Associated varieties of simple affine VOAs $L_k(sl_3)$ and $W$-algebras $W_k(sl_3,f)$
Cuipo Jiang Jingtian Song


















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Associated varieties of simple affine VOAs L_k(sl_3) and W-algebras W_k(sl_3,f)  [Supported by China NSF grants No.12171312.]







	 Cuipo Jiang
		and Jingtian Song

	School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China






 
§ ABSTRACT  
In this paper we first prove that the maximal ideal of the universal affine vertex operator algebra V^k(sl_n)  for k=-n+n-1/q  is generated by two singular vectors of conformal weight 3q if n=3, and by one singular vector of conformal weight 2q if n≥ 4.  We next determine the associated varieties of the simple vertex operator algebras L_k(sl_3) for all the non-admissible levels k=-3+2/2m+1, m≥ 0.  
The varieties of the associated simple affine W-algebras W_k(sl_3,f),  for  nilpotent elements f of sl_3,  are also determined. 
 






§ INTRODUCTION

Let V be a vertex algebra. Recall from <cit.> that the 
 Zhu's C_2-algebra  of V is by definition the quotient space
R_V=V/C_2(V) of V, 
where C_2(V)=span_{a_-2b| a,b∈ V}. 
Then R_V has the Poisson algebra structure given by

    a̅·b̅=a_-1b,    {a̅,b̅}=a_0b,

for a,b ∈ V, where   a̅ := a+C_2(V) is the image of a in R_V.
The associated variety X_V of V is the reduced scheme
X_V=Specm(R_V)
of R_V <cit.>.
This fundamental   invariant of V has been extensively studied in <cit.>.

Let  be a complex simple  finite-dimensional  Lie algebra and V^k() 
the universal affine vertex algebra  at level k ∈. By PBW Theorem, R_V^k() is isomorphic to the polynomial algebra [^*] over ^*. 
Thus the variety X_V  is  the affine space ^* with Kirillov-Kostant Poisson structure.
For a graded quotient V
of V^k(), the variety X_V is a Poisson subscheme of ^*
which is G-invariant and conic, where G is the adjoint group of .

Denote by L_k() the simple quotient of V^k().  
Let h be the Coxeter number of , h^∨ the dual Coxeter number, and r^∨ the lacing number of .
It has been proved in <cit.> that 
L_k()=V^k() if and only if X_L_k() = ^*.   In <cit.>, Gorelik and Kac proved 
that V^k()
is not simple if and only if

    r^∨(k+h^∨) ∈_≥ 0\{1/m| m∈_≥ 1}.

Therefore,

    X_L_k()⊊^*(<ref>) holds.

It is known <cit.> that X_L_k()={0}
 if and only if  k is a non-negative integer. 
For  an admissible   L_k() <cit.>,
or equivalently, for an admissible number k which by definition satisfies 

    k+h^∨=p/q,   p,q∈_≥ 1,  (p,q)=1,p≥h^∨   if (r^∨,q)=1,
    
    h   if (r^∨,q) 1,

the variety X_L_k() is the closure of some nilpotent orbit in . In particular,  if k is a non-degenerate admissible number, that is,
 
    q≥h    if (r^∨,q)=1,
    
     r^∨h^∨   if (r^∨,q)=r^∨,

 then  X_L_k() is the nilpotent cone 𝒩() <cit.>.
 If k=-h^∨, 
 it follows from <cit.> that 
 X_L_k() is also the nilpotent cone.
 It was observed in  <cit.>
 that
there are   
 cases when L_k() is
non-admissible 
and
X_L_k() is the closure of some nilpotent orbit,


 which provide examples for the conjecture in physics <cit.> 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.  
 There are also cases  where X_L_k()
 is neither ^* nor contained in the nilpotent cone 𝒩().
 It was  proved in <cit.>  that X_L_-1(sl_n), n≥ 4, X_L_-m(sl_2m), m≥ 2, X_L_2-r(so_2r), r∈ 2_++1,  are closures of  sheets. In <cit.>, it was shown that the associated variety of a simple affine vertex algebras is contained in the closure of the Dixmier sheet when a chiralization of generalized Grothendieck's simultaneous resolution exists. 
 The variety  X_L_-5/2(sl_4) of L_-5/2(sl_4) was given in <cit.>, which turns out to be the closure of a Jordan class of sl_4. 
 
 In general,
the problem of determining X_L_k() is widely open. Up to now, the  variety X_L_k() has been completely determined for all k∈ only when =sl_2. 
In this paper, we will determine X_L_k(sl_3) for all  non-admissible  numbers k=-3+2/2m+1, m≥ 0. Then together with the known results X_L_k()
for the critical case and admissible cases given in <cit.>, we will obtain a complete characterization of X_L_k(sl_3) for all k∈.  

Another question we are concerned about in this paper is the characterization of the maximal ideal of V^k(sl_n), where k=-n+n-1/q, n≥ 3, q≥ 2, (n-1, q)=1. 
 If q=1, that is,  k=-1, the maximal ideal of V^-1(sl_3) is generated by two singular vectors of conformal weight 3 <cit.>, and the maximal ideal of V^-1(sl_n) is generated by one singular vector of conformal weight 2 <cit.>.
By  using the character formulas given in <cit.>, and the  results for the k=-1 case in <cit.>, we prove our first main result  as follows (also see Theorem <ref>).
  
	Let k=-n+n-1/q with n≥ 3, q≥ 2, (n-1,q)=1. Then 	

    R chL_k(sl_n)=
    	∑_w∈ W∑_γ∈ Q 
    (γ|Λ̅_n-1)≥ 0, (γ|Λ̅_1))≥ 0(-1)^l(w)e^wt_qγ(kΛ_0+≀)-≀.

	Furthermore, we have 
	
		
  * If k=-3+2/2m+1 with m≥ 1, then  the maximal ideal of V^k(sl_3) is generated by  two singular vectors v^1 and v^2 of weights kΛ_0-3(2m+1)δ+2α_1+α_2 and kΛ_0-3(2m+1)δ+α_1+2α_2, respectively. 
		
		
  * If k=-n+n-1/q with n≥ 4, q≥ 2, and (q,n-1)=1, then the maximal ideal of V^k(sl_n) is generated by one singular vector w^1 of weight kΛ_0-2qδ+_1+2_2+⋯+2_n-2+_n-1. 
	


Let θ be the highest root of . 
Our second main result is (also see Theorem <ref>):

	The variety X_L_-1(sl_3) is the closure of the Dixmier sheet   𝕊_l_1=G.ℂ^*λ, i.e.,
	
    X_L_-1(sl_3)=𝕊_ l_1= G.ℂ^*λ=G.ℂ^*λ∪𝒪_min,

	where l_1=h⊕ e_θ⊕ f_θ, λ=h_1-h_2, and 𝒪_min is the minimal non-zero nilpotent orbit of sl_3(ℂ).  In particular, X_L_-1(sl_3) is irreducible and 
	X_L_-1(sl_3)=5.

For k=-3+2/2m+1, m≥ 1, we have our third main result as follows (see also Theorem <ref>).

For k=-3+2/2m+1, m≥ 1, the variety
	
    X_L_k(sl_3)= G.(ℂ^*(h_1-h_2)+f_θ).

	

The proof of  Theorem <ref> is much more complicated than the k=-1 case. The main difficulty is that although we could prove that the maximal ideal of  L_k(sl_3) is generated by two singular vectors v^1 and v^2 of conformal weight 3(2m+1) by Theorem <ref>, it is almost impossible to write these  singular vectors out  as sums of PBW terms for general k=-3+2/2m+1, m≥ 1.  This means that we could not use their structure, as what we do with the  k=-1 case, to obtain the zero locus of the  graded ideal ℐ_k of the C_2-algebra R_V^k(sl_3) determined by these two singular vectors. Our main idea is to analyze coefficients of some critical leading terms.  Then  further prove that 
X_L_k(sl_3)= G.(ℂ^*(h_1-h_2)+f_θ).  Our method here provides a possible way to determine varieties of a simple affine vertex operator algebra without knowing the exact structure of its singular vectors.  We will study the associated varieties of the simple vertex operator algebras V^k(sl_n) for n≥ 4 and general non-admissible numbers k in subsequent papers.

Representations of L_k(sl_3) have been extensively studied in <cit.>, etc. 
We here have the following conjecture for L_k(sl_3) (also see Conjecture <ref>).
 Let k=-3+2/2m+1 with m≥ 0. Then

    {L(tΛ̅_1-2i/3Λ̅_2), L(tΛ̅_2-2i/3Λ̅_1),  L(Λ̅_1-(t+2i/3+1)Λ̅_2),   t∈, i=0,1,⋯, 2m}

provides a complete list of irreducible A(L_k(sl_3))-modules in the category 𝒪.


When m=0, the conjecture is true by <cit.>.  When m=1, we could verify the conjecture by computer programming.

Let f be a nilpotent element of =sl_3. Then f is  minimal or regular.  For k∈, let W^k(sl_3,f) be the universal affine W-vertex algebra associated to f <cit.>. Denote by W_k(sl_3,f) the simple quotient of W^k(sl_3,f). 

