowest L -eigenvalue of L − / ( A ; λ ). By [10], the above sum isﬁnite since there are only ﬁnitely many simple objects in the category O − / .Recall that in general ∆ λ = ( λ | λ + 2 ρ )2( k ♮ + h ∨ g ♮ )which, in the present case, gives∆ λ = λ ( λ + 2)2( k ♮ + 2) = λ ( λ + 2)144 × . Here we have identiﬁed h ∗ with C so that λ = λ̟ where ̟ is the ﬁrst fundamentalweight and λ ∈ C . In particular ρ = 1. We see that ∆ λ is a half-integer for λ = 0 , λ because p − h ∨ A = 3 − A L − / ( A ; ̟ ) = A L − / ( A ) . As a result, m A L − / ( A ;0) + m A L / ( A ; ̟ ) = 2 A L − / ( A ;0) . But m must be at most 1. We conclude that H DS,A ( L − / ( E )) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) . In particular, since H DS,A ( L − / ( E )) is a ﬁnite extension of L − / ( A ) itis simple, from which the desired conclusion follows. • Assume that ( p, q ) = (13 , f ∈ A . In this case we obtain g H DS,A ( L − / ( E )) = g L − / ( A ) and A H DS,A ( L − / ( E )) = 2 A L − / ( A ) . Moreover, we verify that the central charges of H DS,A ( L − / ( E )) and L − / ( A )are both equal to −

25. Arguing as in the previous case, we obtain H DS,A ( L − / ( E )) = M ∆ λ ∈ Z L − / ( A ; λ ) , where ∆ λ is the lowest L -eigenvalue of L − / ( A ; λ ). We require∆ λ = ( λ | λ + 2 ρ )2( k ♮ + 6) = ( λ | λ + 2 ρ )7 ∈ Z OLLAPSING LEVELS OF W -ALGEBRAS 77 O k G.f S O ,f pq = k + h ∨ g k ♮ + h ∨ g ♮ c V g V A V comments E E { } /

13 0 0 113 /

12 0 0 1 E ( a ) E ( a ) { } / E ( a ) A a / / − √ / / −

25 207 228 √ = 128 √ D N A / / −

10 487 17 / / −

48 6 181 √ A + A A a / / −

12 125 15 √ D ( a ) 2 A g sp / − k = − k ♮ = − A + A A + A { } ⊂ A / / A g / / −

10 6 127 √ A A { } ⊂ A × A / / √ A b / − k = − k ♮ = − A unknown in A / / −

25 20 22 √ = 12 √ ≈ O (23) Table 5.

Main asymptotic data in type E where λ = P i =1 λ i ̟ i , assuming that all λ i ∈ Z > , with 1 = p − h ∨ A > P i =1 λ i > λ = 0 or λ = ̟ .On the other hand, by Proposition 4.10, we easily see that A L − / ( A ; ̟ ) = A L − / ( A ) . As a result, m A L − / ( A ;0) + m A L − / ( A ; ̟ ) = 2 A L − / ( A ;0) . But m must be at most 1. We conclude that H DS,A ( L − / ( E )) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) . In particular, since H DS,A ( L − / ( E )) is a ﬁnite extension of L − / ( A )it is simple, whence the expected statement. We argue similarly in all the othercases. (cid:3) Remark . The associated variety of W − / ( E , A ) is not isomorphic to theassociated variety of L − / ( A ). Indeed the former is the nilpotent Slodowy slice S A ,A in E while the latter is the closure of the orbit of A = sl attached withthe partition (2 ). These two varieties cannot be isomorphic since the number We thank Daniel Juteau for this remark. of nilpotent G ♮ -orbits in O (2 ) is 4, while the number of G ♮ -orbits in S A ,A is 3,as we can see from the Hasse diagram of E . Here G ♮ denotes the centraliser in g = E of the sl -triple associated with f in the minimal nilpotent orbit A of E .It is a maximal reductive subgroup of the centraliser of f in g whose Lie algebra is g ♮ ∼ = A .On the other hand we observe that O (2 ) and S A ,A are birationally equivalentsince they are both irreducible with open part the open G ♮ -orbit of dimension 18.We conjecture the following isomorphisms at non admissible level. Conjecture 10.3. W − ( E , A ) ∼ = L − ( G ) , W − ( E , A ) ∼ = L − ( B ) . Theorem 10.4.

