Simplicity of vacuum modules and associated varieties

Simplicity of vacuum modules and associated varieties Tome 8 (2021), p. 169-191.

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SIMPLICITY OF VACUUM MODULES AND ASSOCIATED VARIETIES

by Tomoyuki Arakawa, Cuipo Jiang & Anne Moreau

Abstract. — In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebragis simple if and only if the associated variety of its unique simple quotient is equal tog^∗. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.

Résumé(Simplicité des algèbres vertex affines et variétés associées). — Dans cet article, nous démontrons que l’algèbre vertex affine universelle associée à une algèbre de Lie simplegest simple si et seulement si la variété associée à son unique quotient simple est égale àg^∗. Nous en déduisons un résultat analogue pour la réduction quantique de Drinfeld-Sokolov appliquée à l’algèbre vertex affine universelle.

Contents

1. Introduction. . . 169 2. Universal affine vertex algebras and associated graded vertex Poisson algebras 172 3. Proof of the main result. . . 178 4. W-algebras and proof of Theorem 1.3. . . 187 References. . . 189

1. Introduction LetV be a vertex algebra, and let

V −→(EndV)[[z, z⁻¹]], a7−→a(z) =X

n∈Z

a_(n)z⁻ⁿ⁻¹,

be the state-field correspondence. TheZhuC2-algebra[Zhu96] ofV is by definition the quotient space RV =V /C2(V), where C2(V) = span_C{a₍₋₂₎b |a, b ∈V}, equipped with the Poisson algebra structure given by

a.b=a₍₋₁₎b, {a, b}=a₍₀₎b,

Mathematical subject classification (2020). — 17B69.

Keywords. — Associated variety, affine Kac-Moody algebra, affine vertex algebra, singular vector, affineW-algebra.

T.A. is supported by partially supported by JSPS KAKENHI Grant No. 17H01086 and No. 17K18724.

C.J. is supported by CNSF grants 11771281 and 11531004. A.M. is supported by the ANR Project GeoLie Grant number ANR-15-CE40-0012, and by the Labex CEMPI (ANR-11-LABX-0007-01).

fora, b∈V witha:=a+C2(V). The associated varietyXV ofV is the reduced scheme XV = Specm(RV) corresponding to RV. It is a fundamental invariant of V that captures important properties of the vertex algebraV itself (see, for example, [BFM, Zhu96, ABD04, Miy04, Ara12a, Ara15a, Ara15b, AM18a, AM17, AK18]). Moreover, the associated variety XV conjecturally [BR18] coincides with the Higgs branch of a 4D N = 2 superconformal field theory T, if V corresponds to a theoryT by the 4D/2D duality discovered in [BLL⁺15]. Note that the Higgs branch of a 4D N= 2 superconformal field theory is a hyperkähler cone, possibly singular.

In the case where V is the universal affine vertex algebra V^k(g) at level k ∈ C associated with a complex finite-dimensional simple Lie algebrag, the varietyXV is just the affine spaceg^∗ with Kirillov-Kostant Poisson structure. In the case whereV is the unique simple graded quotient Lk(g) of V^k(g), the variety XV is a Poisson subscheme ofg^∗ which isG-invariant and conic, whereGis the adjoint group ofg.

Note that if the levelkis irrational, then L_k(g) =V^k(g), and henceX_Lk(g)=g^∗. More generally, ifL_k(g) =V^k(g), that is,V^k(g)is simple, then obviouslyX_Lk(g)=g^∗.

In this article, we prove that the converse is true.

Theorem1.1. — The equality Lk(g) =V^k(g) holds, that is,V^k(g) is simple, if and only ifX_Lk(g)=g^∗.

It is known by Gorelik and Kac [GK07] thatV^k(g)is not simple if and only if (1.1) r^∨(k+h^∨)∈Q>0r{1/m|m∈Z>1},

where h^∨ is the dual Coxeter number andr^∨ is the lacing number of g. Therefore, Theorem 1.1 can be rephrased as

(1.2) X_Lk(g)(g^∗ ⇐⇒ (1.1) holds.

Let us mention the cases when the varietyX_Lk(g) is known forksatisfying (1.1).

First, it is known [Zhu96, DM06] that X_Lk(g) = {0} if and only if Lk(g)is inte- grable, that is,kis a nonnegative integer. Next, it is known that ifLk(g)isadmissible [KW89], or equivalently, if

k+h^∨= p

q, p, q∈Z>1, (p, q) = 1, p>

(h^∨ if(r^∨, q) = 1, h if(r^∨, q)6= 1,

wherehis the Coxeter number ofg, thenX_Lk(g)is the closure of some nilpotent orbit ing([Ara15a]). Further, it was observed in [AM18a, AM18b] that there are cases when Lk(g)is non-admissible and X_Lk(g) is the closure of some nilpotent orbit. In fact, it was recently conjectured in physics [XY19] that, in view of the 4D/2D duality, there should be a large list of non-admissible simple affine vertex algebras whose associated varieties are the closures of some nilpotent orbits. Finally, there are also cases [AM17]

whereX_Lk(g) is neitherg^∗ nor contained in the nilpotent coneN(g)ofg.

In general, the problem of determining the varietyXL_k(g)is wide open.

Now let us explain the outline of the proof of Theorem 1.1. First, Theorem 1.1 is known for the critical level k=−h^∨ ([FF92, FG04]). Therefore, sinceR_Vk(g) is a polynomial ringC[g^∗], Theorem 1.1 follows from the following fact.

Theorem1.2. — Suppose that the level is non-critical, that is,k6=−h^∨. The image of any nonzero singular vectorv ofV^k(g)in the ZhuC2-algebraR_Vk(g) is nonzero.

The symbolσ(w) of a singular vectorw in V^k(g) is a singular vector in the cor- responding vertex Poisson algebra grV^k(g) ∼=S(t⁻¹g[t⁻¹]) ∼=C[J_∞g^∗], where J_∞g^∗ is the arc space of g^∗. Theorem 1.2 states that the image of σ(w) of a non-trivial singular vector wunder the projection

(1.3) C[J_∞g^∗]−→C[g^∗] =R_Vk(g)

is nonzero, provided that k is non-critical. Here the projection (1.3) is defined by identifyingC[g^∗]with the ZhuC₂-algebra of the commutative vertex algebraC[J_∞g^∗].