Let f be a minimal nilpotent element of =sl_3.  If k=-3+2m+1/2, m≥ 1, it was shown in <cit.> that X_W_k(sl_3,f)={0} and W_k(sl_3,f) is rational. If k=-1, it has been proved in <cit.> that W_-1(sl_3,f) is isomorphic to the rank one Heisenberg vertex operator algebra. So X_W_-1(sl_3,f) is one-dimensional. 
We have the following result for k=-3+2/2m+1, m≥ 1  (also see Theorem <ref>).

	Let k=-3+2/2m+1, m≥ 1,  and f a minimal nilpotent element of sl_3. Then 

    X_W_k(sl_3,f)={[[          a          b 3(μ^2-a^2);          0       -2aμ          c;          1          0          a ]]|  μ, a,b,c∈,  bc=2(a-μ)(2a+μ)^2 }.

	In particular,  X_W_k(sl_3,f)=3. Furthermore,  W_k(sl_3, f) is not quasi-lisse. 

Let f be a regular nilpotent element of =sl_3. If k=-3+p/q, p,q≥ 3, (p,q)=1, by <cit.>,  X_W_k(sl_3)={0} and W_k(sl_3,f) is rational. Our fifth main result is given as follows (also see Theorem <ref>).
 
	Let k+3=2/2m+1 or 2m+1/2,  m≥ 1,  and f a regular nilpotent element of sl_3. Then
	
    X_W_k(sl_3)={[[      0 3/4μ^2 1/2μ^3;      2      0 3/4μ^2;      0      2      0 ]]|  μ∈}.

	In particular, X_W_k(sl_3)=1. Furthermore, W_k(sl_3,f) is not quasi-lisse.

The rest of this paper is organized as follows. In Section 2, we recollect some concepts and results that will be needed later.  In Section 3, we study singular vectors and maximal ideals of V^k(sl_n) for k=-n+n-1/q with n≥ 3, q≥ 1, (n-1,q)=1. Section 4 is dedicated  to determine the associated varieties of L_k(sl_3) for k=-3+2/2m+1, m≥ 0. In Section 5, we obtain the associated varieties of  the simple affine W-algebras W_k(sl_3,f). 



 §.§ Acknowledgements
 We would like to appreciate Tomoyuki Arakawa, Dražen Adamović, and Anne Moreau for valuable  discussions and comments. In particular, the first author is grateful to Dražen Adamović for showing  her the  very helpful result of Theorem 8.2 in  his  paper <cit.> joint with Pierluigi Möseneder Frajria and Paolo Papi, and to Tomoyuki Arakawara for telling her  the papers  <cit.> and <cit.>,  and for the discussions.
 



§ PRELIMINARIES




 §.§ C_2-algebras and associated varieties

Let V be a vertex algebra over ℂ <cit.>.  Recall that a subset S of V is called a strongly generating set of V if V is linearly spanned by elements of the form:

    {x^1_-m_1⋯ x^s_-m_s 1| x^i∈ S,   m_i∈ℤ_+, s≥ 0,  1≤ i≤ s}.

Furthermore, if S is a finite set, then V is called  finitely strongly generated by S.  

For a vertex algebra V,  define C_2(V)=span_{a_-2b| a,b∈ V} and

    R_V= V/C_2(V).

Then R_V is a Poisson algebra <cit.> with relations:

    a̅·b̅ = a_-1 b,     {a̅, b̅}= a_0b,

for a, b ∈ V, where a̅ = a+C_2(V). It is easy to see that V is finitely strongly generated 
if and only if R_V is finitely generated.  V is called C_2-cofinite if R_V is finite-dimensional <cit.>.

Let V be a finitely strongly generated vertex algebra. By definition <cit.>, the
associated variety of V
is
the reduced scheme 
    X_V=  Specm(R_V).


Recall from <cit.> that V is called lisse if X_V={0}. We have the following result.

	Let V be a finitely strongly generated vertex algebra. Then V is lisse if and only if V is C_2-cofinite.
	

 Since  R_V is a finitely generated Poisson algebra,  the variety X_V is a union of symplectic leaves <cit.>. 
 
 	Let V be a vertex algebra. Then V is 
 	called quasi-lisse if X_V has finitely many symplectic leaves.
	
 
 Let V be a finitely strongly generated vertex algebra. If V is quasi-lisse, then V has finitely many ordinary modules. 
 



 §.§ Affine vertex algebras

Let  be a finite-dimensional simple Lie algebra over ℂ with the following normalized bilinear form:

    (·|·)=1/2h^∨× Killing form of .

Let
=[t,t^-1]⊕ K be the associated affine Kac-Moody algebra 
 with the commutation relations:

    [x⊗ t^m,y⊗ t^n]=[x,y]⊗ t^m+n+m(x|y)δ_m+n,0K,  
    [K,]=0,

where x,y∈ and m,n∈. 
For x ∈ and m ∈, we will write x(m) for x
⊗ t^m.


























For k ∈, set

    V^k()=U()⊗ _U( [t]⊕ K)_k,

where _k is the one-dimensional representation of [t]⊕ K
on which K acts as multiplication by k and [t] acts trivially.
By PBW

Theorem, 
we have

    V^k()≅ U(⊗ t^-1[t^-1]) = U(t^-1[t^-1]).




The space V^k() is naturally graded,

    V^k() =⊕_n∈_≥ 0V^k() _n,

where the grading is defined by

    (x^i_1(-n_1)… x^i_r(-n_r)  1) =  ∑_i=1^r n_i,
       r ≥ 0,   x^i_j∈,

with 1 the image of 1⊗ 1 in V^k().
We have V^k()_0= 1,
and we  may identify  with V^k()_1 via the linear isomorphism
defined by x↦ x(-1) 1.


It is well-known that V^k() has a unique vertex algebra structure
such that 1 is the vacuum vector,

    x(z)= Y(x⊗ t^-1,z)  =∑_n ∈ x(n) z^-n-1,

and

    [T,x(z)]=∂_z x(z)

for x ∈,
where T is the translation operator.
Notice that x(n) acts on V^k() by left multiplication. Then  one can view x(n) as an endomorphism of V^k(). 
The vertex algebra
V^k() is called the universal affine vertex algebra
associated with  at level k <cit.>.


If k+h^∨≠0, the vertex algebra
V^k() is a vertex operator algebra by the Sugawara construction.
More specifically, set

    S=1/2∑_i=1^d
    x_i(-1) x^i(-1)   1,

where  d =, and {x_i i=1,…,d} is the dual
basis of a basis {x^i i=1,…,} of 
with respect to the bilinear form (·|·).



Then the vector
ω=S/k+h^∨
is a conformal vector of V^k() 
with central charge

    c(k)=k /k+h^∨.


Any
graded quotient of V^k()
as -module
has the structure of a quotient vertex algebra.
In particular,
the  unique simple graded quotient L_k()
is a vertex algebra,
and is called the simple affine vertex algebra associated with  at level k. 



 §.§ Zhu's algebra of affine vertex algebras


For a _≥ 0-graded vertex algebra V=⊕_n≥ 0V_n, let O(V) be the subspace of V linearly spanned by elements

    u∘ v=∑_j=0^∞nj u_j-2v,

for u∈ V_n, n∈_≥ 0, v∈ V. The Zhu's algebra <cit.>  

    A(V)=V/O(V)

is a unital associative algebra with the multiplication induced from

    u*v=∑_j=0^∞nj u_j-1v,

for u∈ V_n, n∈_≥ 0, v∈ V.   It is known from <cit.> that the Zhu's algebra of V^k() is the  universal enveloping algebra U() of .  Let ℐ_k be the maximal ideal of V^k(). Then

    L_k()=V^k()/ℐ_k.

Denote by I_k the image of ℐ_k in  A(V^k()), then

    A(L_k())=U()/I_k.


Let  be the Cartan subalgebra of , and =n^+++n^- be a fixed triangular decomposition of . Set

    U()^={u∈ U() | [h, u]=0,   for all h∈}

and let

    p:  U()^→ U()

be the Harish-Chandra projection map,  which is the restriction of the projection map: U()=U()⊕ (^-U()⊕ U()^+)→ U() to U()^. It is known that p is an algebra homomorphism <cit.>.  For a two-sided ideal I of U(), the characteristic variety of I is defined as <cit.>

    𝒱(I)={λ∈^*|p(λ)=0   for all p∈p(I∩ U()^)}.

For λ∈^*, let L(λ) be  the irreducible module of V^k() induced from the simple highest weight -module L(λ̅). Then by <cit.>, L(λ) is a module of L_k() if and only if I_k(L(λ̅))=0. We have the following result from <cit.>.

	L(λ) is an L_k()-module if and only if λ̅∈𝒱(I_k).



 §.§ Associate graded vertex Poisson algebras of affine vertex algebras

It is known by Li <cit.>
that any vertex algebra V admits a canonical filtration F^∙ V,
called the Li filtration of V.

0.2cm
For a quotient V of V^k(),
F^∙ V is described as follows.
The subspace
 F^p V is spanned by the elements

    y_1(-n_1-1)⋯ y_r(-n_r-1)1

with y_i∈,
n_i∈_≥ 0, n_1+⋯ +n_r≥ p.
We have

    V=F^0V⊃ F^1V⊃⋯,   ⋂_pF^pV=0,
    
        TF^pV⊂ F^p+1V,
        a_nF^qV⊂ F^p+q-n-1V  for a∈ F^pV,  n∈,
        a_nF^qV⊂ F^p+q-nV  for a∈ F^pV,  n≥ 0.