The following isomorphisms hold, providing collapsing levels for g = E . W − / ( E , E ) ∼ = C , W − / ( E , E ) ∼ = C , W − / ( E , E ( a )) ∼ = C , W − / ( E , E ) ∼ = L − / ( A ) , W − / ( E , E ) ∼ = L − / ( A ) , W − / ( E , D ) ∼ = L − / ( A ) , W − / ( E , A ) ∼ = C , W − / ( E , A ) ∼ = L ( A ) , W − / ( E , ( A ) ′′ ) ∼ = L − / ( G ) , W − / ( E , E ( a )) ∼ = C , W − / ( E , E ( a )) ∼ = L − / ( A ) , W − / ( E , D ( a )) ∼ = L − / ( A ) , W − / ( E , ( A ) ′ ) ∼ = L − / ( A ) ⊗ L − / ( A ) , W − / ( E , A + A ) ∼ = L ( A ) , W − / ( E , A ) ∼ = L − / ( A ) , W − / ( E , A + A + A ) ∼ = L ( A ) , W − / ( E , A + A ) ∼ = L ( A ) , W − / ( E , ( A ) ′′ ) ∼ = L − / ( G ) , W − / ( E , A + 3 A ) ∼ = L − / ( G ) , W − / ( E , A ) ∼ = L ( A ) ⊗ L − / ( G ) , W − / ( E , A ) ∼ = C , W − / ( E , (3 A ) ′ ) ∼ = L − / ( C ) , W − / ( E , (3 A ) ′′ ) ∼ = L − / ( F ) . Moreover, the following isomorphisms of W -algebras with ﬁnite extensions of ad-missible aﬃne vertex algebras hold. W − / ( E , D ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) , W − / ( E , D ) ∼ = ( L − / ( A ) ⊗ L − / ( A )) ⊕ ( L − / ( A ; ̟ ) ⊗ L − / ( A ; ̟ )) , W − / ( E , A ) ∼ = ( L − / ( A ) ⊗ L − / ( B )) ⊕ ( L − / ( A ; ̟ ) ⊗ L − / ( B ; ̟ )) , W − / ( E , A ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) ⊕ L − / ( A ; ̟ ) , W − / ( E , A ) ∼ = L − / ( D ) ⊕ L − / ( D ; ̟ ) . Remark . (1) The associated variety of W − / ( E , A ) is not isomorphic to the asso-ciated variety of L − / ( A ). Indeed the former is the nilpotent Slodowyslice S A + A ,A in E while the later is the closure of the orbit of A = sl attached with the partition (3 ). The number of nilpotent G ♮ -orbits in O (3 ) is 6 while the number of G ♮ -orbits in S A + A ,A is 5 as we can see fromthe Hasse diagram of E . However we observe that O (3 ) and S A + A ,A are birationally equivalent since they are both irreducible with open partthe open G ♮ -orbit of dimension 24.(2) The associated variety of W − / ( E , A ) is not isomorphic to the associ-ated variety of L − / ( D ). Indeed the former is the nilpotent Slodowyslice S A ,A in E while the later is the closure of the orbit of D = so attached with the partition (3 , , G ♮ -orbits OLLAPSING LEVELS OF W -ALGEBRAS 79 in O , (3 , , is 7 while the number of G ♮ -orbits in S A ,A is 5 as we cansee from the Hasse diagram of E . However we observe that O , (3 , , and S A ,A are birationally equivalent since they are both irreducible withopen part the open G ♮ -orbit of dimension 36.(3) The associated variety of W − / ( E , A ) is not isomorphic to the as-sociated variety of L − / ( A ) ⊗ L − / ( B ). Indeed the former is thenilpotent Slodowy slice S A + A + A ,A in E while the later is the closure ofthe orbit of a × O , (3 , ⊂ A × B . But the number of nilpotent G ♮ -orbitsin a × O , (3 , is 5 × G ♮ -orbits in S A + A + A ,A is6 as we can see from the Hasse diagram of E . However we observe that a × O , (3 , and S A + A + A ,A are birationally equivalent since they areboth irreducible with open part the open G ♮ -orbit of dimension 2+14 = 16.Based on the coincidence of central charges and asymptotic growths and dimen-sions at the non admissible levels k and k ♮ corresponding to k + 18 = p/q = 18 / k ♮ + 4 = 6 /

3, we conjecture:

Conjecture 10.6. W − ( E , A + 3 A ) ∼ = L − ( G ) . Theorem 10.7.