Hence, Theorem 1.2 would follow if the image of any nontrivial singular vector in C[J∞g^∗] under the projection (1.3) is nonzero. However, this is false as there are singular vectors inC[J_∞g^∗]that do not come from singular vectors ofV^k(g)and that belong to the kernel of (1.3) (see Section 3.4). Therefore, we do need to make use of the fact thatσ(w)is the symbol of a singular vectorw in V^k(g). We also note that the statement of Theorem 1.2 is not true ifk is critical (see Section 3.4).

For this reason the proof of Theorem 1.2 is divided roughly into two parts. First, we work in the commutative setting to deduce a first important reduction (Lemma 3.1).

Next, we use the Sugawara construction – which is available only at non-critical levels – in the non-commutative setting in order to complete the proof.

Now, let us consider theW-algebraW^k(g, f)associated with a nilpotent elementf of g at the level k defined by the generalized quantized Drinfeld-Sokolov reduction [FF90, KRW03]:

W^k(g, f) =H_DS,f⁰ (V^k(g)).

Here,H_DS,f^• (M)denotes the BRST cohomology of the generalized quantized Drinfeld- Sokolov reduction associated with f ∈N(g)with coefficients in aV^k(g)-moduleM.

By the Jacobson-Morosov theorem, f embeds into an sl₂-triple (e, h, f). The Slodowy sliceSf atf is the affine spaceSf =f+g^e, whereg^eis the centralizer ofe ing. It has a natural Poisson structure induced from that ofg^∗ (see [GG02]), and we have [DSK06, Ara15a] a natural isomorphismR_Wk(g,f)∼=C[Sf] of Poisson algebras, so that

X_Wk(g,f)=Sf.

The natural surjectionV^k(g)Lk(g)induces a surjectionW^k(g, f)H_DS,f⁰ (Lk(g)) of vertex algebras ([Ara15a]). Hence the varietyX_H0

DS,f(Lk(g))is aC^∗-invariant Poisson subvarieties of the Slodowy sliceSf.

Conjecturally [KRW03, KW08], the vertex algebra H_DS,f⁰ (L_k(g)) coincides the unique simple (graded) quotientWk(g, f)ofW^k(g, f)provided thatH_DS,f⁰ (Lk(g))6= 0.

(This conjecture has been verified in many cases [Ara05, Ara07, Ara11, AvE19].)

As a consequence of Theorem 1.1, we obtain the following result.

Theorem1.3. — Let f be any nilpotent element of g. The following assertions are equivalent:

(1) V^k(g)is simple,

(2) W^k(g, f) =H_DS,f⁰ (Lk(g)), (3) X_H0

DS,f(L_k(g)) =Sf.

Note that Theorem 1.3 implies that V^k(g) is simple if X_Wk(g,f) = Sf and H_DS,f⁰ (L_k(g))6= 0sinceX_H0

DS,f(Lk(g))⊃X_Wk(g,f).

The remainder of the paper is structured as follows. In Section 2 we set up notation in the case of affine vertex algebras that will be the framework of this note. Section 3 is devoted to the proof of Theorem 1.1. In Section 4, we have compiled some known facts on Slodowy slices, W-algebras and their associated varieties. Theorem 1.3 is proved in this section.

Acknowledgements. — T.A. and A.M. like to warmly thank Shanghai Jiao Tong Uni- versity for its hospitality during their stay in September 2019.

2. Universal affine vertex algebras and associated graded vertex Poisson algebras

Letbgbe the affine Kac-Moody algebra associated withg, that is, bg=g[t, t⁻¹]⊕CK,

where the commutation relations are given by

[x⊗t^m, y⊗tⁿ] = [x, y]⊗t^m+n+m(x|y)δm+n,0K, [K,bg] = 0, forx, y∈gandm, n∈Z. Here,

(|) = 1

2h^∨×Killing form ofg

is the usual normalized inner product. Forx∈gandm∈Z, we shall write x(m)for x⊗t^m.

2.1. Universal affine vertex algebras. — Fork∈C, set V^k(g) =U(bg)⊗_U(g[t]⊕CK)Ck,

where Ck is the one-dimensional representation of g[t]⊕CK on which K acts as multiplication bykandg⊗C[t]acts trivially.

By the Poincaré-Birkhoff-Witt Theorem, the direct sum decomposition, we have (2.1) V^k(g)∼=U(g⊗t⁻¹C[t⁻¹]) =U(t⁻¹g[t⁻¹]).

The spaceV^k(g)is naturally graded, V^k(g) = L

∆∈Z>0

V^k(g)_∆,

where the grading is defined by

deg(xⁱ¹(−n1)· · ·x^ir(−nr)1) =

r

X

i=1

ni, r>0, x^ij ∈g,

with 1the image of 1⊗1 in V^k(g). We have V^k(g)0 =C1, and we identify gwith V^k(g)1via the linear isomorphism defined by x7→x(−1)1.

It is well-known thatV^k(g) has a unique vertex algebra structure such that1 is the vacuum vector,

x(z) :=Y(x⊗t⁻¹, z) =X

n∈Z

x(n)z⁻ⁿ⁻¹, and

[T, x(z)] =∂zx(z)

for x ∈ g, where T is the translation operator. Here, x(n) acts on V^k(g) by left multiplication, and so, one can viewx(n)as an endomorphism of V^k(g). The vertex algebraV^k(g)is called theuniversal affine vertex algebraassociated withgat levelk [FZ92, Zhu96, LL04].

