Primitive ideals for W-algebras in type A

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  • Journal of Algebra 359 (2012) 8088

    Contents lists available at SciVerse ScienceDirect

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    rimitive ideals for W-algebras in type A

    an Losev

    rtheastern University, Department of Mathematics, 360 Huntington Avenue, Boston, MA 02115, United States

    r t i c l e i n f o a b s t r a c t

    ticle history:ceived 10 November 2011ailable online 28 March 2012mmunicated by J.T. Stafford

    SC:S99B35

    ywords:-algebrasimitive ideals

    In this note we classify the primitive ideals in nite W-algebras oftype A.

    2012 Elsevier Inc. All rights reserved.

    Introduction

    Let g be a semisimple Lie algebra over an algebraically closed eld K of characteristic 0 and e ga nilpotent element. Then to the pair (g, e) one can assign an associative algebra W called the

    -algebra. This algebra was dened in full generality by Premet in [P1]. Two equivalent denitionsW-algebras are provided in Section 2. For other details on W-algebras the reader is referred to theview [L5].One of the reasons to be interested in W are numerous connections between this algebra and theiversal enveloping algebra U of g. For example, the sets of primitive ideals Pr(W) and Pr(U) of Wd U are closely related. Recall that an ideal in an associative algebra is called primitive if it is thenihilator of some irreducible module. The structure of Pr(U) was studied extensively in 70s ands.One of manifestations of a relationship between Pr(W) and Pr(U) is a map I I : Pr(W)

    (U) constructed in [L1]. One can describe the image of this map. Namely, to each primitive ideal Pr(U) one assigns its associated variety V(U/J ). According to a theorem of Joseph, the associatedriety is the closure of a single nilpotent orbit in g = g. Thanks to [L1, Theorem 1.2.2(vii)], an

    Supported by the NSF grant DMS-0900907.E-mail address: i.loseu@neu.edu.

    21-8693/$ see front matter 2012 Elsevier Inc. All rights reserved.

    tp://dx.doi.org/10.1016/j.jalgebra.2012.03.010

  • I. Losev / Journal of Algebra 359 (2012) 8088 81

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    tioement J Pr(U) is of the form I for some I Pr(W) if and only if O V(U/J ), where O standsr the adjoint orbit of e.In general, the map is not injective. However, the following result holds.

    eorem 1.1. The map : Pr(W) Pr(U) is an injection provided g= sln.

    This theorem provides a classication of primitive ideals in W because the set of primitive ideals U with V(U/J ) =O is known thanks to the work of Joseph, [J].

    W-algebras and the map between ideals

    1. Quantum slice

    Let Y be an ane Poisson scheme equipped with a K-action such that the Poisson bracket hasgree 2. Let Ah be an associative at graded K[h]-algebra (where h has degree 1) such thath,Ah] h2Ah and K[Y ] = Ah/(h) as a graded Poisson algebra. Pick a point Y . Let I bee maximal ideal of in K[Y ] and let I be the inverse image of I in Ah . Consider the completionh := limnAh /In . This is a complete topological Kh-algebra with A

    h /(h) =K[Y ] , where

    the right-hand side we have the usual commutative completion. Moreover, as we have seen in [L4,mma A2],1 the algebra Ah is at over Kh.The cotangent space T Y = I/I2 comes equipped with a natural skew-symmetric form, say .

    x a maximal symplectic subspace V T Y . One can choose an embedding V I such that(u), (v)] = h2(u, v) and whose composition with the projection I T Y is the identity. Thisproved similarly to Proposition 3.3 in [Ka] (or can be deduced from that proposition, compare withe argument of Subsection 7.2 in [L3]). Consider the homogenized Weyl algebra Ah(V ) = T (V )[h]/ v v u h2(u, v)) and its completion A0h (V ) at zero. Similarly to the proof of Lemma 3.2[Ka], one can show that