Here we set F^pV=V for p<0.

Let gr^FV=⊕_p F^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_-1b),
        Tσ_p(a)=σ_p+1(Ta),
       σ_p(a)_nσ_q(b)=σ_p+q- n(a_nb)

for a,b∈ V,
n≥ 0,
where σ_p  F^p(V)→ F^pV/F^p+1V is the principal symbol map.
In particular, the space
gr^F V is a [t]-module by the correspondence

    [t]∋ x(n)⟼σ_0(x)_∈(gr^F V)

for x∈, n≥ 0.






The filtration  F^∙ V
is compatible with the grading:
F^pV=⊕_n∈_≥ 0F^pV_n,
where
 F^p V_n := V_n ∩ F^p V. 








 §.§ Zhu's C_2-algebras and associated varieties
of affine vertex algebras



Let F^∙V be given as above. 
We have <cit.>

    F^p V =  span_{ a_-i-1 b  a ∈ V, i ≥ 1, b ∈ F^p-i V }

for all p ≥ 1. In particular, 

    F^1 V = C_2(V),     R_V = V/C_2(V) = F^0 V / F^1 V ⊂ gr^FV.











It is easily seen that

    F^1 V^k() = C_2(V^k()) = t^-2[t^-1] V^k().





The following map defines an isomorphism of Poisson algebras

    [                    [^*] ≅ S()                             ⟶                      R_V^k(),;                           ∋ x                             ⟼ x (-1)  1
    + t^-2[t^-1] V^k(). ]

So 
    R_V^k()≅[^*]
 and 

    X_V^k()≅^*.

More generally, let V be a quotient of V^k()
by an ideal N, then
we have

    R_V≅[^*]/N

as Poisson algebras,
where N is the image of N in R_V^k()=[^*]. 










Then X_V is just the zero locus of N in
^*.
It is a closed G-invariant conic subset of ^*. Identifying ^* with  via the bilinear form (·|·),
one may view X_V as a subvariety of . 


 §.§ Sheet and Jordan classes


In this subsection, we recall Jordan classes and sheets of a semisimple Lie algebras following <cit.>. 
Let g be as above. Denote by 𝒮 (resp. 𝒩) the set of semisimple (resp. nilpotent) elements of g.  
For x∈g, set 𝔤^x={y∈g|[x, y]=0}, and denote by x_s and  x_n its semisimple and nilpotent components, respectively.  
For x,y∈g, we say that x and y are G-Jordan equivalent if there exists α∈ G such that 

    g^y_s=g^α(x_s)=α(g^x_s),      y_n=α(x_n).

This defines an equivalence relation on g. The equivalence class of x, denoted by J_G(x), is called the Jordan class of x in g.  A Jordan class is clearly a G-stable set. 

For any Lie subalgebra u⊂g, set

    z(u)={x∈u|[x, y]=0,  y∈u}.

Denote by u^reg   the set of elements y∈u such that the dimension of ^y is minimal. We have the following lemma.


	Let x∈g.  
	
		
  * We have
		
    J_G(x)=G(z(g^x_s)^reg+x_n).

		
  * J_G(x) is irreducible in g. 
		
  * J_G(x) is locally closed in g, so it is a subvariety of g. 
	

To a Jordan class J, let  x∈ J,  and  l=g^x_s. Then l  is a Levi subalgebra of g.  Let 𝕆_ l be the nilpotent orbit in l of x_n.  The pair (l, 𝕆_l) does not depend on x∈ J up to G-conjugacy, and there is a one-to-one correspondence between the set of pairs (l, 𝕆_l) and the set of Jordan classes. 

0.3cm
Let n∈ℕ. Denote

    g^(n)={x∈g| g^x=n}.





 For n∈, an irreducible component of g^(n) is called a sheet of g.  

A sheet of  is a disjoint union of Jordan classes. So a sheet 𝕊 contains a unique dense open Jordan class J. The datum (l, 𝕆_ l) of the Jordan class J is called the datum of 𝕊. Then 

    𝕊=J,     𝕊=(J)^reg.

Let 𝕊 be a sheet with datum (l, 𝕆_ l), then the induced nilpotent orbit Ind_ l^(𝕆_ l) of  from 𝕆_ l in l is the unique orbit contained in 𝕊.  The rank of 𝕊 with datum (l, 𝕆_ l) is by definition

    r(𝕊):=𝕊- ( Ind_ l^(𝕆_ l))=z( l).

A sheet 𝕊 with datum (l, 𝕆_ l) is called a Dixmier sheet if 𝕆_ l={0}. 




 §.§ Affine W-algebras

Let f be a nilpotent element of . By the Jacobson-Morozov Theorem,
there is an sl_2-triple (e,h,f) of .


Recall that the Slodowy slice 𝒮_f is the affine space f+^e.
It has a natural Poisson structure induced from that of ^* <cit.>.

The embedding span_{e,h,f}≅sl_2 ↪
exponentiates to a homomorphism
SL_2 → G. By restriction to the one-dimensional
torus consisting of diagonal matrices, we obtain
a one-parameter subgroup ρ^* → G.
For t∈^* and x∈, set

    ρ̃(t)x := t^2ρ(t)(x).

We have
ρ̃(t)f=f, and
the ^*-action of ρ̃ stabilizes 𝒮_f.
Moreover, it contracts to f on 𝒮_f, that is, for all x∈^e,

    lim_t→ 0ρ̃(t)(f+x)=f.


The following proposition is well-known.

	
	The morphism
	
    θ_f  G ×𝒮_f ⟶,
    	   (g,x) ⟼ g.x

	is smooth onto a dense open subset of ^*.



Let W^k(,f) be the affine W-algebra associated with
a nilpotent element f of 
defined by the generalized quantized Drinfeld-Sokolov reduction:

    W^k(,f)=H^0_DS,f(V^k()).

Here H^∙_DS,f(M) denotes the BRST
cohomology of the  generalized quantized Drinfeld-Sokolov reduction
associated with f ∈𝒩() with coefficients in
a V^k()-module M <cit.>.
Recall that we have a natural isomorphism
R_W^k(,f)≅[𝒮_f] of Poisson algebras  <cit.>, so that

    X_W^k(,f)= 𝒮_f.

We write W_k(,f) for the unique simple (graded) quotient of
W^k(,f). Then X_W_k(,f)
is a ^*-invariant Poisson
subvariety of the Slodowy slice 𝒮_f.

Let 𝒪_k be the category 𝒪 of
 at level k.
We have a functor

    𝒪_k⟶ W^k(,f)-Mod
    ,   M⟼
    H^0_DS,f(M),

where
W^k(,f)-Mod denotes the category
of W^k(,f)-modules.

The full subcategory of 𝒪_k consisting of
objects M on which  acts  locally finitely is denoted by KL_k.
Note that both V^k() and L_k() are objects of KL_k.

	 Let 
	 f_θ be a root vector of the highest root θ of . Then  
	 	
	 	
  * H_DS,f_θ^i(M)=0 for all i 0, M∈𝒪_k.
	 	In particular, the functor
	 	
    𝒪_k⟶ W^k(,f_θ)-Mod
    	 		,   M⟼
    	 		H^0_DS,f_θ(M),

 		 is exact.
	 	
  * H^0_DS,f_θ(L(λ))≠ 0 if and only if λ(_0^∨)∉_≥ 0, where _0^∨=-θ^∨+K.  If this is the case, H^0_DS,f_θ(L(λ)) is a simple W^k(,f_θ)-module.
	 	
	

	
	Let k, f be arbitrary. Then
	
		
  * H_DS,f^i(M)=0 for all i 0, M∈KL_k.
		In particular, the functor
		
    KL_k⟶ W^k(,f)-Mod
    			,   M⟼
    			H^0_DS,f(M),

		is exact.
		
  * For any quotient V of V^k(),
		
    X_H^0_DS,f(V)=X_V∩𝒮_f.

		In particular,
		
		
		H_DS,f^0(V)
		 0 if and only if
		G.f⊂ X_V.
		
		
		
	




§ MAXIMAL IDEALS OF V^K(SL_3) FOR K+N=N-1/Q,  N≥ 3, Q≥ 2, (Q,N-1)=1











0.2cm
 Let ,   and V^k() be the same as  in Section 2.   We first recall the following result from <cit.>.
  For k∈, 
	V^k()
	is not simple if and only if
	
    r(k+h^∨) ∈ℚ_≥ 0\{1/m| m∈ℤ_≥ 1},

	where
	r is the lacing number of .


By Theorem <ref>,  V^k(sl_n) is not simple if and only if k=-n+p/q, where p∈_≥ 2, q∈_≥ 1, and (p,q)=1. 

0.2cm
In this section, we will prove that, for  k=-3+2/2m+1, m∈_≥ 1, the maximal ideal of V^k(sl_3) is generated by two singular vectors of conformal weight 3(2m+1), and for n≥ 4, k=-n+n-1/q, q≥ 2, (q,n-1)=1, the maximal ideal of V^k(sl_n) is generated by a singular vector of conformal weight 2q. We reach our results by using   the Kashiwara-Tanisaki characters <cit.> (see also <cit.>), the character formulas for the k=-1 case given in <cit.>,  and the  results  established in <cit.>.