The vertex algebraV^k(g)is a vertex operator algebra, provided that k+h^∨ 6= 0, by theSugawara construction. More specifically, set

S= 1 2

d

X

i=1

xi(−1)xⁱ(−1)1,

where {xi|i= 1, . . . , d}is the dual basis of a basis {xⁱ|i= 1, . . . ,dimg} ofgwith respect to the bilinear form ( | ), with d = dimg. Then for k 6= −h^∨, the vector ω=S/(k+h^∨)is a conformal vector ofV^k(g)with central charge

c(k) = kdimg k+h^∨. Note that, writingω(z) =P

n∈ZLnz⁻ⁿ⁻², we have L₀= 1

2(k+h^∨) ^d

X

i=1

x_i(0)xⁱ(0) +

∞

X

n=1 d

X

i=1

(x_i(−n)xⁱ(n) +xⁱ(−n)xi(n))

,

Ln = 1 2(k+h^∨)

^∞ X

m=1 d

X

i=1

xi(−m)xⁱ(m+n) +

∞

X

m=0 d

X

i=1

xⁱ(−m+n)xi(m)

, ifn6= 0.

Lemma2.1([Kac90]). — We have

[Ln, x(m)] =−mx(m+n), forx∈g, m, n∈Z, andL_n1= 0 forn>−1.

We haveV^k(g)∆={v∈V^k(g)|L0v= ∆v}andT =L−1onV^k(g), provided that k+h^∨6= 0.

Any graded quotient of V^k(g) asbg-module has the structure of a quotient vertex algebra. In particular, the unique simple graded quotient Lk(g)is a vertex algebra, and is called thesimple affine vertex algebra associated withgat level k.

2.2. Associate graded vertex Poisson algebras of affine vertex algebras

It is known by Li [Li05] that any vertex algebra V admits a canonical filtration F^•V, called the Li filtration of V. For a quotientV of V^k(g), F^•V is described as follows. The subspaceF^pV is spanned by the elements

y1(−n1−1)· · ·yr(−nr−1)1 withyi ∈g,ni∈Z>0,n1+· · ·+nr>p. We have

(2.2)

V =F⁰V ⊃F¹V ⊃ · · ·, T

pF^pV = 0, T F^pV ⊂F^p+1V,

a_(n)F^qV ⊂F^p+q−n−1V f or a∈F^pV, n∈Z, a_(n)F^qV ⊂F^p+q−nV f or a∈F^pV, n>0.

Here we have setF^pV =V forp <0.

Let gr^FV = L

pF^pV /F^p+1V be the associated graded vector space. The space gr^FV is a vertex Poisson algebra by

σ_p(a)σ_q(b) =σ_p+q(a₍₋₁₎b), T σp(a) =σp+1(T a), σp(a)_(n)σq(b) =σ_p+q−n(a_(n)b)

for a, b∈ V, n>0, whereσp:F^p(V) →F^pV /F^p+1V is the principal symbol map.

In particular,gr^FV is ag[t]-module by the correspondence (2.3) g[t]3x(n)7−→σ₀(x)_(n)∈End(gr^FV) forx∈g,n>0.

The filtration F^•V is compatible with the grading:F^pV =L

∆∈Z>0F^pV_∆, where

F^pV_∆:=V_∆∩F^pV.

LetU•(t⁻¹g[t⁻¹])be the PBW filtration of U(t⁻¹g[t⁻¹]), that is,Up(t⁻¹g[t⁻¹]) is the subspace of U(t⁻¹g[t⁻¹]) spanned by monomialsy1y2. . . yr with yi ∈ g, r 6p.

Define

GpV =Up(t⁻¹g[t⁻¹])1.

ThenG•V defines an increasing filtration ofV. We have

(2.4) F^pV_∆=G_∆−pG_∆,

where GpV∆ := GpV ∩V∆, see [Ara12a, Prop. 2.6.1]. Therefore, the graded space gr^GV =L

p∈Z_>0GpV /Gp−1V is isomorphic togr^FV. In particular, we have

grV^k(g)∼= grU•(t⁻¹g[t⁻¹])∼=S(t⁻¹g[t⁻¹]).

The action ofg[t]ongrV^k(g) =S(t⁻¹g[t⁻¹])coincides with the one induced from the action ofg[t]ong[t, t⁻¹]/g[t]∼=t⁻¹g[t⁻¹]. More precisely, the elementx(m), forx∈g

andm∈Z>0, acts onS(t⁻¹g[t⁻¹])as follows:

x(m)·1= 0, x(m)·v=

r

X

j=1

X

nj−m>0

y1(−n1)· · ·[x, yj](m−nj)· · ·yr(−nr), (2.5)

ifv=y1(−n1)· · ·yr(−nr)withyi∈g,n1, . . . , nr∈Z>0.

2.3. Zhu’sC₂-algebras and associated varieties of affine vertex algebras We have [Li05, Lem. 2.9]

F^pV = span_C{a_(−i−1)b|a∈V, i>1, b∈F^p−iV} for allp>1. In particular,

F¹V =C₂(V), whereC2(V) = span_C{a₍₋₂₎b|a, b∈V}. Set

RV =V /C2(V) =F⁰V /F¹V ⊂gr^FV.

It is known by Zhu [Zhu96] thatRV is a Poisson algebra. The Poisson algebra structure can be understood as the restriction of the vertex Poisson structure of gr^FV. It is given by

a·b=a₍₋₁₎b, {a, b}=a₍₀₎b, fora, b∈V, where a=a+C2(V).

By definition [Ara12a], theassociated variety ofV is the reduced scheme X_V := Specm(R_V).

It is easily seen that

F¹V^k(g) =C2(V^k(g)) =t⁻²g[t⁻¹]V^k(g).

The following map defines an isomorphism of Poisson algebras C[g^∗]∼=S(g)−→R_Vk(g)

g3x7−→x(−1)1+t⁻²g[t⁻¹]V^k(g).

Therefore,R_Vk(g)∼=C[g^∗]and so,X_Vk(g)∼=g^∗.

More generally, ifV is a quotient ofV^k(g)by some idealN, then we have

(2.6) RV ∼=C[g^∗]/IN

as Poisson algebras, whereI_N is the image of N inR_Vk(g)=C[g^∗]. ThenX_V is just the zero locus ofIN in g^∗. It is a closedG-invariant conic subset ofg^∗.

