SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS

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1 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS TOMOYUKI ARAKAWA AND ANNE MOREAU Abstract. We show that sheet closures appear as associated varieties o aine vertex algebras. We also provide examples o associated varieties that are union o distinct sheet closures, which in particular shows that the associated varieties o vertex algebras need not to be irreducible. Further, we give new examples o non-admissible aine vertex algebras whose associated variety is contained in the nilpotent cone. We also prove some conjectures rom our previous paper and give new examples o lisse aine W -algebras. 1. Introduction It is known [Li05] that every vertex algebra V is canonically iltered and thereore it can be considered as a quantization o its associated graded Poisson vertex algebra gr V. The generating subring R V o gr V is called the Zhu s C 2 -algebra o V [Z96] and has the structure o a Poisson algebra. Its spectrum X V = Spec R V is called the associated scheme o V and the corresponding reduced scheme X V = Specm R V is called the associated variety o V ([Ar12, Ar16b]). Since it is Poisson, the coordinate ring o its arc space J X has a natural structure o a Poisson vertex algebra ([Ar12]), and there is a natural surjective homomorphism C[J XV ] gr V, which is in many cases an isomorphism. We have [Ar12] dim Spec(gr V ) = 0 i and only i dim X V = 0, and in this case V is called lisse or C 2 -coinite. In the case that V is the simple aine vertex algebra V k (g) associated with a inite-dimensional simple Lie algebra g at level k C, X V is a Poisson subscheme o g which is G-invariant and conic, where G is the adjoint group o g. Note that on the contrary to the associated variety o a primitive ideal o U(g), the variety X Vk (g) is not necessarily contained in the nilpotent cone N o g. In act, X Vk (g) = g or a generic k. On the other hand X Vk (g) = {0} i and only i V k (g) is integrable, that is, k is a non-negative integer. Except or a ew cases, the description o X V is airly open even or V = V k (g), although this problem seems to be signiicant in connection with our dimensional superconormal ield theory ([BLL + 15]). In [Ar15a], the irst named author showed that X Vk (g) is the closure o some nilpotent orbit o g in the case that V k (g) is admissible [KW89]. In the previous article [AM15], we showed that X Vk (g) is the minimal nilpotent orbit closure in the case that g belongs to the Deligne exceptional series [De96] and k = h /6 1, where h is the dual Coxeter number o g. Note that the level k = h /6 1 is not admissible or types D 4, E 6, E 7, E Mathematics Subject Classiication. 17B67, 17B69, 81R10. Key words and phrases. sheet, nilpotent orbit, associated variety, aine Kac-Moody algebra, aine vertex algebra, aine W -algebra. 1

2 2 TOMOYUKI ARAKAWA AND ANNE MOREAU In all the above cases X Vk (g) is a closure o a nilpotent orbit O N, or (X Vk (g)) red = g. Thereore it is natural to ask the ollowing. Question 1. Are there cases when X Vk (g) N and X Vk (g) is a proper subvariety o g? For example, are there cases when X Vk (g) is the closure o a non-nilpotent Jordan class (c. 2)? Identiy g with g through a non-degenerate bilinear orm o g. Given m N, let g (m) be the set o elements x g such that dim g x = m, with g x the centralizer o x in g. A subset S g is called a sheet o g i it is an irreducible component o one o the locally closed sets g (m). It is G-invariant and conic and by [ImH05], and it is smooth i g is classical. The sheet closures are the closures o certain Jordan classes and they are parameterized by the G-conjugacy classes o pairs (l, O l ) where l is a Levi subalgebra o g and O l is a rigid nilpotent orbit o l, i.e., which cannot be properly induced in the sense o Lusztig-Spaltenstein [BK79, Bo81] (see also [TY05, 39]). The pair (l, O l ) is called the datum o the corresponding sheet. When O l is zero, the sheet is called Dixmier, meaning that it contains a semisimple element [Di75, Di76]. We will denote by S l the sheet with datum (l, {0}). We reer to 2 or more details about this topic. It is known that sheets appear in the representation theory o inite-dimensional Lie algebras, see, e.g., [BB82, BB85, BB89], and more recently o inite W -algebras, [PT14, Pr14]. Since the sheet closures are G-invariant, conic algebraic varieties which are not necessarily contained in N, one may expect that there are simple aine vertex algebras whose associated variety is the closure o some sheet. This is indeed the case. Theorem 1.1. (1) For n 4, X V 1(sl n) = S l1 as schemes, where l 1 is the standard Levi subalgebra o sl n generated by all simple roots except α 1. Moreover V 1 (sl n ) is a quantization o the ininite jet scheme J S l1 o S l1, that is, as Poisson vertex algebras. (2) For m 2, gr V 1 (sl n ) = C[J S l1 ] X V m(sl 2m) = S l0 as schemes, where l 0 is the standard Levi subalgebra o sl 2m generated by all simple roots except α m. Moreover, V m (sl 2m ) is a quantization o J S l0, that is, as Poisson vertex algebras. gr V m (sl 2m ) = C[J S l0 ] Further, we show that the Zhu s algebras o the above vertex algebras are naturally embedded into the algebra o global dierential operators on Y = G/(P, P ), where P is the connected parabolic subgroup o G corresponding to a parabolic subalgebra p having the above Levi subalgebra l 1 in case (1) and l 0 in case (2) as Levi actor, see Theorem 7.14 and Theorem 8.13.