The following isomorphisms hold, providing collapsing levels for g = E . W − / ( E , E ) ∼ = C , W − / ( E , E ) ∼ = C , W − / ( E , E ( a )) ∼ = C , W − / ( E , E ( a )) ∼ = C , W − / ( E , E ) ∼ = L − / ( A ) , W − / ( E , E ( a )) ∼ = C , W − / ( E , E ( a )) ∼ = L − / ( A ) , W − / ( E , E ( a )) ∼ = C , W − / ( E , E ) ∼ = L − / ( G ) , W − / ( E , E ) ∼ = L − / ( G ) , W − / ( E , E ( a )) ∼ = C , W − / ( E , D ) ∼ = L − / ( C ) , W − / ( A , A ) ∼ = C , W − / ( E , A ) ∼ = L − / ( A ) , W − / ( E , D ) ∼ = L − / ( F ) , W − / ( E , D ) ∼ = L − / ( F ) , W − / ( E , E ( a )) ∼ = C , W − / ( E , E ( a )) ∼ = L − / ( A ) , W − / ( E , E ( a )) ∼ = L − / ( G ) , W − / ( E , D ( a )) ∼ = L − / ( A ) ⊗ L − / ( A ) , W − / ( E , A + A ) ∼ = C , W − / ( E , A + A ) ∼ = L ( A ) , W − / ( E , A ) ∼ = C , W − / ( E , D ( a ) + A ) ∼ = L − / ( A ) , W − / ( E , A + 2 A ) ∼ = C , W − / ( E , A + 2 A ) ∼ = L ( B ) , W − / ( E , A ) ∼ = L − / ( G ) , W − / ( E , A ) ∼ = C , W − / ( E , A ) ∼ = L − / ( F ) , Moreover, the following isomorphisms of W -algebras with ﬁnite extensions of ad-missible aﬃne vertex algebras hold. W − / ( E , E ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) , W − / ( E , D ) ∼ = L − / ( B ) ⊕ L − / ( B ; ̟ ) , W − / ( E , D ) ∼ = L − / ( B ) ⊕ L − / ( B ; ̟ ) , W − / ( E , D ( a )) ∼ = L − / ( D ) ⊕ M i =1 , , L − / ( D ; ̟ i ) , W − / ( E , A ) ∼ = L − / ( E ) ⊕ L − / ( E ; ̟ ) ⊕ L − / ( E ; ̟ ) , W − / ( E , A ) ∼ = L − / ( B ) ⊕ L − / ( B ; ̟ ) , W − / ( E , A ) ∼ = L − / ( E ) ⊕ L − / ( E ; ̟ ) . O k G.f S O ,f pq = k + h ∨ g k ♮ + h ∨ g ♮ c V g V A V comments E E { } /

19 0 0 119 /

18 0 0 1 E ( a ) E ( a ) { } /

14 0 0 1 E ( a ) E a /

13 2 / −

36 3613 113 √ /

12 3 / − √ E ( a ) D a /

11 3 / −

41 3211 244 √ = 144 √

11 ﬁn. ext.19 /

10 12 / −

12 125 15 √ E ( a ) D a × a / / −

18 214 2256 √ = 1256 √ / A A { } ⊂ A / / / √ A ( A ) ′′ N G / / −

84 12 17 E ( a ) E ( a ) { } / E ( a ) a / / − D ( a ) a / / − √ A ) ′ a × a / / − √ / A ) ′′ g sp / / −

34 10 12 √ A + A A + A { } ⊂ A / / s

25 sin π A a / / −

32 325 15 A + A + A A + A + A { } ⊂ A / / A unknown in A × B / / −

44 352 22 √ = 12 √ ≈ a × O , / A + A A + A { } / / √ / A + 3 A g / / − √ / / −

14 143 13 √ A { } × g / / − √ ⊂ A × G / A unknown in A / / −

55 25 33 √ = 13 √ ≈ O (32) A A { } / A ) ′ c / / − (3 A ) ′′ f / / −

20 16 12 A unknown in D / / −

54 36 22 = 12 ﬁn. ext. ≈ O , , Table 6.