    Ah = A0h (V ) Kh Ah,

    here Ah is the centralizer of V in Ah . We remark that the algebra Ah is complete with respect to

    e topology induced by its maximal ideal. The symbol stands for the completed tensor producttopological vector spaces.The argument in the proof of [L6, Proposition 6.6.1, Step 2], shows that any two embeddings

    , 2 : V I satisfying the conditions in the previous paragraph differ by an automorphism ofh of the form exp(

    1h2

    ad(z)) with z (I )3. In particular, the algebra Ah is dened uniquely upa Kh-linear isomorphism.Now let us consider a compatibility of our construction with certain derivations. Suppose Ah isuipped with a derivation D such that Dh = h. The derivation extends to Ah . According to [L4],ere is a derivation D of Ah with the following properties. First, D h = h. Second, we have D D =ad(a) for some element a Ah , where D means the derivation that equals D on Ah and acts byon the symplectic space V generating A0h (V ).An easy special case of the previous construction is when the derivation D comes from a K-tion on Ah preserving . Here we can choose a K-stable V and a K-equivariant : V Id so we get a K-action on Ah . Moreover, the algebra Ah is now dened uniquely up to a K-uivariant Kh-linear isomorphism.

    In [L4] we had an assumption that Y has only nitely many leaves but this assumption is not necessary for many construc-

    ns and results.

  • 82 I. Losev / Journal of Algebra 359 (2012) 8088

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    byNow let Z(Ah) denote the center of Ah . Consider a K-equivariant Kh-linear homomorphism: Z(Ah) K[h] and the corresponding central reduction A,h :=Ah/Ah ker. Until the end of thebsection we assume that lies in the spectrum of A,h/(h).Consider the induced homomorphism Z(Ah) Ah = A0h (V ) Kh Ah . The image is central.e center of A0h (V ) coincides with Kh, so the image of Z(Ah) is contained in Ah . Set A,h =h/Ah ker.Then we have the completion A

    ,h of A,h , and A,h = A

    h /A

    h ker. Furthermore, we have

    e following commutative diagram, where the horizontal arrows are isomorphisms and the verticalrows are the natural quotients.

    Ah A0h (V ) Kh Ah

    A,h A

    0h (V ) Kh A,h

    Now suppose that we have a reductive group Q that acts on Ah rationally by K[h]-algebra au-morphisms xing . Further, suppose that there is quantum comoment map A : q Ah , i.e., a-equivariant linear map with the property that [A(),a] = h2.a, where on the right-hand sideis the derivation of Ah coming from the Q -action. Composing A with a natural homomorphismh Ah we get a quantum comoment map qA

    h again denoted by

    A .We remark that we can choose V to be Q -stable. This gives rise to a Q -action on V by lin-r symplectomorphisms and hence to an action of Q on Ah(V ) by K[h]-linear algebra automor-isms. There is a quantum comoment map A : q Ah(V ) that is the composition q sp(V ) =V Ah(V ).Further, since Q is reductive, we can assume that the embedding : V Ah is Q -equivariant.we get a Q -action on Ah . Let us produce a quantum comoment map for this action. For q

    t () = A() A(). The quantum comoment map conditions for A,A imply that ()mmutes with Ah(V )0 and hence the image of is in Ah . Now it is clear that : q Ah is aantum comoment map for the action of Q on Ah .We remark that the Q -action and the map are dened by the previous construction uniquely upan isomorphism of the form exp( 1

    h2ad(z)), where, in addition to conditions mentioned above, z is

    -invariant. Also we remark that if we have a K-action on Ah as above but additionally commutingith Q and such that A() has degree 2 for each , then the Q -action on Ah also may be assumedcommute with K and () may be assumed to have degree 2.