3 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 3 The vertex algebra V 1 (sl n ) has appeared in the work o Adamović and Perše [AP14], where they studied the usion rules and the complete reducibility o V 1 (sl n )- modules. It would be very interesting to know whether the vertex algebra V m (sl 2m ) has similar properties. Now recall that the associated variety o primitive ideals o U(g) is irreducible [Jo85]. Hence it is also natural to ask the ollowing question. Question 2. Is X Vk (g) always irreducible? Theorem 1.2. Let r be an odd integer, and let l I and l II be the Levi subalgebras o g = so 2r generated by the simple roots α 1,..., α r 2, α r and α 1,..., α r 2, α r 1 respectively. They are non G-conjugate and X V2 r(so 2r) = S l I S l II. In particular the associated variety X V2 r(so 2r) is reducible. We conjecture that X Vk (g) is always equidimensional, and that X Vk (g) is irreducible provided that X Vk (g) N, see Conjecture 1. The vertex algebra V 2 r (so 2r ) has been studied by Perše [Pe13] or all r, and by Adamović and Perše [AP14] or odd r. The proo o Theorem 1.2 uses the act proved in [AP14] that, or odd r, V 2 r (so 2r ) has ininitely many simple objects in the category O. Remarkably, it turned out that the structure o the vertex algebra V 2 r (so 2r ) substantially diers depending on the parity o r. Theorem 1.3. Let r be an even integer such that r 6. Then O min X V2 r(so 2r) O (2 r 2,1 4 ), where O min is the minimal nilpotent orbit o so 2r and O (2 r 2,1 4 ) is the nilpotent orbit o so 2r associated with the partition (2 r 2, 1 4 ) o 2r. In particular, X V2 r(so 2r) is contained in N, and hence, there are only initely many simple V 2 r (g)-modules in the category O. The above theorem gives new examples o non-admissible aine vertex algebras whose associated varieties are contained in the nilpotent cone (c. [AM15]). In act we conjecture 1 that X V2 r(so 2r) = O (2 r 2,1 4 ). This conjecture is conirmed or r = 6, see Theorem 9.6. Notice that or r = 4, X V 2(so 8) = O min = O (2 2,1 4 ) by [AM15]. So the conjecture also holds or r = 4. Our proo o the above stated results is based on the analysis o singular vectors o degree 2 [AM15] and the theory o W -algebras [KRW03, KW04, Ar05, Gi09, Ar11, Ar15a]. This method works or some other types as well, in particular in types B and C, which will be studied in our subsequent paper. We also take the opportunity o this note to clariy some points o [AM15] that are related to the present work. Let us state here the main results. By [Ar15a, Theorem 4.23] (c. Theorem 6.1) we know that the W -algebra W k (g, ) associated with (g, ) at level k ([KRW03]) is lisse i X Vk (g) = G.. For a minimal nilpotent element O min, the converse is also true provided that k Z 0. Theorem 1.4. Suppose that k Z 0, and let O min. W -algebra W k (g, ) is lisse i and only i (X Vk (g)) red = O min. 1 This conjecture is now conirmed in our paper arxiv: Then the minimal

4 4 TOMOYUKI ARAKAWA AND ANNE MOREAU We also positively answer some conjectures o [AM15]. In particular, we show the ollowing. Theorem 1.5. Let g o type G 2. Then W k (g, θ ) is lisse i and only i k is admissible with denominator 3, or an integer equal to or greater than 1. Thus, we obtain a new amily o lisse minimal W -algebras W k (G 2, θ ), or k = 1, 0, 1, 2, Theorem 1.6. Suppose that g is not o type A. holds. That is, X Vk (g) = O min Then Conjecture 2 o [AM15] i and only i the one o the ollowing conditions holds: (1) g is o type C r (r 2), F 4, and k is admissible with denominator 2. (2) g is o type G 2, and k is admissible with denominator 3, or k = 1. (3) g is o type D 4, E 6, E 7, E 8 and k is an integer such that h 6 1 k 1. (4) g is o type D r with r 5, and k = 2, 1. The rest o the paper is organized as ollows. In 2 we recollect some results concerning sheets that will be needed later and give a description o Dixmier sheets o rank one. In 3, we use Slodowy slices and Ginzburg s results on inite W -algebras to state useul lemmas. In 4 we state results and conjectures on the associated variety o a vertex algebra. In 5 we recall some undamental results on Zhu s algebras o vertex algebras. In 6 we recall and state some undamental results on W -algebras. In 7 we study level 1 aine vertex algebras o type A n 1, n 4, and prove Theorem 1.1 (1). In 8 we study level m aine vertex algebras o type A 2m 1, m 2, and prove Theorem 1.1 (2). In 9 we study level 2 r aine vertex algebars o type D 2r, r 5, and prove Theorem 1.2 and Theorem 1.3. In 10, we prove Theorem 1.4, Theorem 1.5 and Theorem 1.6 and some other results related to our previous work [AM15]. In particular we obtain a new amily o lisse minimal W -algebras. Notations. As a rule, or U a g-submodule o S(g), we shall denote by I U the ideal o S(g) generated by U, and or I an ideal in S(g) = C[g ], we shall denote by V (I) the zero locus o I in g. Let ĝ = g[t, t 1 ] CK CD be the aine Kac-Moody Lie algebra associated with g and the inner product ( ) = 1/2h Killing orm (see 4). For λ h (resp. ĥ ), L g (λ) (resp. L(λ)) denotes the irreducible highest weight representation o g (resp. ĝ) with highest weight λ where h is a Cartan subalgebra o g and ĥ = h CK CD. Acknowledgments. We are indebted to the CIRM Luminy or its hospitality during our stay as Research in pairs in October, Some part o the work was done while we were at the conerence Representation Theory XIV, Dubrovnik, June We are grateul to the organizers o the conerence. Results in this paper were presented by the irst named author in part in Vertex operator algebra and related topics, Chendgu, September, He thanks the