Main asymptotic data in type E OLLAPSING LEVELS OF W -ALGEBRAS 81 O k G.f S O ,f pq = k + h ∨ g k ♮ + h ∨ g ♮ c V g V A V comments E E { } E ( a ) E ( a ) { } E ( a ) E ( a ) { } E ( a ) E a / −

24 83 12730 /

19 3 / −

73 5619 276 √ = 176 √

19 ﬁn. ext. E ( a ) E ( a ) { } E ( b ) E ( a ) a /

14 3 / −

25 207 128 √ E ( a ) E ( a ) { } E N G /

12 7 / −

82 12 13 √ /

13 4 / −

168 16813 113 E ( a ) E ( a ) { } /

10 0 0 1 E ( a ) D N B /

11 5 / −

122 10411 22 = 12 ﬁn. ext.31 /

10 3 / −

40 8 15 A A { } ⊂ A / D unknown in B / / −

99 18 22 √ = 12 √

32 ﬁn. ext. ≈ N B A + A A a / / −

36 187 17 √ E ( a ) E ( a ) { } / E ( a ) a / / − √ D ( a ) a × a / / −

12 4 13 / E ( a ) g sp / / −

34 10 12 √ A + A D F ( a ) 30 / / −

312 3127 17 E ( a ) D F ( a ) 31 / / −

164 40 13 A + A A + A { } ⊂ A / / / A A { } ⊂ A / D ( a ) + A a / / − D ( a ) unknown in D / / −

68 22 42 = 12 ﬁn. ext. ≈ O , A unknown in B / / −

125 40 22 = 12 ﬁn. ext. ≈ O , Table 7.

Main asymptotic data in type E O k G.f S O ,f pq = k + h ∨ g k ♮ + h ∨ g ♮ c V g V A V comments2 A + 2 A A + 2 A { } ⊂ B / / / A g × g / / −

20 12 13 / A unknown in E / / −

138 54 33 √ = 13 √ ≈ A + A A A { } ⊂ C / A f / / −

20 16 12 A unknown in B / / −

54 36 22 = 12 ﬁn. ext. ≈ O , A unknown in E / / −

119 70 22 = 12 ﬁn. ext. ≈ A Table 8.

Main asymptotic data in type E (continued) Remark . (1) The associated variety of W − / ( E , D ( a )) is not isomorphic to the as-sociated variety of L − / ( D ). Indeed the former is the nilpotent Slodowyslice S A ,D ( a ) in E while the later is the closure of the orbit O , in D . But the number of nilpotent G ♮ -orbits in O , is 11 while the num-ber of G ♮ -orbits in S A ,D ( a ) is 6. However we observe that O , and S A ,D ( a ) are birationally equivalent since they are both irreducible withopen part an open G ♮ -orbit of dimension 22.(2) The associated variety of W − / ( E , A ) is not isomorphic to the asso-ciated variety of L − / ( B ). Indeed the former is the nilpotent Slodowyslice S A ,A in E while the later is the closure of the orbit O , in B .But the number of nilpotent G ♮ -orbits in O , is 11 while the numberof G ♮ -orbits in S A ,A is 9. However we observe that O , and S A ,A are birationally equivalent since they are both irreducible with open partan open G ♮ -orbit of dimension 40.(3) The associated variety of W − / ( E , A ) is not isomorphic to the asso-ciated variety of L − / ( E ). Indeed the former is the nilpotent Slodowyslice S A +2 A ,A in E while the later is the closure of the orbit 2 A + A in E . But the number of nilpotent G ♮ -orbits in the closure of 2 A + A is 8 while the number of G ♮ -orbits in S A +2 A ,A is 7 as we can see fromthe Hasse diagrams of E and E . However we observe that 2 A + A and S A +2 A ,A are birationally equivalent since the open part is in both sidesan open G ♮ -orbit of dimension 54.(4) The associated variety of W − / ( E , A ) is isomorphic to the associ-ated variety of L − / ( B ). Indeed the former is the nilpotent Slodowyslice S A , A in E while the later is the closure of the orbit O , in B . The number of nilpotent G ♮ -orbits in O , and in S A , A is 4 in OLLAPSING LEVELS OF W -ALGEBRAS 83 O k G.f S O ,f pq = k + h ∨ g k ♮ + h ∨ g ♮ c V g V A V comments G G { } /

12 0 0 14 / G ( a ) G ( a ) { } / A a / / − √ A A m ≈ a / / / / √ π = 12 √ π A A { } ⊂ A / / / / √ Table 9.