    2. W-algebras via quantum slices

    We are going to consider a special case of the construction explained in the previous subsection.Let G be a simply connected semisimple algebraic group and let g be the Lie algebra of G . Set

    := g . We remark that we can identify g with g by means of the Killing form. Pick a nilpotentbit O g and an element e O. We take e for . Let us equip Y with a Kazhdan K-action denedfollows. Pick an sl2-triple (e,h, f ), where h is semisimple. Let :K G be the one-parameterbgroup corresponding to h. We dene a K-action on g by t. = t2 (t), g , t K . Wemark that t. = .For Ah we take the homogenized version Uh of the universal enveloping algebra dened by Uh :=

    (g)[h]/( y y h2[, y]). We can extend the Kazhdan action to Uh . Explicitly, for g with, ] = i we have t. = ti+2 and we set t.h := th.Apply the construction of the previous subsection to Ah, . We get the Kh-algebra Ah acted on

    0

    K together with a K -equivariant isomorphism Ah = Ah (V ) Kh Ah . Here V has the same

  • I. Losev / Journal of Algebra 359 (2012) 8088 83

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    (3(4(5eaning as before but we can describe it explicitly: namely, for V we take the subspace [g, f ] g= Y .Let Ah denote the subalgebra of all K-nite vectors in Ah . It turns out that the algebra Ah/hAh ,

    , more precisely, the corresponding variety is well known in the theory of nilpotent orbits this isso-called Slodowy slice. In more detail, set S := e + zg( f ) and view S as a subvariety in g via theentication g = g . Then S is K-stable and is transverse to Ge at e: g = Te S TeGe. Moreover,e Kazhdan action contracts S to e, meaning that limt t.s = e for all s S . This implies that S isansversal to any G-orbit it intersects g = Ts S + TsGs for all s S . Also the contraction propertyeans that the grading on K[S] induced by the K-action is positive: there are no negative degrees,d the only elements in degree 0 are constants.The contraction property for the K-action on S implies that the subalgebra of K[S] =Ah/hAhnsisting of the K-nite elements coincides with K[S]. Moreover, since the degree of h is positive,is implies that Ah/hAh =K[S].We set A :=Ah/(h 1)Ah . This is a ltered associative algebra whose associated graded is K[S].e algebra Ah can be recovered as the Rees algebra of A, while Ah is the completion A

    h .

    Let us remark that the algebra A comes equipped with a homomorphism Z A, where Z ise center of the universal enveloping algebra U(= Uh/(h 1)Uh) of g. Indeed, set Zh := UGh . Thisthe center of Uh . Consider Zh as a subalgebra of Ah . According to the previous subsection, weve Zh Ah . It is easy to see that Zh consists of K-nite vectors so Zh Ah . This gives risean embedding Z A. We remark that this embedding does not depend on the choice (thatthe embedding V I ) we have made. This is because any two choices are conjugate by antomorphism that commutes with all elements of the center.Consider the subgroup Q := ZG(e,h, f ) G . This group acts on Uh and stabilizes (and S as

    ell). The Q -action commutes with K and there is a quantum comoment map q Ah whoseage consists of functions of degree 2 with respect to the K-action. So we get a Q -action on Ahwell as a quantum comoment map q Ah . Both the action and the quantum comoment mapscend to A.

    3. Equivalence with a previous denition

    Recall that the cotangent bundle T G carries a natural symplectic form . This form is invariantith respect to the natural G G-action. Moreover, for the K-action by berwise dilations we have

    = t1, t K .We remark that we can trivialize T G by using left-invariant 1-forms hence T G = Gg . Consideras a subvariety in g and set X := G S . It turns out that the subvariety X T G is symplectic. Iteasy to see that X is stable with respect to the left G-action, as well as to the Kazhdan K-actionT G given by t.(g,) = (g (t)1, t2 (t)). Clearly, has degree 2 with respect to the Kazhdan

    tion. Also Q acts on T G by q.(g,) = (gq1,q) and X is Q -stable.We remark that G : T G g , G(g,) = g is a moment map, i.e., a G-equivariant map such

    at for any g the derivation {G(), } of K[T G] coincides with the derivation produced by a the G-action. Similarly, Q : T G q , Q (x,) = |q is a moment map.In [L1] the author proved that there is an associative product on K[X][h], where h is an inde-ndent variable, satisfying the following properties:

    ) is G K-equivariant, where G K acts on K[X] as usual, g.h = h, t.h = th.) For f , g K[X] we have f g =i=0 Di( f , g)h2i , where Di is a bi-differential operator of orderat most i.