 §.§ Kashiwara-Tanisaki Characters


In this subsection, we will introduce the Kashiwara-Tanisaki characters following <cit.>.  Let Δ, Δ be the root systems of  and  with respect to  and , respectively. Denote by δ the positive
imaginary root such that any imaginary root is an integral multiple of δ. 
Let W and  be the Weyl groups of  and ,  and Q and  the root lattices of  and , respectively. Let {_1,…,_l} be the simple root system of , and {_0,_1,…,_l}  the simple root system of .  For a real root ∈, set ^∨=2/(|).   Let ρ∈ h^* and ≀∈^* be such that 

    (ρ|_i^∨)=1,   (≀|_0^∨)=0,  (≀|_i^∨)=1,  1≤ i≤ l.

Recall that the twisted (or shifted) action of  on ^* is defined as follows

    w∘λ=w(λ+≀)-≀,

where w∈ and λ∈^*.  Notice that if w∈ W, w∘λ=w(λ+ρ)-ρ.

 For λ∈^*, set 

    (λ)={α∈^re|(λ+≀|^∨)∈},


    _0(λ)={α∈^re|(λ+≀|^∨)=0}.

Notice that (λ) and _0(λ) are subsystems of ^re. Denote the set  of positive roots, the set of negative roots, the set of simple roots  and the Weyl group for  (λ) by
^+(λ),  ^-(λ), Π(λ) and (λ), respectively. Denote those for _0(λ) by  ^+_0(λ), ^-_0(λ), Π_0(λ) and _0(λ).

For a real root ∈,  denote by s_∈ the corresponding reflection. Then  Π(λ) is  the set of ∈^+(λ) such that s_(^+(λ)\{})=^+(λ)\{}, and  ((λ), S(λ)) is a Coxeter group, where   S(λ)={s_: ∈Π(λ)}. 

0.2cm
For w∈(λ), denote by l_λ(w) the length of w. Denote the Bruhat ordering of (λ) by ≥_λ. For y,w∈(λ), denote by P^λ_y,w(q)∈[q] the corresponding Kazhdan-Lusztig polynomial  <cit.>, and by Q^λ_y,w(q)∈[q] the inverse Kazhdan-Lusztig polynomial defined by 

    ∑_x≤_λy≤_λz(-1)^l_λ(y)-l_λ(x)Q^λ_x,y(q)P^λ_y,z(q)=δ_x,z,

for any x,z∈(λ).  Set

    𝒞={λ∈^*| (δ|λ+≀)≠ 0 },


    𝒞^+={λ∈𝒞|(λ+≀|^∨)≥ 0,   for  any ∈^+(λ)},

and

    𝒞^-={λ∈𝒞|(λ+≀|^∨)≤ 0,   for  any ∈^+(λ)}.

Let λ∈𝒞. Then _0(λ) is a finite group <cit.>. 

For λ∈^*, let M(λ) (resp. L(λ)) be the Verma module (resp. simple module) of   with highest weight λ. We have the following result from <cit.>.
<cit.>
	
		
  * Let λ∈𝒞^+,  then for any w∈(λ) which is the longest element of w_0(λ), 
		
    ch(L(w∘λ))=∑_w≤_λy∈(λ)(-1)^l_λ(y)-l_λ(w)Q^λ_w,y(1) ch(M(y∘λ)).

		
  * Let λ∈𝒞^-,  then for any w∈(λ) which is the shortest element of w_0(λ), 
		
    ch(L(w∘λ))=∑_w≥_λy∈(λ)(-1)^l_λ(w)-l_λ(y)P^λ_y,w(1) ch(M(y∘λ)).

	  



 §.§ Maximal ideals of V^k(sl_n) for k=-n+n-1/q(q≥ 2, (q,n-1)=1)

In this subsection, we assume that 
=sl_n(), and k=-n+n-1/q, q≥ 1, (q, n-1)=1. It is obvious that

    Π(kΛ_0)={β_0=qδ-θ, _1, ⋯, _n-1}.

Since (kΛ_0+≀|β_0)=0, it follows that

    Π_0(kΛ_0)={β_0},

and

    _0(kΛ_0)={e, s_β_0}.

In particular, if q=1, then k=-1 and β_0=_0. It follows that
 
    (-Λ_0)=,  _0(-Λ_0)={e, s__0}.

   Let S={s__0, s__1, ⋯, s__n-1}, then (, S) is a Coxeter group. 
We have the following lemma.
 
	For any k=-n+n-1/q, q≥ 1, n≥ 3, we have
	
    ((kΛ_0), S(kΛ_0))≅ (, S),

	as Coxeter groups by σ: (, S)→ ((kΛ_0), S(kΛ_0))
	such that
	
    σ s__0=s_β_0, σ(s__i)=σ(s__i),  i=1,2,⋯,n-1.


 Recall that  the defining relations of (, S) are <cit.>
	
    s__i^2=1,  i=0,1,⋯,n-1,    (s__is__j)^m_ij=1   i,j=0,1,⋯, n-1,

such that 

    m_01=m_10=m_0,n-1=m_n-1,0=3,   m_0j=m_j0=2,  j=2,⋯,n-2,


    m_i,i+1=m_i+1,i=3,  m_ij=m_ji=2,  1≤ i,j≤ n-1,  j≠ i+1.

Notice that S={s__0, s__1, ⋯, s__n-1}, S(kΛ_0)={s_β_0, s__1, ⋯, s__n-1}. 
 Thus it is enough to prove that 

    (s_β_0s__1)^3=(s__1s_β_0)^3=(s_β_0s__n-1)^3=(s__n-1s_β_0)^3=1,

and

    (s_β_0s__j)^2=(s__js_β_0)^2=1,   j=2,⋯, n-2.

We only check 	(s_β_0s__1)^3=1. The computation  for
 the other cases in (<ref>)-(<ref>) is similar. Notice that
 
    s_β_0s__1s_β_0=s_s_β_0(_1)=s_qδ-θ+_1,

 and
 
    s__1s_qδ-θ+_1s__1=s_s__1(qδ-θ+_1)=s_qδ-θ=s_β_0.

 Then we have
 
    (s_β_0s__1)^3=s_β_0s__1s_β_0s__1s_β_0s__1=s_β_0s__1s_qδ-θ+_1s__1=s_β_0s_β_0=1.

	
For γ∈^*, let t_γ∈ End^* be defined by <cit.>

    t_γ(λ)=λ+(λ|δ)γ-((λ|γ)+1/2(λ|δ)(γ|γ))δ,

for λ∈^*. Recall from <cit.>, for any x∈, there exists y∈ W and γ∈ Q such that x=yt_γ.  We have the following lemma.
 
	For γ∈ Q,  σ(t_γ)=t_qγ. 
	
 We first consider γ=θ. Notice that for λ∈^*, 
		
    s_β_0s_θ(λ)=s_β_0(λ-(λ|θ)θ)=λ+(λ|δ)qθ-((λ|θ)q+((λ|δ)q^2)δ=t_qθ(λ).

	This deduces that
	
    s_β_0s_θ=t_qθ.

In particular, if q=1,  s__0s_θ=t_θ (see also <cit.>). So by Lemma <ref>, 

    σ(t_θ)=t_qθ.

Notice that 
    s__1t_θs__1=t_s__1(θ)=t_θ-_1

 and 
 
    s__1t_qθs__1=t_s__1(qθ)=t_q(θ-_1).

 Thus
 
    σ(t_θ-_1)=t_q(θ-_1).

Considering s__2t_θ-_1s__2 and s__2t_q(θ-_1)s__2, we  deduce that

    σ(t_θ-_1-_2)=t_q(θ-_1-_2).

Similarly, we have for 4≤ j≤ n-1, 

    σ(t__j+⋯_n-1)=t_q(_j+⋯_n-1).

Notice that for , β∈ Q, t_t_β=t_+β. Then the lemma follows from  (<ref>)-(<ref>).


Let Λ_i∈^* be such that

    Λ_i(_j^∨)=δ_ij,  i,j=0,1,⋯,n-1,

We have the following character formula from  <cit.>, <cit.>. 

Let n≥ 3, and let L(Λ) be an irreducible level k=-1	sl_n-module with highest weight 

    Λ=-(1+s)Λ_0+sΛ_n-1 ( resp. Λ=-(1+s)Λ_0+sΛ_1),  s∈_≥ 0.

Then the character of L(Λ) is given  by the following formula:

    R chL(Λ)=
    ∑_w∈ W∑_γ∈ Q 
    (γ|Λ̅_n-1(resp. Λ̅_1))≥ 0(-1)^l(w)e^wt_γ(Λ+≀)-≀.

where

    R=∏_∈_+(1-e^-)^ dim_.

In particular, if Λ=-Λ_0, 

    R chL(-Λ_0)=
    ∑_w∈ W∑_γ∈ Q 
    (γ|Λ̅_n-1)≥ 0, (γ|Λ̅_1))≥ 0(-1)^l(w)e^wt_γ(-Λ_0+≀)-≀.

	
We also need the following result from <cit.>,  <cit.>, <cit.>,  <cit.>.
 Let λ∈𝒞^+, and w∈(λ). If L(μ) is a subquotient of M(w∘λ), then there exists y∈(λ) such that μ=yw∘λ.
	


0.2cm
Recall  the following results from <cit.>,  <cit.>, and <cit.>.