Identifying g^∗ with g through the bilinear form ( | ), one may view XV as a subvariety ofg.

2.4. PBW basis. — Let ∆+ = {β1, . . . , βq} be the set of positive roots for g with respect to a triangular decomposition g = n−⊕h⊕n+, where q = (d−`)/2 and

`= rk(g).

Form now on, we fix a basis

{uⁱ, eβ_j, fβ_j |i= 1, . . . , `, j= 1, . . . , q}

ofgsuch that{uⁱ|i= 1, . . . , `}is an orthonormal basis ofhwith respect to(|)and (eβ_i|fβ_i) = 1fori= 1,2, . . . , q. In particular,[eβ_i, fβ_i] =βi fori = 1, . . . , q(see, for example, [Hum72, Prop. 8.3]), where h^∗ and hare identified through (| ). One may also assume thatht(βi)6ht(βj)fori < j, whereht(βi)stands for the height of the positive rootβ_i.

We define the structure constantscα,β by [eα, eβ] =cα,βeα+β,

provided that α,β andα+β are in∆. Our convention is thate_−α stands for fα if α∈∆+. Ifα,β andα+β are in∆+, then from the equalities,

c_−α,α+β = (f_β|[fα, e_α+β]) =−(fβ|[eα+β, f_α]) =−([fβ, e_α+β]|fα) =−c−β,α+β, we get that

(2.7) c−α,α+β=−c−β,α+β.

By (2.1), the above basis of g induces a basis of V^k(g) consisted of 1 and the elements of the form

(2.8) z=z⁽⁺⁾z⁽⁻⁾z⁽⁰⁾1,

with

z⁽⁺⁾:=eβ₁(−1)^a1,1· · ·eβ₁(−r1)^a1,r1· · ·eβ_q(−1)^aq,1· · ·eβ_q(−rq)^aq,rq, z⁽⁻⁾:=f_β1(−1)^b1,1· · ·f_β1(−s₁)^b1,s1· · ·f_βq(−1)^bq,1· · ·f_βq(−s_q)^bq,sq,

z⁽⁰⁾:=u¹(−1)^c1,1· · ·u¹(−t1)^c1,t1· · ·u^{`</sup>(−1)<sup>c`,1}· · ·u<sup>`</sup>(−t`)^c`,t`,

where r1, . . . , rq, s1, . . . , sq, , t1, . . . , t` are positive integers, andal,m, bl,n, ci,j, for l= 1, . . . , q, m = 1, . . . , r<sub>l</sub>, n = 1, . . . , s<sub>l</sub>, i = 1, . . . , `, j = 1, . . . , t_i are nonnegative integers such that at least one of them is nonzero.

Definition2.2. — Each element xof V^k(g) is a linear combination of elements in the above PBW basis, each of them will be called aPBW monomial ofx.

Definition2.3. — For a PBW monomialv as in (2.8), we call the integer depth(v) =

q

X

i=1

^ri

X

j=1

ai,j(j−1) +

s_i

X

j=1

bi,j(j−1)

+

`

X

i=1 t_i

X

j=1

ci,j(j−1)

the depth of v. In other words, a PBW monomial v has depth p means that v ∈ F^pV^k(g)andv6∈F^p+1V^k(g). By convention,depth(1) = 0.

For a PBW monomialv as in (2.8), we calldegree ofv the integer deg(v) =

q

X

i=1

^ri

X

j=1

ai,j+

s_i

X

j=1

bi,j

+

`

X

i=1 t_i

X

j=1

ci,j,

In other words,v has degreepmeans thatv∈G_pV^k(g)andv6∈G_p−1V^k(g)since the PBW filtration ofV^k(g)coincides with the standard filtrationG•V^k(g). By conven- tion,deg(1) = 0.

Recall that a singular vector of a g[t]-representation M is a vector m∈ M such that eα(0).m= 0, for all α∈∆+, andfθ(1)·m= 0, whereθ is the highest positive root ofg. From the identity

L₋₁= 1 k+h^∨

^` X

i=1

∞

X

m=0

uⁱ(−1−m)uⁱ(m)

+ X

α∈∆+

∞

X

m=0

(eα(−1−m)fα(m) +fα(−1−m)eα(m))

, we deduce the following easy observation, which will be useful in the proof of the main result.

Lemma2.4. — If w is a singular vector ofV^k(g), then L−1w= 1

k+h^{∨ `}

X

i=1

uⁱ(−1)uⁱ(0) + X

α∈∆+

eα(−1)fα(0)

w.

2.5. Basis of associated graded vertex Poisson algebras. — Note that grV^k(g) = S(t⁻¹g[t⁻¹]) has a basis consisting of1and elements of the form (2.8). Similarly to Definition 2.2, we have the following definition.

Definition2.5. — Each elementxofS(t⁻¹g[t⁻¹])is a linear combination of elements in the above basis, each of them will be called amonomial ofx.

As in the case ofV^k(g), the spaceS(t⁻¹g[t⁻¹])has two natural gradations. The first one is induced from the degree of elements as polynomials. We shall writedeg(v)for the degree of a homogeneous elementv∈S(t⁻¹g[t⁻¹])with respect to this gradation.

The second one is induced from the Li filtration via the isomorphismS(t⁻¹g[t⁻¹])∼= gr^FV^k(g). The degree of a homogeneous elementv∈S(t⁻¹g[t⁻¹])with respect to the gradation induced by Li filtration will be called the depth of v, and will be denoted bydepth(v).

Notice that any elementv of the form (2.8) is homogeneous for both gradations.

By convention,deg(1) = depth(1) = 0.

As a consequence of (2.5), we get that

(2.9) deg(x(m)·v) = deg(v) and depth(x(m)·v) = depth(v)−m, for m > 0, x∈ g, and any homogeneous element v ∈ S(t⁻¹g[t⁻¹]) with respect to both gradations.

In the sequel, we will also use the following notation, forvof the form (2.8), viewed either as an element ofV^k(g)or of S(t⁻¹g[t⁻¹]):

(2.10) deg⁽⁰⁾₋₁(v) :=

`

X

j=1

c_j,1,

which corresponds to the degree of the element obtained from v⁽⁰⁾ by keeping only the terms of depth0, that is, the termsuⁱ(−1),i= 1, . . . , `.