5 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 5 organizers o this conerences. He is also grateul to Universidade de São Paulo or its hospitality during his stay in November and December, His research is supported by JSPS KAKENHI Grant Numbers and The second named author would like to thank Michaël Bulois or helpul comments about sheets. Her research is supported by the ANR Project GeoLie Grant number ANR-15-CE Jordan classes and sheets Most o results presented in this section come rom [BK79, Bo81] or [Kat82]. Our main reerence or basics about Jordan classes and sheets is [TY05, 39]. Let g be a simple Lie algebra over C and ( ) = 1/2h Killing orm, as in the introduction. We oten identiy g with g via ( ). For a a subalgebra o g, denote by z(a) its center. For Y a subset o g, denote by Y reg the set o y Y or which g y has the minimal dimension with g y the centralizer o y in g. In particular, i l is a Levi subalgebra o g, then z(l) reg := {y g z(g y ) = z(l)}, and z(l) reg is a dense open subset o z(l). For x g, denote by x s and x n the semisimple and the nilpotent components o x respectively. The Jordan class o x is J G (x) := G.(z(g xs ) reg + x n ). It is a G-invariant, irreducible, and locally closed subset o g. To a Jordan class J, we associate its datum which is the pair (l, O l ) deined as ollows. Pick x J. Then l is the Levi subalgebra g xs and O l is the nilpotent orbit in l o 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 o pairs (l, O l ) as above, up to G-conjugacy, and the set o Jordan classes. A sheet is an irreducible component o the subsets g (m) = {x g dim g x = m}, m N. It is a inite disjoint union o Jordan classes. So a sheet S contains a unique dense open Jordan class J and we can deine the datum o S as the datum (l, O l ) o the Jordan class J. We have S = J and S = (J) reg. A sheet is called Dixmier i it contains a semisimple element o g. A sheet S with datum (l, O l ) is Dixmier i and only i O l = {0}. We shall simply denote by S l the Dixmier sheet with datum (l, {0}). A nilpotent orbit is called rigid i it cannot be properly induced in the sense o Lusztig-Spaltenstein. A Jordan class with datum (l, O l ) is a sheet i and only i O l is rigid in l. So we get a one-to-one correspondence between the set o pairs (l, O l ), up to G-conjugacy, with l a Levi subalgebra o g and O l a rigid nilpotent orbit o l, and the set o sheets. Each sheet contains a unique nilpotent orbit. Namely, i S is a sheet with datum (l, O l ) then the induced nilpotent orbit Ind g l (O l) o g rom O l in l is the unique nilpotent orbit contained in S. Note that a nilpotent orbit O is itsel a sheet i and only i O is rigid. For instance, outside the type A, the minimal nilpotent orbit O min is always a sheet.

6 6 TOMOYUKI ARAKAWA AND ANNE MOREAU The rank o a sheet S with datum (l, O l ) is by deinition rank(s) := dim S dim Ind g l (O l) = dim z(l). I S = S l is Dixmier, then O l = 0 and we have S = G.[p, p] = G.(z(l) + p u ) and S = (G.[p, p] ) reg. where p = l p u is a parabolic subalgebra o g with Levi actor l and nilradical p u (c. [TY05, Proposition ]). Let h be a Cartan subalgebra o g. For S a sheet with datum (l, O l ), one can assume without loss o generality that h is a Cartan subalgebra o l. In particular, z(l) h. Lemma 2.1. (1) Let S l be a Dixmier sheet o rank one, that is, z(l) = Cλ with λ h \ {0}. Then and S l = G.C λ = G.(Cλ + p u ) = G.C λ Ind g l (0), S l = G.C λ Ind g l (0). (2) Let S l1,..., S ln be Dixmier sheets o rank one, that is, z(l i ) = Cλ i with λ i h \ {0}, such that dim Ind g l i (0) = dim Ind g l j (0) or all i, j. Let X be a G-invariant, conic, Zariski closed subset o g such that n n X h = Cλ i, X N Ind g l i (0). Then X = n i=1 S l i. i=1 Part (1) o the lemma is probably well-known. We give a proo or the convenience o the reader. Proo. (1) The equalities S l = G.C λ = G.(Cλ + p u ) are clear by [TY05, Corollaries and ]. Let us prove that S l = G.C λ Ind g l (0). The inclusion G.C λ Ind g l (0) S l is known [TY05, Proposition ] (and its proo). So it suices to prove that Cλ + p u G.C λ Ind g l (0) since S is G-invariant. Let x = cλ + y Cλ + p u with c C and y p u. Assume that c C. Then x s and cλ are G-conjugate. Since x S l, dim g x dim g λ. But dim g x dim g λ i and only i x n = 0 since g x = (g xs ) xn. Hence x is G-conjugate to cλ, and so x G.C λ. I c = 0, then x p u and so x is nilpotent. But (Cλ + p u ) N S l N = Ind g l (0), whence the statement. It remains to prove that S l = G.C λ Ind g l (0). We have S l = (G.(Cλ + p u )) reg, and the inclusion G.C λ Ind g l (0) S l is clear. So it suices to prove that (Cλ + p u ) reg G.C λ Ind g l (0) since S l and S l are G-invariant. The above argument shows that or x (Cλ + p u ) reg \ p u, x G.C λ. And i x (p u ) reg, then x (p u ) reg N S l N = Ind g l (0), whence the statement. (2) The inclusion i S l i X is clear. Conversely, let x X. I x is nilpotent, then x i S l i by the assumption. Assume that x is not nilpotent, that is x s 0. Since X is G-stable, we can assume that x s h. I x n = 0 then x s X h n i=1 S l i by hypothesis. i=1

7 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 7 Assume that x n 0 and let (e, h, ) be an sl 2 -triple o g with e = x n. We can assume that h h so that [h, x s ] = 0. Let γ : C G be the one-parameter subgroup generated by h. Since X is G-invariant, or any t C, the element γ(t).x = x s + t 2 x n belongs to X. Since X is closed, we deduce that x s X. So, by the assumption, x s = cλ i or some i and c C. Thereore, because X is a cone, we can assume that x s = λ i. Thus l i = g xs and x n l i. For any t C, the element t 2 γ(t 1 ).x = t 2 (λ i + t 2 x n ) = t 2 λ i + x n belongs to X. This shows that C λ i + x n X. Then G.(C λ i + x n ) = G.(z(l i ) reg + x n ) = J G (x) X, whence J G (x) X because X is closed. Let O li,x n be the nilpotent orbit o x n in l i. One knows that Ind g l i (O li,x n ) J G (x) [Bo81]. So Ind g l i (O li,x n ) X, and the assumption gives that Ind g l i (O li,x n ) j Indg l j (0). In particular, dim Ind g l i (O li,x n ) = dim Ind g l j (0) or any j (this makes sense by our assumption on the Levi subalgebras l j ), whence codim li (O li,x n ) = codim lj (0) = codim li (0) = dim l i by the properties o induced nilpotent orbits. So O li,x n = {0}, that is x n = 0, and x = λ i S li. Let P be the connected parabolic subgroup o G with Lie algebra p = l p u. The G-action on Y := G/(P, P ), where (P, P ) is the commutator-subgroup o P, induces an algebra homomorphism rom U(g) to the algebra D Y ψ Y : U(g) D Y o global dierential operators on Y. Let J Y := ker ψ Y be the kernel o this homomorphism. It is a two-sided ideal o U(g). It has been shown by Borho and Brylinski [BB82, Corollary 3.11 and Theorem 4.6] that grjy is the deining ideal o the Dixmier sheet closure determined by P, that is, S l. Furthermore, J Y = Ann U(g) U(p) C λ. λ z(l) Here, or λ p, C λ stands or the one-dimensional representation o p corresponding to λ, and we extend a linear orm λ z(l) to p by setting λ(x) = 0 or x [l, l] p u. Identiying g with g through ( ), z(l) identiies with z(l). In particular, i z(l) = Cλ or some nonzero semisimple element λ g, we get J Y = t C Ann U(g) U(p) C tλ. In act (1) J Y = Ann U(g) U(p) C tλ. t Z