	
		
  * <cit.>	Let
		
    [                                                                      u^1= [-e__1(-1)^2e__2(-1)+e__1(-1)e__1+_2(-1)h_2(-1)+e__1+_2(-1)^2f__2(-1)] 1,; ;                                                                      u^2=                            [e__1(-1)e__2(-1)^2+e__2(-1)e__1+_2(-1)h_1(-1);                                                                                                -2e_α_2(-1)e__1+_2(-2)
    			-e__1+_2(-1)^2f__1(-1)] 1. ]

		Then u^1, u^2 are  singular vectors of V^-1(sl_3). 
		
		
  * <cit.> Let
		
    u=e_θ(-1)e__2+⋯_n-2(-1) 1-e__1+⋯_n-2(-1)e__2+⋯_n-1(-1) 1.

		Then u is a singular vector of V^-1(sl_n), n≥ 4. 
		
		
  * <cit.> The ideal of V^-1(sl_3) generated by u^1 and u^2 is the maximal ideal of V^-1(sl_3). 
		
		
		
  * <cit.> The ideal  generated by u in (2) is the maximal ideal of V^-1(sl_n), n≥ 4.
	
	


We are now in a position to state the main result of this section.
  Let k=-n+n-1/q, n≥ 3, q≥ 2, (n-1,q)=1. Then 	

    R chL_k(sl_n)=
    ∑_w∈ W∑_γ∈ Q 
    (γ|Λ̅_n-1)≥ 0, (γ|Λ̅_1))≥ 0(-1)^l(w)e^wt_qγ(kΛ_0+≀)-≀.

Furthermore, we have

	
  * If k=-3+2/2m+1, m≥ 1, then  the maximal ideal of V^k(sl_3) is generated by  two singular vectors v^1 and v^2 of weights kΛ_0-3(2m+1)δ+2α_1+α_2 and kΛ_0-3(2m+1)δ+α_1+2α_2, respectively. 
	
	
  * If k=-n+n-1/q satisfying n≥ 4, q≥ 2, and (q,n-1)=1, then the maximal ideal of V^k(sl_n) is generated by a singular vector w^1 of weight kΛ_0-2qδ+_1+2_2+⋯+2_n-2+_n-1. 


 Let =sl_n, n≥ 3. 
Notice that if q=1, then k=-1 and ((-Λ_0), S(-Λ_0))=(, S). So  ((-Λ_0), S(-Λ_0))≅ ((kΛ_0), S(kΛ_0)), k=-n+n-1/q, q≥ 2, (q,n-1)=1.
By Lemma <ref>, 
and the definitions of P^λ_y,w(q) and Q^λ_y,w(q) <cit.>, we have for  y,w∈,  

    w≥_-Λ_0y   if  and  if σ(w)≥_kΛ_0σ(y),
, 

    l_-Λ_0(y)=l_kΛ_0σ(y),  l_-Λ_0(w)=l_kΛ_0σ(w),

and

    P^-Λ_0_y,w(1)=P^kΛ_0_σ(y),σ(w)(1),  Q^-Λ_0_y,w(1)=Q^kΛ_0_σ(y),σ(w)(1).

It is obvious that -Λ_0,  kΛ_0∈𝒞^+,  for k=-n+n-1/q, q≥ 2, (q,n-1)=1.  Notice that for  any w∈(-Λ_0) which is the longest element of w_0(-Λ_0), σ(w) is the longest element of σ(w)_0(kΛ_0). Then by Theorem <ref> and (<ref>)-(<ref>), 

    ch(L(σ(w)∘ kΛ_0))=∑_w≤_-Λ_0y∈(-Λ_0)(-1)^l_-Λ_0(y)-l_-Λ_0(w)Q^-Λ_0_w,y(1) ch(M(σ(y)∘ (kΛ_0))),

which is equivalent to 

    R ch(L(σ(w)∘ kΛ_0))=∑_w≤_-Λ_0y∈(-Λ_0)(-1)^l_-Λ_0(y)-l_-Λ_0(w)Q^-Λ_0_w,y(1)e^σ(y)∘ kΛ_0.

Then by Lemma <ref>, Lemma <ref>, and Theorem <ref>, 

    R chL(kΛ_0)=
    ∑_w∈ W∑_γ∈ Q
    (γ|Λ̅_n-1)≥ 0, (γ|Λ̅_1))≥ 0(-1)^l(w)e^wt_qγ(kΛ_0+≀)-≀.

Furthermore, (<ref>) together with  Lemma <ref>  means that  L(y∘ (-Λ_0))  for some y∈(-Λ_0) is isomorphic to a subquotient  of M(w∘ (-Λ_0)) if and only if L(σ(y)∘ kΛ_0) is a subquotient of M(σ(w)∘ kΛ_0). 

Case 1    n=3,  k=-3+2/2m+1, m≥ 1.  Let u^1 and u^2 be the singular vectors given in (1) of Theorem <ref>. The weights of u^1 and u^2 with respect to  are -Λ_0-3δ+2_1+_2 and -Λ_0-3δ+_1+2_2, respectively. It is easily checked that 

    s__2∘ (t__1(-Λ_0+≀)-≀)=-Λ_0-3δ+2_1+_2,

and

    s__1∘ (t__2(-Λ_0+≀)-≀)=-Λ_0-3δ+_1+2_2.

Thus V^k(sl_3) has two singular vectors v^1 and v^2 of wights s__2∘ (t_(2m+1)_1(kΛ_0+≀)-≀) and s__1∘ (t_(2m+1)_2(kΛ_0+≀)-≀), respectively.  It is easy to check  that

    s__2∘ (t_(2m+1)_1(kΛ_0+≀)-≀)=kΛ_0-3(2m+1)δ+2_1+_2,
and

    s__1∘ (t_(2m+1)_2(kΛ_0+≀)-≀)=kΛ_0-3(2m+1)δ+_1+2_2,

By (3) of Theorem <ref>, the ideal generated by u^1 and u^2 is the maximal ideal of V^-1(sl_3). It follows that the ideal generated by v^1 and v^2 is the maximal ideal of V^k(sl_3). We complete the proof of (1).

0.2cm
Case 2  n≥ 4, k=-n+n-1/q, q≥ 2, (q, n-1)=1.  Let u be the singular vector in (2) of Theorem <ref>. The weight of u with respect to the Cartan subalgebra  is 
-Λ_0-2δ+θ+β, where β=_2+⋯ +_n-2. It is easily checked that

    -Λ_0-2δ+θ+β=(s_θs__1s__n-1)∘ (t_-β(-Λ_0+≀)-≀).

Then V^k(sl_n) has a singular vector w^1 of weight (s_θs__1s__n-1)∘ (t_-qβ(kΛ_0+≀)-≀).
It can be checked directly that 

    (s_θs__1s__n-1)∘ (t_-qβ(kΛ_0+≀)-≀)=kΛ_0-2qδ+θ+β.

By (4) of Theorem <ref>, the ideal generated by u is the maximal ideal of V^-1(sl_n). It follows that the ideal generated by w^1 is the maximal ideal of V^k(sl_n).


 The  method here can be used to  the subregular case  for  of other  types. 



§ VARIETIES OF L_K(SL_3) FOR K=-3+2/2M+1, M≥ 0

Let  and V^k() be as above in Section 3. 
For x∈ V^k(g), we denote by x̅ the image of x in R_V^k(). Recall that   R_V^k()≅[^*] by

    x_1(-1)⋯ x_n(-1) 1↦ x_1⋯ x_n,

for x_1,⋯,x_n∈. 
As in Section 2, we denote by ℐ_k  the maximal ideal of V^k(), then 

    L_k()=V^k()/ℐ_k

and

    R_L_k()=[^*]/ℐ_k,

where ℐ_k is the image of ℐ_k in R_V^k(). 


 §.§ The associated variety of L_-1(sl_3)

Let Δ be the root system of sl_3(ℂ) and 

    {h_1, h_2, e__1, e__2, e__1+_2, f__1,f__2, f__1+_2}

be a Chevalley basis of sl_3(ℂ) with structure constants c_,, for ,∈Δ such that 
    c__1,_2=1.

Then we have 

    c__2,-_1-_2=c_-_1-_2,_1=1.

0.2cm
Recall from Theorem <ref> <cit.> that 

    [                                                                      u^1= [-e__1(-1)^2e__2(-1)+e__1(-1)e__1+_2(-1)h_2(-1)+e__1+_2(-1)^2f__2(-1)] 1,; ;                                                                      u^2=                            [e__1(-1)e__2(-1)^2+e__2(-1)e__1+_2(-1)h_1(-1);                                                                                                   -2e_α_2(-1)e__1+_2(-2)
    -e__1+_2(-1)^2f__1(-1)] 1. ]

are  singular vectors of V^-1(sl_3). Then  we have
 There exist x,y∈ℐ_-1 such that

    [                                                              x≡      -e__1+_2(-1)h_1(-1)h_2(-1) 1-e__1(-1)e__1+_2(-1)f__1(-1) 1; ;                                                                   -e__1+_2(-1)^2f__1+_2(-1) 1
    +e__1(-1)e__2(-1)(2h_1+h_2)(-1) 1; ;                                                                             +2e__2(-1)e__1+_2(-1)f__2(-1) 1
    (  F^1(V^-1(sl_3))); ;                                                              y≡ h_1(-1)h_2(-1)(h_1(-1)+h_2(-1)) 1+e__1(-1)e__2(-1)f__1+_3(-1) 1; ;                                                                       -3e__1+_2(-1)f__1(-1)f__2(-1) 1-e__1(-1)f__1(-1)h_1(-1) 1; ;                                                                                      +e__1+_2(-1)f__1+_2(-1)(h_1(-1)+h_2(-1)) 1; ;                                                                                 -e__2(-1)f__2(-1)h_2(-1) 1 (  F^1(V^-1(sl_3))). ]


 Let x=f__1(0)u^1, and   y=f__1(0)f__1+_2(0)u^1∈ℐ_-1. Then the lemma follows.	
	