    ) f g f g mod h2.) f g g f h2{ f , g} mod h4.

    ) The maps G : gK[X][h], Q : qK[X][h] are quantum comoment maps.

  • 84 I. Losev / Journal of Algebra 359 (2012) 8088

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    stFogride last property was established in [L2]. By denition, the W-algebra W is the quotient of thevariant subalgebra K[X][h]G by h 1. The quantum comoment map : g K[X][h] gives rise toe homomorphism Z = U (g)G W .

    oposition 2.1. We have a ltration preserving Q -equivariant isomorphism W A intertwining the em-ddings of Z and the quantum comoment maps from q.

    oof. Similarly to the above we have a star-product on T G having the properties analogous to)(5). Set x := (1,) X T G . Consider the completions K[T G]hGx ,K[X]hGx of the cor-sponding algebras (with respect to star-products) at the ideals of Gx. According to Theorem 2.3.1om [L2], we have a G K-equivariant (where we consider the Kazhdan K-actions) topologicalh-algebra isomorphism

    K[T G

    ]hGx A0h (V ) Kh K[X]hGx ,

    d this isomorphism intertwines the quantum comoment maps for the G-action and Q -actionhe Weyl algebra component of the quantum comoment map for G on the right-hand side is 0).e algebra of G-invariants of the left-hand side is Uh , while on the right-hand side we get0

    h (V ) Kh Wh , see [L2]. So we can take W

    h for Ah . It follows that we have Q -equivariant

    omorphisms Wh =Ah of graded K[h]-algebras and W =A of ltered algebras. Both isomorphismstertwine the quantum comoment maps from q. Moreover, the embedding UGh Uh =K[T G]hGduced by the quantum comoment map is just the inclusion. This completes the proof of the propo-tion. 4. Map between the set of ideals

    Let us construct the map mentioned in Theorem 1.1. We will start with the general settingplained in Section 2.1.Consider the set Idh(Ah) of all K-stable h-saturated ideals Jh Ah , where h-saturated means

    at Ah/Jh is at over K[h]. Similarly, consider the set Idh(Ah) of all D -stable h-saturated ideals inh . The discussion of D

    in Section 2.1 implies that an h-saturated I h Ah is D -stable if and only if0h Kh I h A

    h is D-stable. In particular, the set Idh(Ah) does not depend on the choice of D .

    We have maps between Idh(Ah) and Idh(Ah) constructed as follows. Take an ideal Jh Ah andrm its closure J h A

    h . This ideal is D-stable but also one can check that it is actually h-

    turated. As such, the ideal J h has the form A0h Kh I h for a unique two-sided ideal I h in Ah .e ideal I h is automatically D -stable and h-saturated. We consider the map : Idh(Ah) Idh(Ah)nding Jh to I h .Let us produce a map in the opposite direction. Take I h Idh(Ah). Then Jh :=Ah A0h Kh I ha K-stable h-saturated ideal in Ah . Consider the map : Idh(Ah) Idh(Ah) sending I h to Jh .Now suppose that the grading on K[Y ] induced by the K-action is positive. Then Idh(Ah) is in

    jection with the set Id(A) of two-sided ideals in A :=Ah/(h 1). Under this bijection, the ideal incorresponding to Jh Idh(Ah) is Jh/(h 1)Jh .Similarly, suppose that D is also induced from some K-action such that Ah is the projective limitsome positively graded algebras (this is the case in the situation considered in Section 2.2). Thene have natural identications Idh(Ah) = Idh(Ah) = Id(A). So we have maps between Id(A),Id(A)at still will be denoted by ,.In a special case we have some additional information about the maps ,. Suppose that Y is

    ill equipped with a contracting K-action and, moreover, has only nitely many symplectic leaves.r a symplectic leaf L let IdL(A) denote the subset of Id(A) consisting of all ideals J such that(A/J ) is supported on the closure of L. The maximal elements in IdL(A) are precisely prime

    eals. By [L4], any prime ideal in IdL(A) is primitive.