Notice that a nonzero depth-homogeneous element ofS(t⁻¹g[t⁻¹])has depth0 if and only if its image in

R_Vk(g)=V^k(g)/t⁻²g[t⁻¹]V^k(g) is nonzero.

3. Proof of the main result This section is devoted to the proof of Theorem 1.1.

3.1. Strategy. — Let Nk be the maximal graded submodule of V^k(g), so that L_k(g) =V^k(g)/N_k. Our aim is to show that ifV^k(g)is not simple, that is,N_k6={0}, then X_Lk(g) is strictly contained in g^∗ ∼= g, that is, the image Ik := IN_k of Nk in R_Vk(g)=C[g^∗]is nonzero.

Fork=−h^∨, it follows from [FG04] thatIk is the defining ideal of the nilpotent cone N(g) of g, and so X_Lk(g) = N(g) (see [Ara12b] or Section 3.4 below). Hence, there is no loss of generality in assuming thatk+h^∨6= 0.

Henceforth, we suppose that k+h^∨ 6= 0 and that V^k(g) is not simple, that is, N_k 6={0}. Then there exists at least one non-trivial (that is, nonzero and different from 1) singular vectorw inV^k(g). Theorem 1.2 states that the image of win I_k is nonzero, and this proves Theorem 1.1. The rest of this section is devoted to the proof of Theorem 1.2.

Letwbe a nontrivial singular vector ofV^k(g). One can assume thatw∈F^pV^k(g)r F^p+1V^k(g)for some p∈Z>0.

The image

w:=σ(w)

of this singular vector in S(t⁻¹g[t⁻¹])∼= gr^FV^k(g) is a nontrivial singular vector of S(t⁻¹g[t⁻¹]). Here σ: V^k(g) → gr^FV^k(g) stands for the principal symbol map. It follows from (2.9) that one can assume thatw is homogeneous with respect to both gradations on S(t⁻¹g[t⁻¹]). In particular w has depth p. It is enough to show that p= 0, that is,whas depth zero. Write

w=X

j∈J

λjw^j,

whereJ is a finite index set,λj are nonzero scalar for allj∈J, and wj are pairwise distinct PBW monomials of the form (2.8). LetI⊂J be the subset ofi∈J such that

depthwⁱ = p= depthw. Since w ∈ F^pV^k(g)rF^p+1V^k(g), the set I is nonempty.

Here, wⁱ stands for the image ofwⁱ in gr^FV^k(g)∼=S(t⁻¹g[t⁻¹]).

More specifically, for anyj∈I, write

(3.1) w^j= (w^j)⁽⁺⁾(w^j)⁽⁻⁾(w^j)⁽⁰⁾1, with

(w^j)⁽⁺⁾:=e_β1(−1)^a(j)1,1· · ·e_β1(−r₁)^a(j)1,r1· · ·e_βq(−1)^a(j)q,1· · ·e_βq(−r_q)^a(j)q,rq (w^j)⁽⁻⁾:=f_β1(−1)^b(j)1,1· · ·f_β1(−s₁)^b(j)1,s1· · ·f_βq(−1)^b(j)q,1· · ·f_βq(−s_q)^b(j)q,sq, (w^j)⁽⁰⁾:=u¹(−1)^c(j)1,1· · ·u¹(−t1)^c

(j)

1,t1· · ·u^{`</sup>(−1)<sup>c(j)`,1}· · ·u<sup>`</sup>(−t`)^c

(j)

`,t`,

wherer₁, . . . , r_q, s₁, . . . , s_q, , t₁, . . . , t<sub>`</sub> are nonnegative integers, anda<sup>(j)</sup><sub>l,m</sub>, b<sup>(j)</sup><sub>l,n</sub>, c<sup>(j)</sup><sub>i,p</sub>, for l = 1, . . . , q, m= 1, . . . , rl, n= 1, . . . , sl, i= 1, . . . , `, p= 1, . . . , ti, are nonnegative integers such that at least one of them is nonzero.

The integers rl’s, for l = 1, . . . , q, are chosen so that at least one of the a^(j)_l,r

l’s is nonzero for j running through J if for some j ∈J, (w^j)⁽⁺⁾6= 1. Otherwise, we just set (w^j)⁽⁺⁾:= 1. Similarly are defined the integerssl’s andtm’s, forl = 1, . . . , q and m= 1, . . . , `. By our assumption, note that for alli∈I,

q

X

n=1

^rn

X

l=1

a⁽ⁱ⁾_n,l+

sn

X

l=1

b⁽ⁱ⁾_n,l

+

`

X

n=1 tn

X

l=1

c⁽ⁱ⁾_n,l= deg(w)

q

X

n=1

^rn

X

l=1

a⁽ⁱ⁾_n,l(l−1) +

s_n

X

l=1

b⁽ⁱ⁾_n,l(l−1)

+

`

X

n=1 t_n

X

l=1

c⁽ⁱ⁾_n,l(l−1) = depth(w) =p.

3.2. A technical lemma. — In this paragraph we remain in the commutative setting, and we only deal with w∈S(t⁻¹g[t⁻¹])and its monomialswⁱ’s, fori∈I.

Recall from (2.10) that,

deg⁽⁰⁾₋₁(wⁱ) =

`

X

j=1

c⁽ⁱ⁾_j,1 fori∈I. Set

d⁽⁰⁾₋₁(I) := max{deg⁽⁰⁾₋₁(wⁱ)|i∈I}, and

I₋₁⁽⁰⁾:={i∈I|deg⁽⁰⁾₋₁(wⁱ) =d⁽⁰⁾₋₁(I)}.

If(wⁱ)⁽⁰⁾= 1for alli∈I, we just setd⁽⁰⁾₋₁(I) = 0 and thenI₋₁⁽⁰⁾ =I.