Main asymptotic data in type G both sides. But the dimension of the G ♮ -orbits are diﬀerent: in O , itis 36, 32, 20, 0 while in S A , A it is 36, 22, 20, 0. However, we observethat O , and S A , A are birationally equivalent since they are bothirreducible with open part an open G ♮ -orbit of dimension 36.(5) The associated variety of W − / ( E , A ) is not isomorphic to the associ-ated variety of L − / ( E ). Indeed the former is the nilpotent Slodowyslice S A ,A in E while the later is the closure of the orbit 4 A in E .But the number of nilpotent G ♮ -orbits in the closure of 4 A is 6 while thenumber of G ♮ -orbits in S A ,A is 5 as we can see from the Hasse diagramsof E and E . However we observe that 4 A and S A ,A are birationallyequivalent since the open part is in both sides an open G ♮ -orbit of dimension70.(6) The associated variety of S A ,D is birationally equivalent to the nilpotentcone of B . We think they are actually isomorphic (they both have 7 G ♮ -orbits of respective dimensions 18, 16, 14, 12, 10, 8, 0).In type E we make the following conjecture concerning non admissible levels. Conjecture 10.9.

The following isomorphisms hold. W − / ( E , E ) ∼ = L − / ( G ) , W − ( E , E ( a )) ∼ = L − ( G ) , W − / ( E , A + 2 A ) ∼ = C , W − / ( E , A + A + A ) ∼ = C . For example in the case of f of Bala-Carter type A + 2 A the conjecture ismotivated by the fact that the central charge is 0 and k ♮ = k ♮ = 0. Similarly for f of type A + A + A we ﬁnd k ♮ = 0. Theorem 10.10.

The following isomorphisms hold, providing collapsing levels for G . W − / ( G , G ) ∼ = C , W − / ( G , G ) ∼ = C W − / ( G , G ( a )) ∼ = C , W − / ( G , ˜ A ) ∼ = L − / ( A ) , W − / ( G , A ) ∼ = C , W − / ( G , A ) ∼ = L ( A ) . Moreover, the following isomorphism holds. W − / ( G , A ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) . Remark . The associated variety of W − / ( G , A ) is not isomorphic to theassociated variety of L − / ( A ). Indeed the former is the nilpotent Slodowy slice S ˜ A ,A in G while the later is the minimal nilpotent orbit a in A . It is knownthat S ˜ A ,A is isomorphic to the variety m in the notation of [52, § m and a are birationally equivalentsince they are both irreducible with open part an open G ♮ -orbit 2. Theorem 10.12.