  • I. Losev / Journal of Algebra 359 (2012) 8088 85

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    coNow let L be the leaf containing . Then denes a map IdL(A) Idh, f in(Ah), where, bynition, the target set consists of all D -stable ideals I h such that Ah/I h is free of nite ranker Kh. For a prime ideal J IdL(A) and any minimal prime ideal I h of J we have J = (I h),e [L4, Lemma A4]. Corollary 3.17 from [ES] implies that there are nitely many prime ideals inh, f in(Ah) and so IdL(A) also contains nitely many prime ideals.We are interested in the special case when A is a central reduction of U at some central charac-

    r . Consider a unique K-equivariant K[h]-linear homomorphism Z(Uh) K[h] specializing to h = 1. This homomorphism will also be denoted by . So Ah = U,h is the Rees algebra of U . Thederlying variety Y is the nilpotent cone N of g and so contains nitely many symplectic leavesnilpotent orbits). Using the usual K-action we get an identication Id(U) = Idh(U,h). Simi-

    rly, using the Kazhdan action we get an identication Id(W) = Idh(W,h). We remark that thezhdan action differs from the usual one by inner automorphisms and so an h-saturated ideal in

    ,h is stable under the usual K-action if and only if it is stable under the Kazhdan K-action.nce the Kazhdan action on S is contracting, we get Idh(W,h) = Idh(W,h ).So we get maps : Id(U) Id(W), : Id(W) Id(U) that rst appeared in [L1]. Theiroperties are summarized in the following proposition.For a nilpotent orbit O let PrO (U) denote the subset in Pr(U) consisting of all ideals J such

    at the associated variety of U/J is O . Similarly, for a locally closed subvariety L S we canne a subset PrL (W) Pr(W).

    oposition 2.2. Let g= sln.) The sets Pr(W),Pr(U) are nite.) The map : Pr(W) Pr(U) is surjective. More precisely, for J PrO (U) there is I PrL(W)with I = J . Here L is an irreducible component of S O (in fact, below we will see that S O isirreducible).

    ) The restriction of to Pr (W) is a bijection Pr (W) PrO(U).

    We remark that sends Pr(W) to Pr(U). This follows, for example, from [L1, Theorem 1.2.2(ii)].

    oof. The rst claim for U is well known. For W it follows from the main result of [L4]. Indeed,e associated graded algebra grW is nothing else but K[SN ], where N is the nilpotent cone in g.e Poisson variety S N has nitely many symplectic leaves (according to Proposition 3.1 below hich is independent of the present proposition these leaves are in bijection with nilpotent orbits1 g with OO1). So the main theorem of [L4] does apply to W .Assertion (2) follows from [L1, Theorem 1.2.2(vii)]. In fact, the associated variety of any primitive

    eal in W is irreducible thanks to a theorem of Ginzburg, [Gi].Finally (3) follows from [L2, Conjecture 1.2.1] (proved in that paper) because the action of themponent group on Pr (W) is trivial. We remark that Pr(U) = Pr(U), where the union is taken over all central characters. A similar

    aim holds for W . The map : Pr(W) Pr(U) is obtained from the maps : Pr(W) Pr(U).Proof of the main theorem

    1. Classical level

    Here we are going to prove a quasi-classical analog of Theorem 1.1 that seems to be of somedependent interest. Instead of primitive ideals in associative algebras we will consider symplecticaves of the corresponding Poisson varieties.From the description of the Poisson structure on S given in [GG], symplectic leaves are irreduciblemponents of S O1, where O1 are (co)adjoint orbits in g= g .