Lemma3.1. — If i ∈ I₋₁⁽⁰⁾, then (wⁱ)⁽⁻⁾ = 1. In other words, for i ∈ I₋₁⁽⁰⁾, we have wⁱ= (wⁱ)⁽⁰⁾(wⁱ)⁽⁺⁾1.

Proof. — Suppose the assertion is false. Then for some positive rootsβj₁, . . . , βj_t ∈

∆+, one can write for any i∈I₋₁⁽⁰⁾,

(3.2) (wⁱ)⁽⁻⁾=fβj1(−1)^b(i)j1,1· · ·fβj1(−sj1)^b

(i)

j1,sj1· · ·fβ_jt(−1)^b(i)jt,1· · ·fβ_jt(−sjt)^b

(i) jt,sjt,

so that for anyl∈ {1, . . . , t}, {b⁽ⁱ⁾_jl,s

jl |i∈I₋₁⁽⁰⁾} 6={0}.

Set

K₋₁⁽⁰⁾={i∈I₋₁⁽⁰⁾|b⁽ⁱ⁾_j

1,sj1 >0}.

Sincewis a singular vector ofS(t⁻¹g[t⁻¹])and s_j1−1∈Z>0, we have eβ_j1(sj₁−1)·w= 0.

On the other hand, using the action ofg[t]onS(t⁻¹g[t⁻¹])as described by (2.5), we see that

(3.3) 0 =eβ_j1(sj₁−1)·w= X

i∈K₋₁⁽⁰⁾

λib⁽ⁱ⁾_j

1,s_j1vⁱ+v, where fori∈K₋₁⁽⁰⁾,

vⁱ:= (wⁱ)⁽⁰⁾β_j1(−1)f_βj

1(−1)^b(i)j1,1· · ·f_βj

1(−s_j1)^b

(i) j1,sj1

−1

· · ·f_βjt(−1)^b(i)jt,1· · ·f_βjt(−sjt)^b

(i)

jt,sjt(wⁱ)⁽⁺⁾1, andv is a linear combination of monomialsxsuch that

deg⁽⁰⁾₋₁(x)6d⁽⁰⁾₋₁(I).

Indeed, fori∈K₋₁⁽⁰⁾, it is clear that

eβ_j1(sj₁−1)·wⁱ=b⁽ⁱ⁾_j

1,s_j1vⁱ+yⁱ,

whereyⁱis a linear combination of monomialsysuch thatdeg⁽⁰⁾₋₁(y)6d⁽⁰⁾₋₁(I)because ht(βj₁)6ht(βj_l)for alll∈ {1, . . . , t}. Next, fori∈I₋₁⁽⁰⁾rK₋₁⁽⁰⁾,eβ_j1(sj₁−1)·wⁱis a linear combination of monomials z such thatdeg⁽⁰⁾₋₁(z)6d⁽⁰⁾₋₁(I)becauseb⁽ⁱ⁾_j

1,s_j1 = 0.

Finally, fori∈IrI₋₁⁽⁰⁾, we havedeg⁽⁰⁾₋₁(wⁱ)< d⁽⁰⁾₋₁(I)and, hence,eβ_j1(sj₁−1)·wⁱ is a linear combination of monomialsz such thatdeg⁽⁰⁾₋₁(z)6d⁽⁰⁾₋₁(I)as well.

Now, note that for eachi∈K₋₁⁽⁰⁾,

deg⁽⁰⁾₋₁(vⁱ) = deg⁽⁰⁾₋₁(wⁱ) + 1 =d⁽⁰⁾₋₁(I) + 1.

Hence by (3.3) we get a contradiction because all monomialsvⁱ, forirunning through K₋₁⁽⁰⁾, are linearly independent while λib⁽ⁱ⁾_j

1,s_j1 6= 0, for i ∈ K₋₁⁽⁰⁾. This concludes the

proof of the lemma.

3.3. Use of Sugawara operators. — Recall that w = P

j∈Jλjw^j. Let J1 ⊆ J be such that fori∈J₁, (wⁱ)⁽⁻⁾= 1. Then by Lemma 3.1,

∅6=I₋₁⁽⁰⁾⊆J1. SoJ16=∅. Set

d⁽⁰⁾₋₁:=d⁽⁰⁾₋₁(J1) = max{deg⁽⁰⁾₋₁(wⁱ)|i∈J1},

and

J₋₁⁽⁰⁾:={i∈J₁|deg⁽⁰⁾₋₁(wⁱ) =d⁽⁰⁾₋₁}.

Thend⁽⁰⁾₋₁(I)6d⁽⁰⁾₋₁. Set

d⁺:= max{deg(wⁱ)⁽⁺⁾|i∈J₋₁⁽⁰⁾} and let

J⁺={i∈J₋₁⁽⁰⁾|deg(wⁱ)⁽⁺⁾=d⁺} ⊆J₋₁⁽⁰⁾.

Our next aim is to show that for i ∈J⁺, wⁱ has depth zero, whencep= 0 sincep is by definition the smallest depth of the w^j’s, and so the image of w in R_Vk(g) = F⁰V^k(g)/F¹V^k(g)is nonzero.

This will be achieved in this paragraph through the use of the Sugawara construc- tion.

Recall that by Lemma 2.4,

L₋₁w=Le₋₁w sincewis a singular vector ofV^k(g), where

Le−1:= 1 k+h^∨

^` X

i=1

uⁱ(−1)uⁱ(0) + X

α∈∆+

eα(−1)fα(0)

.

Lemma3.2. — Let z be a PBW monomial of the form (2.8). ThenLe₋₁z is a linear combination of PBW monomials xsatisfying all the following conditions:

(a) deg(x⁽⁺⁾)6deg(z⁽⁺⁾) + 1anddeg(x⁽⁰⁾)6deg(z⁽⁰⁾) + 1, (b) if z⁽⁻⁾6= 1, thenx⁽⁻⁾6= 1.

(c) ifx⁽⁻⁾=z⁽⁻⁾, then either deg(x⁽⁰⁾) = deg(z⁽⁰⁾) + 1, orx⁽⁰⁾=z⁽⁰⁾. (d) if deg(x⁽⁰⁾) = deg(z⁽⁰⁾) + 1, thenx⁽⁻⁾=z⁽⁻⁾ anddeg(x⁽⁺⁾)6deg(z⁽⁺⁾).