The following isomorphisms hold, providing collapsing levels for g = F . W − / ( F , F ) ∼ = C , W − / ( F , F ) ∼ = C W − / ( F , F ( a )) ∼ = C , W − / ( F , C ) ∼ = L − / ( A ) W − / ( F , B ) ∼ = L − / ( A ) , W − / ( F , B ) ∼ = L ( A ) W − / ( F , F ( a )) ∼ = C , W − / ( F , C ( a )) ∼ = L − / ( A ) W − / ( F , B ) ∼ = L − / ( A ) ⊗ L − / ( A ) , W − / ( F , ˜ A )) ∼ = L − / ( G ) W − / ( F , A + ˜ A ) ∼ = C , W − / ( F , A )) ∼ = L − / ( A ) , W − / ( F , A )) ∼ = C . Moreover, the following isomorphisms of W -algebras with ﬁnite extensions of ad-missible aﬃne vertex algebras holds. W − / ( F , C ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) , W − / ( F , B ) ∼ = ( L − / ( A ) ⊗ L − / ( A )) ⊕ ( L − / ( A ; ̟ ) ⊗ L − / ( A ; ̟ )) , W − / ( F , A + ˜ A ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) , W − / ( F , A ) ∼ = L − / ( A ) ⊕ L − / ( A ; ̟ ) ⊕ L − / ( A ; ̟ ) , W − / ( F , ˜ A ) ∼ = L − / ( A ) ⊕ M i =1 , , L − / ( A ; ̟ i ) . Remark . (1) The associated variety of W − / ( F , ˜ A ) is not isomorphic to the asso-ciated variety of L − / ( A ). Indeed the former is the nilpotent Slodowyslice S A + ˜ A , ˜ A in F while the later is the nilpotent cone of A = sl . Butthe number of nilpotent G ♮ -orbits in N A is 5 while the number of G ♮ -orbitsin S A + ˜ A , ˜ A is 4 as we can see from the Hasse diagram of F . However weobserve that N A and S A + ˜ A , ˜ A are birationally equivalent since they areboth irreducible with open part the open G ♮ -orbit of dimension 12.(2) The associated variety of W − / ( F , A + ˜ A ) is not isomorphic to the as-sociated variety of L − / ( A ). Indeed the former is the nilpotent Slodowyslice S ˜ A + A ,A + ˜ A whose singularity at f ∈ A + ˜ A has type m (see [52, § A = sl (type a ). Howeverwe observe that S ˜ A + A ,A + ˜ A and N A are birationally equivalent. Conjecture 10.14.

The following isomorphisms hold. W − / ( F , ˜ A ) ∼ = L − / ( G ) , W − ( F , ˜ A ) ∼ = L − ( G ) . Writing k = − / − / k ♮ = − / − /

6, we ﬁndthat W − / ( F , ˜ A ) and L − / ( G ) have the same central charge, and that the(conjectural) formula for asymptotic growth and asymptotic dimension coincide;see Table 10. Similarly, writing k = − / k ♮ = − W − ( F , ˜ A ) and L − ( G ) have the same central charge, and that the (conjectural)formula for asymptotic growth and asymptotic dimension coincide; see Table 10. OLLAPSING LEVELS OF W -ALGEBRAS 85 O k G.f S O ,f pq = k + h ∨ g k ♮ + h ∨ g ♮ c V g V A V comments F F { } /

18 0 0 19 /

13 0 0 1 F ( a ) F ( a ) { } /

12 0 0 1 F ( a ) C a / / −

18 187 17 √ /

10 3 / −

17 145 22 √ = 12 √ B a / / −

18 187 17 √ B B { } ⊂ A / / √ A N G / / −

98 494 not admissible F ( a ) F ( a ) { } / C ( a ) a / / − √ B a × a / / −

12 4 13 √ / / / −

34 285 45 / = 25 / ﬁn. ext.3 / A g sp / / −

34 10 12 √ / / −

98 143 13 √ A + A ˜ A g / / −

14 143 13 √ A + ˜ A m / / √ π = 12 √ π A unknown 10 / / −

325 325 3 × / (cid:18) sin π (cid:19) sin 2 π ≈ N A = 2 / (cid:18) sin π (cid:19) sin 2 π A + ˜ A A + ˜ A { } ⊂ A / / A a +2 / / − ˜ A unknown 13 / / −

33 12 42 = 12 ﬁn. ext. ≈ N A A A { } ⊂ C / / Table 10.

Main asymptotic data in type F Appendix A. Centralisers of sl -triples in simple exceptional Liealgebras In this appendix we collect the data relative to each sl -triples in simple ex-ceptional Lie algebras. Our results are obtained using the software GAP4 and aresummarised in Tables 11, 12, 13, 15 and 17.

In the Tables, nilpotent orbits are given by the Bala-Carter classiﬁcation (ﬁrstcolumn). We indicate in the second column whether the nilpotent orbit is even ornot. The third column gives the type of g ♮ , and the last column gives the valuesof the k ♮i ’s. Obviously, the order follows the order of the simple factors of g ♮ asappearing in the third column. OLLAPSING LEVELS OF W -ALGEBRAS 87 G.f even g ♮ = L i g ♮i k ♮i G yes { } G ( a ) yes { } ˜ A no A k ♮ = k + A no A k ♮ = 3 k + 5 Table 11.