    The main result of this section characterizes nilpotent symplectic leaves of S .

  • 86 I. Losev / Journal of Algebra 359 (2012) 8088

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    seinoposition 3.1. Suppose g = sln. Let O1 be a nilpotent orbit with O O1 . Then the intersection O1 S isreducible.

    oof. We will check that the intersection S O1 is normal. Since S O1 is K-stable and hencennected, the normality implies that S O1 is irreducible. Being an open subvariety in S O1, theriety S O1 is also irreducible.Again, thanks to the contracting K-action it is enough to show that the completion (S O1)ethe point e is normal. Recall that the intersection S O is transversal at e. This implies that Oe1composes into the direct product Oe (S O1)e . According to Kraft and Procesi, [KP], the variety1 is normal. Hence the formal scheme O

    e1 is normal as well. The direct product decomposition now

    plies that (S O1)e is normal. mark 3.2. In fact, the techniques used below to prove Theorem 1.1 allow one to show that S O1irreducible for any (not necessarily nilpotent) orbit O1.

    2. Quantum level

    Pick some nilpotent orbit O g whose closure contains O, let O S , and let W be therresponding W-algebra. We claim that the inequalities

    PrOS(W)

    Pr (W) (3.1)

    or all possible O) imply that the map sends PrOS (W) to PrO(U) and is a bijection between

    e two sets.Indeed, thanks to Proposition 2.2(3), the map : Pr (W) PrO(U) is a bijection. On the othernd, the preimage of J Pr

    O(U) in Pr(W) contains at least one element from PrOS (W), asser-

    on (2). Finally, by assertion (1), both sets Pr(W),Pr(U) are nite. So inequalities (3.1) imply theaim of the previous paragraph.To prove the inequality |Pr

    OS (W)| |Pr (W)| we will apply the general construction of Sec-on 2.1 to Y = SN ,A=W and the point SO. Let Ah be the quantum slice algebra producedthat construction.

    mma 3.3.Ah = W,h , where W,h is the homogenized version (= the Rees algebra) of the central reduction

    of W .

    oof. Set V1 := T (S O) and let V2 be the skew-orthogonal complement of V1 in T O. We canrm the corresponding completed homogenized Weyl algebras Ah(V1)0 ,Ah(V2)0 . Then we have

    U,h = Ah(V1 V2)0 Kh W

    ,h , (3.2)

    W,h = Ah(V1)0 Kh Ah. (3.3)

    .2) follows from the construction of the W-algebras, while (3.3) is essentially a denition of Ah .Applying the construction used in the proof of Proposition 2.1 to the point (1, ) G S T G we

    e that Uh = Ah(V2)0 KhWh . From here, the description of the embedding Z W provided

    Section 2.3 and the commutative diagram in Section 2.1 one can see that

    0 U,h

    = Ah(V2) Kh W,h . (3.4)

  • I. Losev / Journal of Algebra 359 (2012) 8088 87

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    Regicombining (3.3) and (3.4), we get

    U,h

    = Ah(V1 V2)0 Kh Ah. (3.5)

    om Section 2.1, we see that there is a Kh-linear isomorphism W,h

    =Ah . Now we have two derivations of W,h , the derivation D induced by the Kazhdan action dened for

    e nilpotent element , and the derivation D coming from an isomorphism W,h

    =Ah . Both satisfyh = D h = h. Consider the sets Prn,h(W,h ),Prn,h(W

    ,h ) that consist of all prime (= maximal)

    saturated ideals I h W,h such that W

    ,h/I h is of nite rank over Kh and such that I h is,

    spectively, D- and D -stable. The set Prn,h(W1h) is in natural bijection with Prn(W). On theher hand, by the results recalled in Section 2.4, the cardinality of Prn,h(W