8 8 TOMOYUKI ARAKAWA AND ANNE MOREAU or any Zariski dense subset Z o C ([BJ77]). In this paper, we shall consider sheets in Lie algebras o classical types A r and D r. Let us introduce more speciic notations. Let n N, and denote by P(n) the set o partitions o n. As a rule, we write an element λ o P(n) as a decreasing sequence λ = (λ 1,..., λ s ) omitting the zeroes. Case sl n. According to [CMa93, Theorem 5.1.1], nilpotent orbits o sl n are parametrized by P(n). For λ P(n), we denote by O λ the corresponding nilpotent orbit o sl n. In sl n, all sheets are Dixmier and each nilpotent orbit is contained in exactly one sheet. The Levi subalgebras o sl n, and so the (Diximer) sheets, are parametrized by compositions o n. More precisely, i λ P(n), then the (Dixmier) sheet associated with λ is the unique sheet containing Ot λ where t λ is the dual partition o λ. Case so n. Set P 1 (n) := {λ P(n) ; number o parts o each even number is even}. According to [CMa93, Theorem and Theorem 5.1.4], nilpotent orbits o so n are parametrized by P 1 (n), with the exception that each very even partition λ P 1 (n) (i.e., λ has only even parts) corresponds to two nilpotent orbits. For λ P 1 (n), not very even, we denote by O λ the corresponding nilpotent orbit o so n. For very even λ P 1 (n), we denote by O I λ and OII λ the two corresponding nilpotent orbits o so n. In act, their union orm a single O n -orbit. Contrary to the sl n case, it may happen in the so n case that a given nilpotent orbit belongs to dierent sheets, and not all sheets are Dixmier. 3. Some useul lemmas Let be a nilpotent element o g that we embed into an sl 2 -triple (e, h, ) o g and let S := χ + (g ) be the Slodowy slice associated with (e, h, ) where χ := ( ) g. Denote by g(h, i) the i-eigenspace o ad(h) or i Z. Choose a Lagrangian subspace L g(h, 1) and set m := L i 2 g(h, i), J χ := x m C[g ](x χ(x)). Let M be the unipotent subgroup o G corresponding to m. Let µ : g m be the moment map or the M-action, which is just a restriction map. By [GG02], the adjoint action map gives the isomorphism and thus, M S µ 1 (χ), S = µ 1 (χ)/m.

9 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 9 In particular, C[S ] = C[µ 1 (χ)] M = (C[g ]/J χ ) M. Let HC be the category o initely generated (C[g ], G)-modules, that is, the category o initely generated C[g ]-modules K equipped with the G-module structure such that g.(.m) = (g()).g.m or g G, C[g ], m K. Theorem 3.1 ([Gi09], see also [Ar15b]). (1) The unctor H : HC C[S ] -mod, K (K/J χ K) M, is exact. (2) For any K HC, supp C[S ] H (K) = (supp C[g ] K) S. Lemma 3.2. Let K HC. Then G. supp C[g ] K i and only i K J χ K. Proo. Since S admits a C -action contracting to, Theorem 3.1(2) implies that G. supp C[g ] K i and only i H (K) 0. However H (K) 0 i and only i K/J χ K 0 by [Gi09, Proposition 3.3.6]. Let I be an ad g-invariant ideal o C[g ], so that C[g ]/I HC. Lemma 3.2 to K = C[g ]/I we obtain the ollowing assertion. Applying Lemma 3.3. Let I be an ad g-invariant ideal o C[g ]. Then G. V (I) i and only i where V (I) is the zero locus o I in g. C[g ] = I + J χ Let l be a Levi subalgebra o g and h a Cartan subalgebra o g contained in l. Thus z(l) h. Let p be a parabolic subalgebra o g with Levi actor l and nilradical p u. Assume that e (p u ) reg and h h. Identiying g with g through ( ), we get S = + g e, and by [Kat82, Lemma 3.2] (see also [Bu11, Proposition 3.2]), we have S l = G.( + z(l)). Note that we have the ollowing decomposition: g(h, 0) = [, g(h, 2)] (g(h, 0) g e ), and since ad() induces a bijection rom g(h, 2) to [, g(h, 2)], or any x g(h, 0) there is a well-deined element η(x) g(h, 2) such that x [, η(x)] g(h, 0) g e. Lemma 3.4. Assume that z(l) is generated by a nonzero element λ o h and that g(h, i) = 0 or i > 2. (1) The set {exp(ad η(tλ))( + tλ) t C} is an irreducible component o S l S. Moreover, i g is classical, then S l S is irreducible and S l S = S l S = {exp(ad η(tλ))( + tλ) t C}. (2) I g is classical then, as schemes, S l S = Spec C[z]. Here we endow Sl with its natural structure o irreducible reduced scheme.