Centralisers of sl -triples in type G G.f even g ♮ = L i g ♮i k ♮i F yes { } F ( a ) yes { } F ( a ) yes { } C yes A k ♮ = k + 6 B yes A k ♮ = 8 k + 60 F ( a ) yes { } C ( a ) no A k ♮ = k + ˜ A + A no A k ♮ = 3 k + 17 B no A × A k ♮ = k + k ♮ = k + A + ˜ A no A k ♮ = 6 k + ˜ A yes G k ♮ = k + 4 A yes A k ♮ = 2 k + 10 A + ˜ A no A × A k ♮ = k + 4 k ♮ = 8 k + 40˜ A no A k ♮ = k + 3 A no C k ♮ = k + Table 12.

Centralisers of sl -triples in type F G.f even g ♮ = L i g ♮i k ♮i E yes { } E ( a ) yes { } D yes C k ♮ = k + E ( a ) yes { } D ( a ) no C k ♮ = 12 k + 117 A no A k ♮ = k + A + A no C k ♮ = 15 k + 144 D yes A k ♮ = 2 k + 18 A yes C × A k ♮ = k + 72 k ♮ = k + 8 D ( a ) yes C k ♮ = k + 9 A + A no C × A k ♮ = k + 9 k ♮ = k + A + A no A k ♮ = 3 k + 23 A no C × B k ♮ = 3 k + 27 k ♮ = k + 7 A + 2 A no C × A k ♮ = 6 k + 45 k ♮ = 6 k + 452 A no G k ♮ = k + 6 A + A no C × A k ♮ = 6 k + 45 k ♮ = k + 6 A yes A × A k ♮ = k + 6 k ♮ = k + 63 A no A × A k ♮ = k + k ♮ = 2 k + 122 A no C × B k ♮ = 3 k + 18 k ♮ = k + 4 A no A k ♮ = k + 3 Table 13.

Centralisers of sl -triples in type E G.f even g ♮ = L i g ♮i k ♮i E yes { } E ( a ) yes { } E ( a ) yes { } E ( a ) yes { } E yes A k ♮ = 3 k + 48 E ( a ) yes C k ♮ = k + 16 D no A k ♮ = k + E ( a ) yes { } D ( a ) no A k ♮ = k + 14 D + A no A k ♮ = 2 k + 30 A yes A k ♮ = 7 k + 108 E ( a ) yes { } D yes A × A k ♮ = k + 14 k ♮ = 2 k + 30 E ( a ) yes A k ♮ = 3 k + 44 D ( a ) yes A k ♮ = k + D ( a ) + A yes A k ♮ = 8 k + 116 A + A no A k ♮ = 3 k + 41( A ) ′ no A × A k ♮ = k + k ♮ = 3 k + 44 A + A yes A k ♮ = 15 k + 216 D ( a ) no C × A k ♮ = 2 k + 29 k ♮ = k + 13 A + A no C k ♮ = k + 16 D + A no B k ♮ = k + ( A ) ′′ yes G k ♮ = k + 12 A + A + A yes A k ♮ = 24 k + 320 A yes C × A k ♮ = k + k ♮ = k + 12 A + A no C × A k ♮ = k + k ♮ = k + 12 Table 14.

Centralisers of sl -triples in type E G.f even g ♮ = L i g ♮i k ♮i D ( a ) + A no A × A k ♮ = k + 12 k ♮ = k + 12 D yes C k ♮ = k + 12 A + 2 A no A × A k ♮ = k + k ♮ = 2 k + 6 D ( a ) yes A × A × A k ♮ = k + 12 k ♮ = k + 12 k ♮ = k + 12( A + A ) ′ no A × A × A k ♮ = k + k ♮ = k + 12 k ♮ = 2 k + 242 A + A no A × A k ♮ = 3 k + 36 k ♮ = 3 k + 35( A + A ) ′′ no B k ♮ = k + 10 A + 3 A yes G k ♮ = 2 k + 222 A yes A × G k ♮ = 3 k + 36 k ♮ = k + 10 A no A × B k ♮ = k + 12 k ♮ = k + 10 A + 2 A no A × A × A k ♮ = k + 10 k ♮ = 2 k + 22 k ♮ = 6 k + 66 A + A no C × A k ♮ = k + 11 k ♮ = k + 94 A no C k ♮ = k + A yes A k ♮ = k + 8(3 A ) ′ no A × C k ♮ = k + k ♮ = k + 8(3 A ) ′′ yes F k ♮ = k + 62 A no A × B k ♮ = k + 8 k ♮ = k + 6 A no D k ♮ = k + 4 Table 15.