Let 

    [                                                                                    p_1=                                               -e__1^2e__2+e__1e__1+_2h_2+e__1+_2^2f__2,; ;                                                                                    p_2=                                                e__1e__2^2+e__2e__1+_2h_1-e__1+_2^2f__1,; ;                                                                                    p_3= -e__1+_2h_1h_2-e__1e__1+_2f__1-e__1+_2^2f__1+_2
     +e__1e__2(2h_1+h_2)
    +2e__2e__1+_2f__2,; ;                                                                                    p_4=                          h_1h_2(h_1+h_2)+e__1e__2f__1+_3
    -3e__1+_2f__1f__2-e__1f__1h_1,; ;                                                                                                                                           +e__1+_2f__1+_2(h_1+h_2)-e__2f__2h_2. ]

Then by Lemma <ref>,  p_i∈ℐ_-1, 1≤ i≤ 4.


	 Denote λ=h_1-h_2. Then 

    X_L_-1(sl_3)=𝕊_ l_1= G.ℂ^*λ=G.ℂ^*λ∪𝒪_min,

which is the closure of the Dixmier sheet 𝕊_ l_1=G.ℂ^*λ, where l_1=h⊕ e_θ⊕ f_θ.  In particular, X_L_-1(sl_3) is irreducible and 

    X_L_-1(sl_3)=5,

where 𝒪_min is the minimal non-zero nilpotent orbit of sl_3(ℂ).

  It is known that for any simple Lie algebra  over ,  is the finite disjoint union of its Jordan classes <cit.>.  In particular, for =sl_3(), we have 
	
    =G· (^*λ)∪ G· (^*λ+f_θ)∪ G· ( h^reg)∪ G· f_θ∪ G· (f__1+f__2)∪{0},

	where G is the connected adjoint group of , and θ=_1+_2.	
	Notice that
	 
    p_1(f__1+f__2)=-1,  p_3(tλ+f_θ)=-t^2.
  
	 This implies that  G· (f__1+f__2) and G· (^*λ+f_θ) could not belong in X_L_-1(sl_3). 
Furthermore, for regular semisimple vector h, all
(h_1|h),  (h|h_2), (h|h_1+h_2) are  non-zero. 
So

    p_4(h)=(h|h_1)(h|h_2)(h|h_1+h_2)≠ 0.
 We see that for any h∈ h^reg, h∉ X_L_-1(sl_3). 
We deduce that

    X_L_-1(sl_3)⊆G.ℂ^*λ=G.ℂ^*λ∪𝒪_min.

By (3) of Theorem <ref>,  the ideal generated by u^1 and u^2 is the maximal ideal of V^-1(sl_3). This means that 

    X_L_-1(sl_3)= G.ℂ^*λ=G.ℂ^*λ∪𝒪_min.

It follows that X_L_-1(sl_3) is irreducible and  X_L_-1(sl_3)=5. 



 §.§ Associated varieties of L_k(sl_3) for k=-3+2/2m+1, m≥ 1

In this subsection we always  assume that k=-3+2/2m+1, m≥ 1. 


0.2cm
Recall from Theorem <ref> that 
		V^k(sl_3) has two singular vectors v^1 and v^2 with weights kΛ_0-3(2m+1)δ+2α_1+α_2 and kΛ_0-3(2m+1)δ+α_1+2α_2, respectively.  
  Then for i=1,2, 

    e__1(0)v^i=e__2(0)v^i=f__1(1)v^i=f__2(1)v^i=0.

Notice that each element in V^k(sl_3) is a linear combination of elements of the following form:
 
    z = z^(+)  z^(-)  z^(0) 1,

with

    z^(+) :=     e__1(-1)^a_11⋯
    e__1 (- r_1)^a_1r_1e__2(-1)^a_21⋯
    e__2 (- r_2)^a_2r_2
        e__1+_2(- 1)^a_31⋯ e__1+_2(- r_3)^a_3r_3 , 
    
     z^(-) : =    f__1(-1)^b_11⋯ f__1(- s_1)^b_1s_1f__2(-1)^b_21⋯ f__2(- s_2)^b_2s_2
        f__1+_2(- 1)^b_31⋯ f__1+_2(- s_3)^b_3s_3 , 
    
      z^(0) :=     h_1(- 1)^c_11⋯ h_1(- t_1)^c_1t_1
    h_2(- 1)^c_21⋯ h_2(- t_2)^c_2t_2,

where r_1,r_2,r_3, s_1,s_2,s_3,,t_1,t_2
are positive integers, and a_lp,b_ln,c_ij, for l =1,2,3,
p=1,…,r_l, n=1,…,s_l,  i = 1, 2, j=1,…,t_i, 
are nonnegative integers such
that at least one of them is non-zero. Recall  that

    depth(z^(+))=∑_i=1^3∑_j=1^r_i-1(j-1)a_ij,   depth(z^(-))=∑_i=1^3∑_j=1^s_i-1(j-1)b_ij,


    depth(z^(0))=∑_i=1^2∑_j=1^t_i-1(j-1)c_ij;


    deg(z^(+))=∑_i=1^3∑_j=1^r_ia_ij,   deg(z^(-))=∑_i=1^3∑_j=1^s_ib_ij,  deg(z^(0))=∑_i=1^2∑_j=1^t_ic_ij;


    depth(z)=depth(z^(+))+depth(z^(-))+depth(z^(0));


    deg(z)=deg(z^(+))+deg(z^(-))+deg(z^(0)).



 Let V^1 be the subspace of V^k(sl_3) linearly spanned by elements z=z^(+)  z^(-)  z^(0) 1 of weight kΛ_0-3(2m+1)δ+2_1+_2 such that deg(z^(0))≤ 6m-2 or depth(z^(0))≥ 1.  Then we  may assume that 

    [                                                                            v^1=                       ∑_i=0^6m+1a_ie__1(-1)e__1+_2(-1)h_1(-1)^ih_2(-1)^6m+1-i 1; ;                                                                                 +∑_i=0^6m[x_ie__1(-1)^2e__2(-1)+y_ie__1+_2(-1)^2f__2(-1)+b_ie__1(-2)e__1+_2(-1); ;                                                                                                                 +c_ie__1(-1)e__1+_2(-2)]h_1(-1)^ih_2(-1)^6m-i 1; ;                                                                                   +∑_i=0^6m-1[d_ie__1+_2(-1)e__1+_2(-2)f__2(-1)
    +z_ie__1(-1)e__2(-1)e__1+_2(-1); ;                                                                                              f__2(-1)
    +l_ie__1(-1)^2e__1+_2(-1)f__1(-1)
    +n_ie__1(-2)e__1+_2(-2); ;                                                                                     +k_ie__1(-1)^2e__2(-2)+g_ie__1(-1)e__1(-2)e__2(-1)+p_ie__1+_2(-1)^2f__2(-2); ;                                                                                                     +q_ie__1(-1)e__1+_2(-1)^2f__1+_2(-1)+m_ie__1(-1)e__1+_2(-3); ;                                                                                                          +r_ie__1(-3)e__1+_2(-1)]h_1(-1)^ih_2(-1)^6m-1-i 1+u^1, ]

where u^1∈ V^1. 

0.3cm
The  following result comes from Lemma 3.1 of  <cit.>. 
 Not all a_i, x_j, 0≤ i≤ 6m+1, 0≤ j≤ 6m in v^1 are zero. 
	
	


We now have the following lemma.
 

    a_i-y_i=0,   a_6m+1=0,   0≤ i≤ 6m,


    -2(i+1)a_i+1+(6m+1-i)a_i+x_i+l_i-1=0,   0≤ i≤ 6m,


    (i+1)a_i+1-2(6m+1-i)a_i-2x_i+z_i=0,  0≤ i≤ 6m-1,


    (6m+1)a_6m+1-2a_6m-2x_6m=0,


    (-4+2/2m+1)a_i+2y_i-b_i-1=0,   0≤ i≤ 6m,


    (-4+2/2m+1)a_6m+1-b_6m=0,    (-4+2/2m+1)a_0+2y_0=0,


    (-4+2/2m+1)a_i-2x_i-c_i-1-c_i=0,  0≤ i≤ 6m,


    a_0=y_0=0,


    a_i+x_i=0,  0≤ i≤ 6m,


    -(i+1)a_i+1-(6m-i)a_i+q_i+q_i-1=0,  0≤ i≤ 6m.