10 10 TOMOYUKI ARAKAWA AND ANNE MOREAU Proo. (1) The irst assertion results rom [ImH05, Lemma 2.9] and the proo o [Bu11, Lemma 3.4] (which is a reormulation o [Kat82, Lemma 3.2 and Lemma 5.1]). In the case that g is classical, S l S is irreducible (see [ImH05, Theorem 5.2 and Theorem 6.2]) and so S l S = {exp(ad η(tλ))( + tλ) t C}. On the other hand, by Lemma 2.1(1), and S l = G.C λ = G.C λ G. S l = G.C λ G.. So, S l S = {exp(ad η(tλ))( + tλ) t C} since G. S = {} and {exp(ad η(tλ))( + tλ) t C}. In particular, S l S is an irreducible variety o dimension one. (2) According to [Gi09, Corollary 1.3.8(1)], S l S is a reduced complete intersection in S l since S l is reduced, whence S l S = Spec C[z] as a scheme by (1). Remark 3.5. Assume that g is classical. Deine a one-parameter subgroup γ : C G by: t C, x g, γ(t).x := t 2 γ(t).x where γ(t) is the one-parameter o G generated by ad(h). In the notation o Lemma 3.4, the sets S l and S are both stabilized by γ(t), and we have: S l S = S l S = { γ(t).µ t C }, or any nonzero semisimple µ in S l S, or example µ = exp(ad η(z))( + λ). Let Ω be the Casimir element o g and denote by I Ω the ideal o C[g ] generated by Ω. Lemma 3.6. Assume that g is classical. Let I be a homogeneous ad g-invariant ideal o C[g ], S l a Dixmier sheet o rank one, p a parabolic subalgebra o g with Levi actor l and nilradical p u. Let (e,, h) be an sl 2 -triple o g such that e (p u ) reg. Further, assume that the ollowing conditions are satisied: (1) g(h, i) = 0 or i > 2, (2) supp C[g ](C[g ]/I) = S l, (3) I + I Ω is the deining ideal o G., (4) H (C[g ]/I) = C[z] as algebras, (5) Ω(λ) 0, Then I is prime, that is, I = I. Condition (2) implies that I is deining ideal o S l since S l is irreducible. In particular, I is prime. Also, condition (3) means that I + I Ω is prime. Note that conditions (2) and (3) imply that I I + IΩ. since G. S l.

11 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 11 Proo. Set J := I. Then J/I HC. Since the sequence is exact, we get an exact sequence 0 J/I C[g ]/I C[g ]/J 0 0 H (J/I) H (C[g ]/I) H (C[g ]/J) 0 by Theorem 3.1 (1). Furthermore by Theorem 3.1 (2), Spec H (C[g ]/J) = Spec(C[g ]/J) S = Spec C[z]. The last equality comes rom Lemma 3.4 (2) since J is the deining ideal o S l. Hence by condition (4), we get H (J/I) = 0. By [Gi09, Proposition 3.3.6], H (J/I) 0 i and only i supp C[g ](J/I) G.. However, supp C[g ](J/I) supp C[g ](C[g ]/I) = S l and any G-invariant closed cone o S l which strictly contains G. contains G.. Thereore, supp C[g ](J/I) G. since supp C[g ](J/I) is a G-invariant closed cone o g. In particular, supp C[g ](J/I) is contained in the nilpotent cone N. Since Ω is a nonzero homogeneous element in the deining ideal o N, we deduce that Ω acts nilpotently on J/I. Hence or n suiciently large, (2) Ω n J/I = 0. We can now achieve the proo o the lemma. We have to show that J I. Let a J. Since J I + I Ω, or some b 1 I and 1 C[g ], we have a = b 1 + Ω 1. Since J is prime and Ω J by condition (5), 1 J. Applying what oregoes to the element 1 o J, we get that or some b 2 I and 2 J, a = b 1 + Ω(b 2 + Ω 2 ) = c 2 + Ω 2 2, with c 2 := b 1 + Ωb 2 I. A rapid induction shows that or any n Z >0, there exist c n I and n J such that a = c n + Ω n n. But c n + Ω n n I or n big enough by (2), whence a I. 4. Associated variety and singular support o aine vertex algebras Let ĝ = g[t, t 1 ] CK CD be the aine Kac-Moody Lie algebra associated with g and ( ), with the commutation relations [x(m), y(n)] = [x, y](m + n) + m(x y)δ m+n,0 K, [D, x(m)] = mx(m), [K, ĝ] = 0, or m, n Z and x, y g, where x(m) = x t m. For k C, set V k (g) = U(ĝ) U(g[t] CK CD) C k, where C k is the one-dimensional representation o g[t] CK CD on which g[t] CD acts trivially and K acts as multiplication by k. The space V k (g) is naturally a

12 12 TOMOYUKI ARAKAWA AND ANNE MOREAU vertex algebra, and it is called the universal aine vertex algebra associated with g at level k. By the PBW theorem, V k (g) = U(g[t 1 ]t 1 ) as C-vector spaces. The vertex algebra V k (g) is naturally graded: V k (g) = V k (g) d, V k (g) d = {a V k (g) Da = da}, d Z 0 Let V k (g) be the unique simple graded quotient o V k (g). As a ĝ-module, V k (g) is isomorphic to the irreducible highest weight representation o ĝ with highest weight kλ 0, where Λ 0 is the dual element o K. A V k (g)-module is the same as a smooth ĝ-module o level k. As in the introduction, let X V be the associated variety [Ar12] o a vertex algebra V, which is the maximum spectrum o the Zhu s C 2 -algebra R V := V/C 2 (V ) o V. In the case that V is a quotient o V k (g), V/C 2 (V ) = V/g[t 1 ]t 2 V and we have a surjective Poisson algebra homomorphism C[g ] = S(g) R V = V/g[t 1 ]t 2 V, x x( 1) + g[t 1 ]t 2 V, where x( 1) denotes the image o x( 1) in the quotient V. Then X V is just the zero locus o the kernel o the above map in g. It is G-invariant and conic. Conjecture 1. Let V = d 0 V d be a simple, initely strongly generated (i.e., R V is initely generated), positively graded conormal vertex operator algebra such that V 0 = C. (1) X V is equidimensional. (2) Assume that X V has initely many symplectic leaves. Then X V is irreducible. In particular X Vk (g) is irreducible i X Vk (g) N. For a scheme X o inite type, let J m X be the m-th jet scheme o X, and J X the ininite jet scheme o X (or the arc space o X). Recall that the scheme J m X is determined by its unctor o points: or every C-algebra A, there is a bijection Hom(Spec A, J m X) = Hom(Spec A[t]/(t m+1 ), X). I m > n, we have a natural morphism J m X J n X. This yields a projective system {J m X} o schemes, and the ininite jet scheme J X is the projective limit lim J mx in the category o schemes. Let π m : J m X X, m > 0, and π : J X X be the natural morphisms. I X is an aine Poisson scheme then its coordinate ring C[J X] is naturally a Poisson vertex algebra ([Ar12]). Let V = F 0 V F 1 V... be the canonical decreasing iltration o the vertex algebra V deined by Li [Li05]. The associated graded algebra gr V = p 0 F p V/F p+1 V is naturally a Poisson vertex algebra. In particular, it has the structure o a commutative algebra. We have F 1 V = C 2 (V ) by deinition, and by restricting the Poisson vertex algebra structure o gr V we obtain the Poisson structure o R V = V/F 1 V gr V. There is a surjection C[J X V ] gr V o Poisson vertex algebras ([Li05, Ar12]). By deinition [Ar12], the singular support o V is the subscheme SS(V ) = Spec(gr V )