Proof. — Parts (a)–(c) are easy to see. We only prove (d). Assume thatdeg(x⁽⁰⁾) = deg(z⁽⁰⁾) + 1. Either xcomes from the termP`

i=1uⁱ(−1)uⁱ(0)z, or it comes from a termeα(−1)fα(0)z for someα∈∆+.

Ifx comes from the term P`

i=1uⁱ(−1)uⁱ(0)z, then it is obvious that x⁽⁻⁾=z⁽⁻⁾ andx⁽⁺⁾=z⁽⁺⁾.

Assume thatxcomes from eα(−1)fα(0)zfor some α∈∆+. We have e_α(−1)f_α(0)z=e_α(−1)[f_α(0), z⁽⁺⁾]z⁽⁻⁾z⁽⁰⁾1+e_α(−1)z⁽⁺⁾[f_α(0), z⁽⁻⁾]z⁽⁰⁾1

+eα(−1)z⁽⁺⁾z⁽⁻⁾[fα(0), z⁽⁰⁾]1.

Clearly, any PBW monomialsxfrom

e_α(−1)z⁽⁺⁾[f_α(0), z⁽⁻⁾]z⁽⁰⁾1 or e_α(−1)z⁽⁺⁾z⁽⁻⁾[f_α(0), z⁽⁰⁾]1

satisfies thatdeg(x⁽⁰⁾)6deg(z⁽⁰⁾). Then it is enough to consider PBW monomials in eα(−1)[fα(0), z⁽⁺⁾]z⁽⁻⁾z⁽⁰⁾1.

The only possibility for a PBW monomialxineα(−1)[fα(0), z⁽⁺⁾]z⁽⁻⁾z⁽⁰⁾1to satisfy deg(x⁽⁰⁾) = deg(z⁽⁰⁾) + 1is that it comes from a term[fα(0), eα(−n)] =−α(−n)for

some n∈Z>0, whereeα(−n)is a term inz⁽⁺⁾. But then, for PBW monomialsxin eα(−1)[fα(0), z⁽⁺⁾]z⁽⁰⁾1such thatdeg(x⁽⁰⁾) = deg(z⁽⁰⁾) + 1, we havex⁽⁻⁾=z⁽⁻⁾and

deg(x⁽⁺⁾)6deg(z⁽⁺⁾).

We now consider the action ofLe−1 on particular PBW monomials.

Lemma3.3. — Let z be a PBW monomial of the form (2.8)such thatz⁽⁻⁾= 1 and depth(z⁽⁺⁾) = 0, that is, either z⁽⁺⁾ = 1, or for some j₁, . . . , j_t ∈ {1, . . . , q} (with possible repetitions),

z=eβ_j1(−1)eβ_j2(−1)· · ·eβ_jt(−1)z⁽⁰⁾1.

Then Le₋₁z is a linear combination of PBW monomials y satisfying one of the fol- lowing conditions:

(1)y⁽⁻⁾= 1,depth(y⁽⁺⁾)>1,deg(y⁽⁺⁾)6deg(z⁽⁺⁾),y⁽⁰⁾=z⁽⁰⁾,

(2)y⁽⁻⁾= 1,depth(y⁽⁺⁾) = 0,deg(y⁽⁺⁾)6deg(z⁽⁺⁾)−1, anddeg(y⁽⁰⁾)>deg(z⁽⁰⁾), deg⁽⁰⁾₋₁(y) = deg⁽⁰⁾₋₁(z),

(3)y⁽⁻⁾= 1,depth(y⁽⁺⁾)>1,deg(y⁽⁺⁾)6deg(z⁽⁺⁾)−1, anddeg⁽⁰⁾₋₁(y) = deg⁽⁰⁾₋₁(z)+1, (4)y⁽⁻⁾6= 1.

Proof. — First, we have

`

X

i=1

uⁱ(−1)uⁱ(0)z=

t

X

r=1

eβ_j1(−1)· · · ^`

X

i=1

uⁱ(−1)uⁱ(0), eβ_jr(−1)

· · ·eβ_jt(−1)z⁽⁰⁾1, and

`

X

i=1

uⁱ(−1)uⁱ(0), eβ_jr(−1) =

`

X

i=1

uⁱ(−1)[uⁱ(0), eβ_jr(−1)] + [uⁱ(−1), eβ_jr(−1)]uⁱ(0)

=βj_r(−1)eβ_jr(−1) +eβ_jr(−2)βj_r(0).

So (3.4)

`

X

i=1

uⁱ(−1)uⁱ(0)z

=

t

X

r=1

e_βj

1(−1)· · ·(β_jr(−1)e_βjr(−1) +e_βjr(−2)β_jr(0))· · ·e_βjt(−1)z⁽⁰⁾1.

Second, we have X

α∈∆₊

eα(−1)fα(0)z= X

α∈∆₊ t

X

r=1

eα(−1)eβ_j1(−1)· · ·[fα(0), eβ_jr(−1)]· · ·eβ_jt(−1)z⁽⁰⁾1

+ X

α∈∆+

eα(−1)eβ_j1(−1)eβ_j2(−1)· · ·eβ_jt(−1)[fα(0), z⁽⁰⁾]1.

It is clear that any PBW monomialyin X

α∈∆+

e_α(−1)eβj1(−1)eβj2(−1)· · ·e_βjt(−1)[fα(0), z⁽⁰⁾]1 satisfies

(3.5) y⁽⁻⁾6= 1.

We now consider ur:= X

α∈∆+

eα(−1)eβ_j1(−1)· · ·[fα(0), eβ_jr(−1)]· · ·eβ_jt(−1)z⁽⁰⁾1, for16r6t.

– Ifβjr =α+βfor someα, β∈∆+, then there is a partial sum of two terms inur: c_−α,α+βe_α(−1)e_βj

1(−1)· · ·e_β(−1)· · ·e_βjt(−1)z⁽⁰⁾1

+c_−β,α+βeβ(−1)eβ_j1(−1)· · ·eα(−1)· · ·eβ_jt(−1)z⁽⁰⁾1.