Centralisers of sl -triples in type E (contin-ued) OLLAPSING LEVELS OF W -ALGEBRAS 89 G.f even g ♮ = L i g ♮i k ♮i E yes { } E ( a ) yes { } E ( a ) yes { } E ( a ) yes - { } E ( a ) yes { } E no A k ♮ = k + E ( b ) yes { } E ( a ) yes { } E ( a ) no A k ♮ = k + 26 E ( b ) yes { } D no A k ♮ = 2 k + E ( a ) yes { } E ( a ) no A k ♮ = k + E + A no A k ♮ = 3 k + 77 D ( a ) no C k ♮ = 4 k + 106 E ( b ) yes { } E ( a ) no A k ♮ = k + 25 E ( a ) + A no C k ♮ = 2 k + 51 A no A k ♮ = 4 k + D ( a ) no C k ♮ = k + 26 E yes G k ♮ = k + 24 D no B k ♮ = k + D + A yes C k ♮ = 3 k + 76 E ( a ) yes A k ♮ = k + 24 E ( a ) no A k ♮ = k + 24 A + A no A k ♮ = 7 k + 180 D ( a ) no A × A k ♮ = k + 48 k ♮ = k + 48 A yes A × A k ♮ = k + 24 k ♮ = k + 24 E ( a ) yes { } D + A no A × A k ♮ = k + k ♮ = 2 k + 48 E ( a ) no A k ♮ = k + E ( a ) + A no A k ♮ = 3 k + 71 D ( a ) no A × A k ♮ = k + k ♮ = k + D ( a ) + A no A k ♮ = 6 k + 285 / A + A no A × A k ♮ = k + k ♮ = 3 k + 14 A + A no A k ♮ = 10 k + 238 D yes B k ♮ = k + 22 E ( a ) yes G k ♮ = k + 22 D + A yes A k ♮ = 2 k + 46 Table 16.

Centralisers of sl -triples in type E G.f even g ♮ = L i g ♮i k ♮i A + A + A no A k ♮ = 15 k + 350 D ( a ) + A no A × A k ♮ = k + 22 k ♮ = 8 k + 184 A no A × G k ♮ = k + k ♮ = k + 22 A + A yes A × A k ♮ = 15 k + 350 k ♮ = k + 22 A + 2 A no C × A k ♮ = k + k ♮ = 2 k + 45 D ( a ) no A k ♮ = k + 212 A no B k ♮ = 2 k + A + A no C × A k ♮ = k + k ♮ = k + 21 D ( a ) + A yes A k ♮ = 6 k + 132 D + A no C k ♮ = k + A + A + A no A × A k ♮ = k + k ♮ = 24 k + 528 A yes A k ♮ = k + 20 A + A no C × B k ♮ = k + 22 k ♮ = k + 20 D ( a ) + A no A × A × A k ♮ = k + 20 k ♮ = k + 20 k ♮ = k + 20 A + 2 A no A × B k ♮ = 2 k + 40 k ♮ = k + A + 2 A no B k ♮ = 3 k + 59 D yes F k ♮ = k + 18 D ( a ) no D k ♮ = k + 18 A + A no A × B k ♮ = k + k ♮ = k + 182 A + A no A × G k ♮ = 3 k + 59 k ♮ = k + A yes G × G k ♮ = k + 18 k ♮ = k + 18 A + 3 A no A × G k ♮ = k + k ♮ = 2 k + 36 A no B k ♮ = k + 16 A + 2 A no A × B k ♮ = 6 k + 108 k ♮ = k + 16 A + A no A k ♮ = k + 154 A no C k ♮ = k + A yes E k ♮ = k + 123 A no A × F k ♮ = k + k ♮ = k + 122 A no B k ♮ = k + 10 A no E k ♮ = k + 6 Table 17.

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