    ,h ) is bigger than or

    ual to that of PrSO(W). So it remains to show that the two sets coincide.It is enough to check that any derivation d of Ah = W

    ,h with d(h) = h xes any maximal

    saturated ideal of nite corank. Consider the quotient (Ah)(n) of Ah by the ideal generated bye elements s2n(x1, . . . , x2n) =S2n sgn( )x(1) . . . x(2n), x1, . . . , x2n Ah . This ideal is clearly d-able. Also consider the analogous quotient A(n)h of Ah := W,h . It follows from Section 7.2 of [L3]at A(n)h has nite rank over Kh. But (Ah)(n) is the completion of A(n)h at . So (Ah)(n) has nitenk over Kh. Therefore the localization (Ah)(n)[h1] is a nite dimensional K[h1, h-algebra.Maximal h-saturated ideals of nite corank in Ah are in a natural one-to-one correspondence

    ith maximal ideals of nite codimension in the K[h1, h-algebra Ah[h1]. Clearly Ah[h1](n) =h)

    (n)[h1]. Thanks to the AmitsurLevitzki theorem, every maximal ideal of nite codimension inh[h1] is the preimage of an ideal in Ah[h1](n) for some n. Of course, d induces a K[h1, h-linearrivation of Ah[h1](n) . Now it remains to use a fact that a maximal ideal in a nite dimensionalgebra is stable under any derivation of this algebra. For readers convenience we will provide a proofre.Let A be a nite dimensional algebra over some eld K and let m be a maximal ideal of A.placing A with A/

    i=1mi , we may assume that m is a nilpotent ideal and hence the radical of A.

    complete the proof apply Lemma 3.3.3 from [D].

    3. Proof of Theorem 1.1

    For readers convenience, let us summarize the proof of the theorem. The map : Pr(W) = Pr(W) Pr(U) =

    Pr(U) maps Pr(W) to Pr(U). The sets Pr(U),Pr(W) are nite. More-

    er, every ideal J PrO(U) has the form I for I PrL (W), where L is an irreducible compo-

    nt of O S (provided, of course, O is contained in the closure of O). See Proposition 2.2 for all theseaims. According to Proposition 3.1, O S is irreducible. So Theorem 1.1 will follow if one checksat the cardinalities of Pr

    O(U) and PrOS (W) coincide, while from the above one only knows that

    rO(U)| |PrOS (W)|. But, again according to Proposition 2.2, PrO(U) = Pr (W), where W is

    e W-algebra constructed from O. So one only needs to show |Pr (W)| |PrOS (W)|. This isne in two steps. Lemma 3.3 together with general results of Section 2.4 produces a surjection fromcertain set of ideals in W

    ,h to PrOS (W). The discussion after the lemma shows that the set ofeals of W

    ,h under consideration is, in fact, isomorphic to Pr (W).

    mark 3.4. Let us nish by explaining where exactly we use the assumption that g= sln . First, for aven orbit O g we want the intersections SO to be irreducible for all orbits O intersecting S . One

    ndition that guarantees the irreducibility is the normality of the closure of O that often fails outside

  • 88 I. Losev / Journal of Algebra 359 (2012) 8088

    type A. Another feature of type A that we need is that the centralizers of nilpotent elements areconnected (modulo the center of the whole group), this guarantees |Pr (W)| = |PrO(U)|. Outsidetype A, the centralizers are often non-connected.

    Acknowledgments

    I would like to thank Jon Brundan for communicating this problem to me.

    References

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    Primitive ideals for W-algebras in type A1 Introduction2 W-algebras and the map between ideals2.1 Quantum slice2.2 W-algebras via quantum slices2.3 Equivalence with a previous denition2.4 Map between the set of ideals

    3 Proof of the main theorem3.1 Classical level3.2 Quantum level3.3 Proof of Theorem 1.1

    AcknowledgmentsReferences