13 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 13 o J X V. Theorem 4.1. Let V be a quotient o the vertex algebra V k (g). X V = G.C x or some x g. Then SS(V ) red = J X V = J G.C x = J G.C x. Suppose that Proo. By [Ar12, Lemma 3.3.1], XV = π (SS(V )). We know that SS(V ) J XV, so that SS(V ) red J X V. Let us prove the other inclusion. Set X = X V and U = G.C x. Since U is an irreducible open dense subset o X, πm 1 (U) = J m U or any m > 0 [EM09, Lemma 2.3], and πm 1 (U) = πm 1 (X reg ) is an irreducible component o J m X. Hence J X = π 1 (U) = J U because J X is irreducible [Ko73] and closed. Thereore, it is enough to prove that SS(V ) red contains J U since SS(V ) red is closed. The map µ: G (C x) G.C x, (g, tx) g.(tx) is a submersion at each point, so it is smooth (c. [Ha76, Ch. III, Proposition 10.4]). Hence by [EM09, Remark 2.10], we get that the induced map µ : J G J C x J G.C x is surjective and ormally smooth, and so µ (J G J C.x) = J G.C x. Since SS(V ) red is J G-invariant and J C -invariant, we deduce that J G.C x = J U is contained in SS(V ) red. The closures o nilpotent orbits satisy the conditions o Theorem 4.1, and also Dixmier sheets o rank one by Lemma 2.1 (1). Corollary 4.2. Let V be a quotient o the aine vertex algebra V k (g). (1) Suppose that X V = O or some nilpotent orbit O o g. Then SS(V ) red = J O = J O. (2) Suppose that X V = S or some Dixmier sheet S o g o rank one. Then SS(V ) red = J S = J S. See [Ar15a] and [AM15] or examples o aine vertex algebras V k (g) satisying the condition o Corollary 4.2 (1). 5. Zhu s algebra o aine Vertex algebras For a Z 0 -graded vertex algebra V = d V d, let A(V ) be the Zhu s algebra o V, where V V is the C-span o the vectors A(V ) = V/V V, a b := i 0 ( i ) a (i 2) b

14 14 TOMOYUKI ARAKAWA AND ANNE MOREAU or a V, Z 0, b V, and V (End V )[[z, z 1 ]], a n Z a (n)z n 1, denotes the state-ield correspondence. The space A(V ) is a unital associative algebra with respect to the multiplication deined by a b := ( ) a i (i 1) b i 0 or a V, Z 0, b V. Let M = d d 0 M d, M d0 0, be a positive energy representation o V. Then A(V ) naturally acts on its top weight space M top := M d0, and the correspondence M M top deines a bijection between isomorphism classes o simple positive energy representations o V and simple A(V )-modules ([Z96]). The vertex algebra V is called a chiralization o an algebra A i A(V ) = A. For instance, consider the universal aine vertex algebra V k (g). The Zhu s algebra A(V k (g)) is naturally isomorphic to U(g) ([FZ92], see also [Ar16a, Lemma 2.3]), and hence, V k (g) is a chiralization o U(g). Let Ĵk be the unique maximal ideal o V k (g), so that V k (g) = V k (g)/ĵk. We have an exact sequence A(Ĵk) U(g) A(V k (g)) 0 since the unctor A(?) is right exact and thus A(V k (g)) is the quotient o U(g) by the image I k o A(Ĵk) in U(g): A(V k (g)) = U(g)/I k. Fix a triangular decomposition g = n h n + o g. Then ĥ = h CK CD is a Cartan subalgebra o ĝ. A weight λ ĥ is called o level k i λ(k) = k. The top degree component o L(λ) is L g ( λ), where λ is the restriction o λ to h. Hence, by Zhu s Theorem, the level k representation L(λ) is a V k (g)-module i and only i I k L g ( λ) = 0. Set U(g) h := {u U(g) [h, u] = 0 or all h h} and let Υ: U(g) h U(h) be the Harish-Chandra projection map which is the restriction o the projection map U(g) = U(h) (n U(g) + U(g)n + ) U(h) to U(g) h. It is known that Υ is an algebra homomorphism. For a two-sided ideal I o U(g), the characteristic variety o I (without ρ-shit) is deined as V(I) = {λ h p(λ) = 0 or all p Υ(I h )} where I h = I U(g) h, c. [Jo77]. Identiying g with g through ( ), and thus h with h, we view V(I) as a subset o h. Proposition 5.1 ([Ar16a, Proposition 2.5]). For a level k weight λ ĥ, L(λ) is a V k (g)-module i and only i λ V(I k ). The Zhu s algebra A(V ) is related with the Zhu s C 2 algebra R V as ollows. The grading o V induces a iltration o A(V ) and the associated graded algebra gr A(V ) is naturally a Poisson algebra ([Z96]). There is a natural surjective homomorphism η V : R V gr A(V ) o Poisson algebras ([DSK06, Proposition 2.17(c)], [ALY14, Proposition 3.3]). In particular, Spec(gr A(V )) is a subscheme o X V.