	

 We consider e__1(0)v^1, e__2(0)v^1, f__1(1)v^1, and  f__1+_2(1)v^1. By the definition of V^1, there are no monomials 
z=z^(+)  z^(-)  z^(0) 1   in  e__1(0)u^1, e__2(0)u^1, f__1(1)u^1, f__2(1)u^1,  and  f__1+_2(1)u^1 such that deg(z^(0))≥ 6m, depth(z)=0, 
and z^(-)=1. Then it is easy to deduce that 
the coefficients of e__1(-1)^2e__1+_2(-1)h_1(-1)^ih_2(-1)^6m-i 1 in e__1(0)v^1, 0≤ i≤ 6m,   are 

    -2(i+1)a_i+1+(6m+1-i)a_i+x_i+l_i-1
    ,   0≤ i≤ 6m.

Then (<ref>) holds.

0.2cm
The coefficients of e__1+_2(-1)^2h_1(-1)^ih_2(-1)^6m+1-i 1 in e__2(0)v^1, 0≤ i≤ 6m,  are 

    -a_i+y_i,   0≤ i≤ 6m,

and the coefficient of e__1+_2(-1)^2h_1(-1)^6m+1-i 1 is a_6m+1. Then  (<ref>) follows. 

0.2cm
The coefficients of e__1(-1)e__2(-1)e__1+_2(-1)h_1(-1)^ih_2(-1)^6m-i 1, 0≤ i≤ 6m-1, in e__2(0)v^1 are

    (i+1)a_i+1-2(6m+1-i)a_i-2x_i+z_i,  0≤ i≤ 6m-1,

and the coefficient  of e__1(-1)e__2(-1)e__1+_2(-1)h_1(-1)^6m 1  in e__2(0)v^1 is

    (6m+1)a_6m+1-2a_6m-2x_6m.

Then (<ref>) and (<ref>) hold. 

0.2cm
The coefficients of e__1+_2(-1)h_1(-1)^ih_2(-1)^6m+1-i 1, 0≤ i≤ 6m+1,
 in f__1(1)v^1 are
 
    (-4+2/2m+1)a_i+2y_i-b_i-1,   1≤ i≤ 6m,

 and
 
    (-4+2/2m+1)a_6m+1-b_6m(i=6m+1),    (-4+2/2m+1)a_0+2y_0(i=0).

 Then (<ref>)and (<ref>) hold.

0.2cm
The coefficients of e__1(-1)h_1(-1)^ih_2(-1)^6m+1-i 1, 0≤ i≤ 6m, in f__1+_2(1)v^1 are 

    (-4+2/2m+1)a_i-2x_i-c_i-1-c_i,  0≤ i≤ 6m.

This imlpies (<ref>).

0.2cm
By (<ref>), (<ref>), and the fact that m≥ 1, 

    a_0=y_0=0.

Then (<ref>) follows.

0.2cm
Notice that (_2|2_1+_2)=0 and e__2(0)v^1=0. It follows that 
f__2(0)v^1=0. 
Considering the coefficients of e__1(-1)^2h_1(-1)^ih_2(-1)^6m+1-i, 0≤ i≤ 6m, we obtain that 

    a_i+x_i=0,  0≤ i≤ 6m,

which is  (<ref>). 

0.2cm
The coefficients of e__1(-1)e__1+_2(-1)^2h_1(-1)^ih_2(-1)^6-i 1, 0≤ i≤ 6m, in e_+_2(0)v^1 are

    -(i+1)a_i+1-(6m+1-i)a_i+y_i+q_i+q_i-1,  0≤ i≤ 6m.

Then (<ref>) follows from (<ref>).

By Lemma  <ref>, not all a_i, x_j, 0≤ i≤ 6m+1, 0≤ j≤ 6m are zero, and by Lemma <ref>, a_i=-x_i, 0≤ i≤ 6m,  a_6m+1=0. Then  not all a_i, 0≤ i≤ 6m+1 are zero. By (<ref>), a_0=0. Thus we may assume that 

    a_0=⋯=a_r-1=0,  a_r≠ 0,

for some r≥ 1. 
We have the following lemma.
 For k=-3+2/2m+1,  m≥ 1, 
r=2m.
	
  
	It can be checked directly that the coefficient of 
    e__1(-1)e__1+_2(-1)h_1(-1)^r-1h_2(-1)^6m+1-r
 in h_1(1)v^1
	is 

    -5ra_r+2r(-3+2/2m+1)a_r+4x_r-1+2y_r-1+2b_r-1+c_r-1+c_r-2-z_r-1+4l_r-2+2(q_r-1+q_r-2),
	
which should be zero.
By (<ref>) and (<ref>), 

    x_r-1=y_r-1=0.

Then by (<ref>) and (<ref>), we have 

    l_r-2=2ra_r,  z_r-1=-ra_r.

By (<ref>) and (<ref>),

    b_r-1=-4m/2m+1a_r.

By (<ref>) and (<ref>),

    c_0=0,  c_j+c_j-1=0,  1≤ j≤ r-1.

Then we have

    c_0=⋯=c_r-1=0.

By (<ref>), 

    q_r-1+q_r-2=ra_r.

Thus (<ref>) becomes

    (4r/2m+1-8m/2m+1)a_r=0.

This deduces that r=2m.

Let U^0 be the sl_3-module generated by v^1. The following lemma is easy to check. 
  
	With respect to the Cartan subalgebra h= h_1⊕ h_2,  the weight zero subspace of U^0 is one-dimensional, which is linearly spanned by f__1(0)f__1+_2(0)v^1. 
	
Let W^1 be the subspace of V^k(sl_3) linearly spanned by monomials z=z^(+)z^(-)z^(0) 1 such that z∈ F^0(V^k(sl_3)) and z^(-)≠ 1.  Then  it can be  checked directly  that

    f__1(0)f__1+_2(0)v^1
    =∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1+w^1,

for some w^1∈ W^1+F^1(V^k(sl_3)). We denote

    v^3=f__1(0)f__1+_2(0)v^1,

and without loss of generality we may assume that 

    a_2m=1.

We have the following lemma.

	
    ∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1=h_1(-1)^2m+1h_2(-1)^2m+1(h_1+h_2)(-1)^2m+1 1.
	
That is,	

    [                                                                                                 v^1-e__1(-1)e__1+_2(-1)h_1(-1)^2mh_2^2m+1(h_1+h_2)(-1)^2m; ;                                                                              +e__1(-1)^2e__2(-1)h_1(-1)^2mh_2(-1)^2m(h_1+h_2)(-1)^2m∈ W^1+F^1(V^k(sl_3)). ]
	
and	

    v^3=h_1(-1)^2m+1h_2(-1)^2m+1(h_1+h_2)(-1)^2m+1 1+w^1,

where w^1 is the same as in (<ref>).


	It suffices to prove that 

    ∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1=h_1(-1)^2m+1h_2(-1)^2m+1(h_1+h_2)(-1)^2m+1 1.

Denote

    π_2=exp f__2(0)exp(-e__2(0))exp f__2(0),

    π_3
    =exp f__1+_2(0)exp(-e__1+_2(0))exp f__1+_2(0).
 
Then  by Lemma <ref>, there exists 0≠ c_i∈, i=2,3 such that 

    π_i(v^3)=c_iv^3.

On the other hand, it is  obvious that

    π_i(W^1+F^1(V^k(sl_3)))⊆ W^1+F^1(V^k(sl_3)),   i=2,3.

So for i=2,3, we have

    [                                                               π_i(∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1); ;                                                            = c_i(∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1). ]

Notice that 

    [                                                             π_2(∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1); ;                                                           =  ∑_i=2m^6m+1a_i(h_1+h_2)(-1)^i+1(-h_2(-1))^6m+1-ih_1(-1) 1. ]

This means that 
    ∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1
 has a factor (h_1+h_2)(-1)^2m+1.  Also
 
    [                                                             π_3(∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1); ;                                                           =    -∑_i=2m^6m+1a_ih_2(-1)^i+1h_1(-1)^6m+1-i(h_1+h_2)(-1) 1. ]

This implies that ∑_i=2m^6m+1a_ih_1(-1)^i+1h_2(-1)^6m+1-i(h_1+h_2)(-1) 1 has a factor h_2(-1)^2m+1. Thus (<ref>) holds. 

For the singular vector v^2, we can similarly obtain the following lemma.
  There exists a non-zero number c such that 
	
    [                                                                                                cv^2-e__2(-1)e__1+_2(-1)h_1(-1)^2m+1h_2^2m(h_1+h_2)(-1)^2m; ;                                                                              -e__1(-1)e__2(-1)^2h_1(-1)^2mh_2(-1)^2m(h_1+h_2)(-1)^2m∈ W^1+F^1(V^k(sl_3)). ]
	

We are now in a position to give the  second main result of this section. 
 For k=-3+2/2m+1, m≥ 1, X_L_k(sl_3)= G.(ℂ^*(h_1-h_2)+f_θ), where θ=_1+_2.

 
Notice that

    =G·^*(h_1-h_2)∪ G· (^*(h_1-h_2)+f_θ)∪ G· ( h^reg)∪𝒩,

and

    G·^*(h_1-h_2)∪𝒩⊆G· (^*(h_1-h_2)+f_θ).
 