Rewriting the above sum to a linear combination of PBW monomials, and noticing that

c_−α,α+βeα(−1)eβ(−1) +c_−β,α+βeβ(−1)eα(−1) =c_−α,α+βcα,βeα+β(−2), due to (2.7), we deduce that it is a linear combination of PBW monomialsysuch that (3.6) y⁽⁻⁾=z⁽⁻⁾= 1, y⁽⁰⁾=z⁽⁰⁾, depth(y⁽⁺⁾)>1, deg(y⁽⁺⁾)6deg(z⁽⁺⁾), wherec_−α,α+β, c_−β,α+β, c_α,β∈R^∗.

– Ifα−βj_r ∈∆+ for someα∈∆+, then there is a term inur: (3.7) c_−α,βjre_α(−1)e_βj

1(−1)· · ·e_βjr−1(−1)f_α−βjr(−1)e_βjr+1(−1)· · ·e_βjt(−1)z⁽⁰⁾1.

It is easy to see that (3.7) is a linear combination of PBW monomialsy such that y satisfies one of the following:

y⁽⁻⁾= 1, depth(y⁽⁺⁾)>1, deg(y⁽⁺⁾)6deg(z⁽⁺⁾), y⁽⁰⁾ =z⁽⁰⁾, (3.8)

y⁽⁻⁾= 1, depth(y⁽⁺⁾) = 0, deg(y⁽⁺⁾)6deg(z⁽⁺⁾)−1, (3.9)

deg(y⁽⁰⁾)>deg(z⁽⁰⁾), deg⁽⁰⁾₋₁(y) = deg⁽⁰⁾₋₁(z), y⁽⁻⁾6= 1.

(3.10)

Notice also that withα=β_jr, there is a term inu_r:

−eβ_jr(−1)eβj1(−1)· · ·e_βjr−1(−1)βjr(−1)eβ_jr+1(−1)· · ·e_βjt(−1)z⁽⁰⁾1.

Together with (3.4), we see that

`

X

i=1

uⁱ(−1)uⁱ(0)z+

t

X

r=1

eβ_jr(−1)eβj1(−1)· · ·[fβ_jr(0), eβ_jr(−1)]· · · ·eβ_jt(−1)z⁽⁰⁾1

=

t

X

r=1

e_βj

1(−1)· · ·(β_jr(−1)e_βjr(−1) +e_βjr(−2)β_jr(0))· · ·e_βjt(−1)z⁽⁰⁾1

−

t

X

r=1 r−1

X

s=1

eβ_j1(−1)· · ·[eβ_jr(−1), eβ_js(−1)]

· · ·eβ_jr−1(−1)βj_r(−1)eβ_jr+1(−1)· · ·eβ_jt(−1)z⁽⁰⁾1

−

t

X

r=1

eβj1(−1)· · ·eβ_jr−1(−1)eβ_jr(−1)βjr(−1)eβ_jr+1(−1)· · ·eβ_jt(−1)z⁽⁰⁾1

is a linear combination of PBW monomialsy satisfying one of the following:

y⁽⁻⁾= 1, depth(y⁽⁺⁾)>1,deg(y⁽⁺⁾)6deg(z⁽⁺⁾), y⁽⁰⁾=z⁽⁰⁾, (3.11)

y⁽⁻⁾= 1, depth(y⁽⁺⁾)>1,deg(y⁽⁺⁾)6deg(z⁽⁺⁾)−1, (3.12)

deg⁽⁰⁾₋₁(y) = deg⁽⁰⁾₋₁(z) + 1.

Then the lemma follows from (3.5), (3.6), (3.8)–(3.12).

Lemma3.4. — Letzbe a PBW monomial of the form (2.8)such thatz⁽⁻⁾= 1. Then Le−1z=cz⁽⁺⁾(γ−

q

X

j=1

aj,1βj)(−1)z⁽⁰⁾+y¹, wherecis a nonzero constant,γ=Pq

j=1

Prj

s=1aj,sβj, andy¹ is a linear combination of PBW monomialsy such that

deg⁽⁰⁾₋₁(y) = deg⁽⁰⁾₋₁(z) + 1, deg(y⁽⁺⁾)6deg(z⁽⁺⁾)−1, or

deg⁽⁰⁾₋₁(y)6deg⁽⁰⁾₋₁(z).

Proof. — Since the proof is similar to that of Lemma 3.3, we left the verification to

the reader.

Lemma3.5. — Fori∈J⁺, we have thatdepth((wⁱ)⁽⁺⁾) = 0.

Proof. — First we have

w= X

j∈J⁺

λ_jw^j+ X

j∈J₋₁⁽⁰⁾rJ⁺

λ_jw^j+ X

j∈J₁rJ₋₁⁽⁰⁾

λ_jw^j+ X

j∈JrJ₁

λ_jw^j.

Then by Lemma 3.2(b) and Lemma 3.4, we have (k+h^∨)eL₋₁w= X

i∈J⁺

(wⁱ)⁽⁺⁾

γ_i−

q

X

j=1

a⁽ⁱ⁾_j,1β_j

(−1)(wⁱ)⁽⁰⁾

+ X

i∈J1rJ⁺

(wⁱ)⁽⁺⁾

γi−

q

X

j=1

a⁽ⁱ⁾_j1βj

(−1)(wⁱ)⁽⁰⁾+y¹,

where γi = Pq j=1

P^r

(i) j

s=1a⁽ⁱ⁾_j,sβi, for i ∈ J1, and y¹ is a linear combination of PBW monomialsy satisfying one of the following conditions:

deg⁽⁰⁾₋₁(y) =d⁽⁰⁾₋₁+ 1, deg(y⁽⁺⁾)6d⁺−1, deg⁽⁰⁾₋₁(y)6d⁽⁰⁾₋₁,

y⁽⁻⁾6= 1.

On the other hand, by Lemma 2.4

L−1w=Le−1w.