15 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 15 For V = V k (g) this means the ollowing. We have V (gr I k ) X Vk (g), where V (gr I k ) is the zero locus o gr I k C[g ] in g. The map η Vk (g) is not necessarily an isomorphism. However, conjecturally [Ar15b] we have V (gr I k ) = X Vk (g). 6. Aine W -algebras For a nilpotent element o g, let W k (g, ) be the W -algebra associated with (g, ) at level k, deined by the generalized quantized Drineld-Sokolov reduction [FF90, KRW03]: W k (g, ) = H 2 +0 (V k (g)). Here H 2 +0 (M) is the corresponding BRST cohomology with coeicients in a ĝ- module M. Let (e,, h) be an sl 2 -triple associated with. The W -algebra W k (g, ) is conormal provided that k h, which we assume in this paper, and the central charge c (k) o W k (g, ) is given by c (k) = dim g(h, 0) 1 2 dim g(h, 1) 12 ρ k + h h 2 k + h 2, where ρ is the hal sum o the positive roots o g. We have W k (g, ) = W k (g, ), 1 2 Z 0 W k (g, ) 0 = C, W k (g, ) 1/2 = 0, W k (g, ) 1 = g, where g is the centralizer in g o the sl 2 -triple (e,, h). We have [DSK06, Ar15a] a natural isomorphism R Wk (g,) = C[S ] o Poisson algebras, so that X Wk (g,) = S. Let W k (g, ) be the unique simple quotient o W k (g, ). Then X Wk (g,) is a C - invariant, Poisson subvariety o S. Since it is C -invariant, W k (g, ) is lisse i and only i X Wk (g,) = {}. Let O k be the category O o ĝ at level k. We have a unctor O k W k (g, ) -Mod, M H 2 +0 (M), where W k (g, ) -Mod denotes the category o W k (g, )-modules. Let KL k be the ull subcategory o O k consisting o objects M on which g acts locally initely. Note that V k (g) and V k (g) are objects o KL k. Theorem 6.1 ([Ar15a], k, arbitrary). (1) H 2 +i (M) = 0 or all i 0, M KL k. In particular, the unctor KL k W k (g, ) -Mod, M H 2 +0 (M), is exact. (2) For any quotient V o V k (g) we have R +0 H 2 = H (R V ), (V )

16 16 TOMOYUKI ARAKAWA AND ANNE MOREAU where H (R V ) is deined in Theorem 3.1. Hence, X +0 H 2 is isomorphic (V ) to the scheme theoretic intersection X V g S. (3) H 2 +0 (V ) 0 i and only i G. X V. (4) H 2 +0 (V ) is lisse i X V = G.. Let θ be a root vector o the highest root θ o g, which is a minimal nilpotent element so that O min where O min is the minimal nilpotent orbit o g. Theorem 6.2 ([Ar05], = θ ). (1) H 2 +i θ (M) = 0 or all i 0, M O k. In particular, the unctor O k W k (g, ) -Mod, M H 2 +0 (M), is exact. (2) H 2 +0 θ (L(λ)) 0 i and only i λ(α0 ) N 0. I this is the case, H 2 +0 θ (L(λ)) is a simple W k (g, )-module, where α0 = θ + K. Recall that is a short nilpotent element i g = g(h, 2) g(h, 0) g(h, 2). I this is the case, we have 1 2 h P, where P is the coroot lattice o g, and 1 2 h deines an element o the extended aine Weyl group W = W P o ĝ, which we denote by t 1 2 h. Here W is the Weyl group o g. Let t 1 2 h be a Tits liting o t 1 2 h to an automorphism o ĝ. Set D h = D h, and put M d,h = {m M D h m = dm} or a ĝ-module M. The operator D h extends to the grading operator o W k (g, ) ([KRW03, Ar05]). The subalgebra t 1 2 h (g) acts on each homogeneous component M d,h because [D h, t 1 2 h (g)] = 0. Note that t 1 2 h (g) is the subalgebra o ĝ generated by root vectors o roots t 1 2 h (α) = α 1 2 α(h)δ, α, where δ ĥ is the dual element o D. In particular, t 1 2 h (g) contains l := g(h, 0) as a Levi subalgebra. We regard M d,h as a g-module through the isomorphism t 1 2 h (g) = g. Since α(d h ) 0 or all positive roots α o ĝ by the assumption that is short, we have L(λ) = L(λ) d,h. d λ(d h ) Let O l k be the ull subcategory o category O k o ĝ consisting o objects on which l acts locally initely. Theorem 6.3. Let be a short nilpotent element as above. (1) H 2 +i (M) = 0 or all i 0, M Ok l. In particular, the unctor Ol k W k (g, ) -Mod, M H 2 +0 (M), is exact.

17 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 17 (2) Let L(λ) Ok l. Then H 2 +0 (L(λ)) 0 i and only i Dim L(λ) λ(dh ),h = 1/2 dim G.(= dim g(h, 2)), where Dim M is the Geland-Kirillov dimension o the g-module M. I this is the case, H 2 +0 (L(λ)) is almost irreducible over W k (g, ), that is, any nonzero submodule o H 2 +0 (L(λ)) intersects its top weight component non-trivially (c. [Ar11]). (3) Suppose that g is o type A. Then, or L(λ) Ok l, H 2 +0 (L(λ)) is zero or irreducible. Proo. The above theorem is just a restatement o main results o [Ar11] since the unctor Ok l Wk (g, ) -Mod, M H 2 +0 (M), is identical to the -reduction unctor H0 BRST (?) studied in [Ar11] (in the case o short nilpotent elements). 7. Level 1 aine vertex algebra o type A n 1, n 4 We assume in this section that g = sl n with n 4. Let = {ε i ε j i, j = 1,..., n, i j} be the root system o g. Fix the set o positive roots + = {ε i ε j i, j = 1,..., n, i < j}. Then the simple roots are α i = ε i ε i+1 or i = 1,..., n 1. The highest root is θ = ε 1 ε n = α α n 1. Denote by (e i, h i, i ) the Chevalley generators o g, and ix the root vectors e α, α, α + as ollows. e εi ε j = e i,j and εi ε j = e j,i or i < j, where e i,j is the standard elementary matrix associated with the coeicient (i, j). For α +, denote by h α = [e α, α ] the corresponding coroot. In particular, h i = e i,i e i+1,i+1 or i = 1,..., n 1. Let g = n h n + be the corresponding triangular decomposition. Denote by ϖ 1,..., ϖ n 1 the undamental weights o g, ϖ i := (ε ε i ) i n (ε ε n ). Identiy g with g through ( ). Thus, the undamental weights are viewed as elements o g. Let θ 1 be the highest root o the root system generated by the simple roots perpendicular to θ, i.e., Set and put θ 1 = α α n 2 = ε 2 ε n 1, β := α α n 2 = ε 1 ε n 2, γ := α α n 1 = ε 2 ε n, v 1 = e θ e θ1 e β e γ S 2 (g), where S d (g) denotes the component o degree d in S(g) or d 0. Then v 1 is a singular vector with respect to the adjoint action o g and generates an irreducible inite-dimensional representation W 1 o g in S 2 (g) isomorphic to L g (θ + θ 1 ). Recall that we have a g-module embedding [AM15, Lemma 4.1], σ d : S d (g) V k (g) d, x 1... x d 1 (3) x σ(1) ( 1)... x σ(d) ( 1). d! σ S d