By  Theorem <ref>,  the ideal generated by v^1 and v^2 is the maximal ideal of V^k(sl_3). Recall that W^1 is the subspace of V^k(sl_3) linearly spanned by monomials z=z^(+)z^(-)z^(0) 1 such that z∈ F^0(V^k(sl_3)) and z^(-)≠ 1. For v∈ V^k(sl_3), denote by v̅ the image of v in R_V^k(sl_3). Then for any w∈ W^1, the  
value of w̅ at h_1-h_2+f_θ is zero. 
Let h be a semisimple regular element in the Cartan subalgebra h= h_1+ h_2. Then 
all the (h|h_1), (h|h_2), and (h|h_1+h_2) are non-zeros. Thus by Lemma <ref>, 

    h∉ X_L_k(sl_3).

Then  by Lemmas <ref>-<ref>, the zero locus of ℐ_k⊆ R_V^k(sl_3) is exactly G· (^*(h_1-h_2)+f_θ).  It follows that

    X_L_k(sl_3)=G· (^*(h_1-h_2)+f_θ).


Recall that if k∈_≥ 0, X_L_k(sl_3)={0} <cit.>,  X_L_k(sl_3)=𝒪_min  if k=-3+2m+1/2, m≥ 1 <cit.>, and X_L_k(sl_3)=𝒩  if k=-3 or k=-3+p/q, p,q≥ 3, (p,q)=1 <cit.>, <cit.>.  Then we have 
 
	
		
  * For k∈, the associated variety X_L_k(sl_3) of L_k(sl_3) is one of the following:
		
    ^*,  {0},  𝒩,  𝒪_min,   G·^*(h_1-h_2),  G· (^*(h_1-h_2)+f_θ).
		
		
		
  * Let h be a regular semi-simple vector of sl_3, then there is no k∈ such that 
		X_L_k(sl_3)= G·^*h.
	
	
For k=-3+2/2m+1, m≥ 0,  let ℐ_k be the maximal ideal of V^k(sl_3), and 𝒱(I_k) be defined as in Section 2. We have the following conjecture.

	
    𝒱(I_k)={tΛ̅_1-2i/3Λ̅_2, tΛ̅_2-2i/3Λ̅_1,  tΛ̅_1-(t+2i/3+1)Λ̅_2,    t∈, i=0,1,⋯, 2m}.



	If m=0, that is k=-1,  the conjecture is true by Proposition 5.5 of <cit.>. When m=1, we could verify the conjecture  by computer programming. 



§ VARIETIES OF SIMPLE AFFINE W-ALGEBRAS W_K(SL_3,F)

 
We first have the following results from <cit.> and <cit.>.
 Let  be a simple complex Lie algebra, 
	 f  a regular  nilpotent element of ,  and k=-h^∨+p/q a non-degenerate admissible number. Then the simple W-algebra W_k(g, f) is rational and lisse. 


		Let f be a minimal nilpotent vector of sl_3 and k=-3+2m+1/2, m≥ 1. Then the simple W-algebra W_k(sl_3) is rational and lisse. 
		
Let k=-1 and f a minimal nilpotent element of sl_3. It was proved in  <cit.> that  H^0_DS,f(L_-1(sl_3))=W_-1(sl_3,f), and  is isomorphic to the rank one Heisenberg algebra M(1). So  its associated variety  is one-dimensional. 


	 Let k=-3+2/2m+1, m≥ 1,  and f a minimal nilpotent vector of sl_3, then 
	 
	 	
  * 
    X_W_k(sl_3,f)={[[          a          b 3(μ^2-a^2);          0        -2a          c;          1          0          a ]]| μ, a,b,c∈,  bc=2(a-μ)(2a+μ)^2 }.
	
	 	In particular, 
	 	
    X_W_k(sl_3,f)=3.

	 	
	 	
  * W_k(sl_3) is not quasi-lisse.
	 
	 
	
 By Theorem <ref>,  for a minimal vector f∈ sl_3, and k+3=2/2m+1, m∈_≥ 0, 
	
    H^0_DS,f(L_k(sl_3))=W_k(sl_3, f).
	
	By Theorem <ref> and (<ref>),
	
    X_W_k(sl_3))=X_L_k(sl_3)∩𝒮_f,

for k+3=2/2m+1, m∈_≥ 0. We may assume that f=f_θ. Then

    𝒮_f=f_θ+^e_θ,
	
and 

    ^e_θ=(_1-_2)⊕ e_θ⊕ e__1⊕ e__2.

Then we have
	
    f_θ+^e_θ={[[   a   b   d;   0 -2a   c;   1   0   a ]]|  a,b,c,d∈}.

It is obvious that for  μ∈, 

    A_μ=[[   μ   0   0;   0 -2μ   0;   1   0   μ ]]∈G· (^*(h_1-h_2)+f_θ).

In general, 

    B=[[   a   b   d;   0 -2a   c;   1   0   a ]]∈G· (^*(h_1-h_2)+f_θ)∩𝒮_f_θ

	if and only if B  is similar to A_μ for some μ∈. Then we deduce that
	
    d=3(μ^2-a^2),

	and 
	
    bc=2(a-μ)(μ+2a)^2.
	
Thus (<ref>) and (<ref>) hold. 

By Theorem 8.2 of <cit.>, L(tΛ̅_1) and L(tΛ̅_2), t∈_≥ 0, are irreducible ordinary modules of L_k(sl_3). This  means that L_k(sl_3) has infinitely many irreducible ordinary modules. 
 Then by Theorem 5.4 of <cit.>, 
for t∈_≥ 0, the minimal quantum hamiltonian reductions ℋ_j_t+1, 1Δ_t+1,1 and  ℋ_j_1,t+1Δ_1,t+1
of L(tΛ̅_1) and L(tΛ̅_2) are  irreducible ordinary W_k(sl_3,f)-modules,  
where

    j_t+1,1=t/3,   j_1,t+1=2t/3,  Δ_t+1,1=t^2+3t/3(k+3)-2t/3,  Δ_1,t+1=t^2+3t/3(k+3)-t/3.

This means that W_k(sl_3,f) has infinitely many irreducible modules. By Theorem <ref>, W_k(sl_3,f) is not quasi-lisse.

We now assume that f is a regular nilpotent vector of sl_3.  We may assume that 

    f=[[ 0 0 0; 2 0 0; 0 2 0 ]],   e=[[ 0 1 0; 0 0 1; 0 0 0 ]],  h=[[  2  0  0;  0  0  0;  0  0 -2 ]].

Then

    ^e= e⊕ e_θ,

and 
	
    f+^e={[[ 0 a b; 2 0 a; 0 2 0 ]]|  a,b∈}.

Denote  the associated  universal affine W-algebras and the simple quotients by W^k(sl_3) and W_k(sl_3), respectively.  
 Let k+3=2/2m+1 or 2m+1/2, then
	
    X_W_k(sl_3)={[[      0 3/4μ^2 1/2μ^3;      2      0 3/4μ^2;      0      2      0 ]]|  μ∈}.

In particular, X_W_k(sl_3)=1.  Furthermore, W_k(sl_3) is not quasi-lisse.

 Since k=-3+2/2m+1, m≥ 1, it follows that
	
    (kΛ_0+≀|)∉,   for ∈{-_1+δ, -_2+δ, -(_1+_2)+δ, -(_1+_2)+2δ}.

	Then by Theorem 9.1.4 of <cit.>, 
	
    H^0_DS,f(L_k(sl_3))=W_k(sl_3, f).

	Thus
	
    X_W_k(sl_3))=X_L_k(sl_3)∩𝒮_f.

	By the Feigin-Frenkel Langlands duality <cit.>, <cit.>, 
	
    W^k(sl_n)≅ W^k'(sl_n),

	for k+n=p/q and k'+3=q/p. In particular,  for k+3=2/2m+1 and k'+3=2m+1/2,
	
    W_k(sl_3)≅ W_k'(sl_3).

	Let 
	
    B=[[ 0 a b; 2 0 a; 0 2 0 ]]∈ f+^e.

Then 

    f_B(λ)=|λ I-B|=|[  λ -a -b; -2  λ -a;  0 -2  λ ]|=λ^3-4aλ-4b.
	
Notice that 

    X_L_k(sl_3)= G· (^*(h_1-h_2)+f_θ).
	
So B∈ X_W_k(sl_3) if and only if B is similar to

    A_μ=μ(h_1-h_2)+f_θ=[[ -μ  0  0;  0 2μ  0;  1  0 -μ ]].

for some μ∈.  Then it is easy to deduce that B∈ X_W_k(sl_3) if and only if
 
    a=3/4μ^2,  b=1/2μ^3.

 Thus
	
    X_W_k(sl_3)={[[      0 3/4μ^2 1/2μ^3;      2      0 3/4μ^2;      0      2      0 ]]|  μ∈}.


Let L(tΛ̅_1) and L(tΛ̅_2), t∈_≥ 0 be the irreducible ordinary modules of L_k(sl_3) as in the proof of Theorem <ref>, then by  Theorem 5.4 of <cit.>, 
for t∈_≥ 0, the principal  quantum hamiltonian reductions 𝒲_h_t+1, 1w_t+1,1 and  𝒲_h_1,t+1w_1,t+1
of L(tΛ̅_1) and L(tΛ̅_2) are  irreducible ordinary W_k(sl_3)-modules,  
where

    h_t+1,1=t^2+3t/3(k+3)=h_1,t+1,

and

    w_t+1,1=-2t√(3)/3(k+3)^3(2t/3-k-2)(t/3-k-2)=-w_1,t+1.

It follows  from Theorem <ref> that W_k(sl_3) is not quasi-lisse. 



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