18 18 TOMOYUKI ARAKAWA AND ANNE MOREAU We will denote simply by σ this embedding or d = 2. Proposition 7.1 ([Ad03]). For l 0, The vector σ(v 1 ) l+1 is a singular vector o V k (g) i and only i k = l 1. Theorem 7.2. The vecor σ(v 1 ) generates the maximal submodule o V 1 (g), that is, V 1 (g) = V 1 (g)/u(ĝ)σ(v 1 ). Set Ṽ 1 (g) = V 1 (g)/u(ĝ)σ(v 1 ). To prove Theorem 7.2, we shall use the minimal W -algebra W k (g, θ ). Let g be the centralizer in g o the sl 2 -triple (e θ, h θ, θ ). Then g = g 0 g 1, where g 0 is the one-dimensional center o g and g 1 = [g, g ]. Note that g 0 = C(h 1 h n 1 ) and g 1 = e αi, αi i = 2,..., n 2 = sl n 2. There is an embedding, [KW04], [AM15, 7], V k 0 (g0 ) V k 1 (g1 ) W k (g, θ ) o vertex algebras, where k 0 = k + n/2 and k 1 = k + 1. Note that V k 0 0 (g 0) is isomorphic to the rank one Heisenberg vertex algebra M(1) provided that k 0 0. For k = 1, we have k 1 = 0 and V k 1(g 1 ) = V k 1 (g)/u(ĝ1 )e θ1 ( 1)1 = C. By Theorem 6.2 the exact sequence 0 U(ĝ)σ(v 1 ) V 1 (g) Ṽ 1(g) 0 induces an exact sequence (4) 0 H 2 +0 θ (U(ĝ)σ(v 1 )) W 1 (g, θ ) H 2 +0 θ (Ṽ 1(g)) 0. Lemma 7.3. The image o σ(v 1 ) in W 1 (g, θ ) coincides with the image o the singular vector e θ1 ( 1)1 o V 0 (g 1 ) W 1 (g, θ ). Proo. Since it is singular, σ(v 1 ) deines a singular vector o W k (g, θ ). Its image in R W k (g, θ ) is the image o v 1 in C[S θ ] = (C[g ]/J χ ) M, where χ = ( θ ), see 3. Since v 1 e θ1 (mod J χ ), we have σ(v 1 ) e θ1 ( 1)1 (mod C 2 (W k (g, θ ))). Then σ(v 1 ) and e θ1 ( 1)1 must coincide since they both have the same weight and are singular vectors with respect to the action o ĝ 1. Theorem 7.4. We have H 2 +0 θ (Ṽ 1(g)) = M(1), the rank one Heisenberg vertex algebra. In particular H 2 +0 θ (Ṽ 1(g)) is simple, and hence, isomorphic to W 1 (g, θ ). Proo. By Theorem 6.2, H 2 +0 θ (V 1 (g)) is isomorphic to W 1 (g, θ ). Since V 1 (g) is a quotient o Ṽ 1(g), W 1 (g, θ ) is a quotient o H 2 +0 θ (Ṽ 1(g)) by the exactness result o Theorem 6.2. In particular, H 2 +0 θ (Ṽ 1(g)) is nonzero. Recall that W k (g, θ ) is generated by ields J {a} (z), a g, G {v} (z), v g 1/2, and L(z) described in [KW04, Theorem 5.1]. Since V 0 (g 1 )/U(ĝ 1 )e θ1 ( 1)1 = C, Lemma 7.3 implies that J {a} (z), a g 1, are all zero in H 2 +0 θ (Ṽ 1(g)). Then, since [J {a} λ G{v} ] = G [a,v] and g 1/2 is a direct sum o non-trivial irreducible initedimensional representations o g 1, it ollows that G (v) (z) = 0 or all v g 1/2.

19 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 19 Finally, [KW04, Theorem 5.1 (e)] implies that L coincides with the conormal vector o V n 2 2 (g 1 ) = M(1) in H 2 +0 θ (Ṽ 1(g)). We conclude that H 2 +0 θ (Ṽ 1(g)) is generated by J {a} (z), a g 0, which proves the assertion. Proo o Theorem 7.2. Suppose that Ṽ 1(g) is not simple. Then there is at least one singular vector v o weight, say µ. Then µ sϖ 1 Λ 0 or sϖ l Λ 0 or some s N (mod Cδ) by [AP08, Proposition 5.5]. Let M be the submodule o Ṽ 1 (g) generated by v. Since M is a Ṽ 1(g)-module, H 2 +0 θ (M) is a module over H 2 +0 θ (Ṽ 1(g)) = M(1). Because µ(α0 ) < 0, Theorem 6.2 implies that the image o v in H 2 +0 θ (M) is nonzero, and thus, it generates the irreducible highest weight representation M(1, µ(h1 hn 1) ) o M(1) with highest weight µ(h1 hn 1). Now the exactness o 2(n 2) 2(n 2) the unctor H 2 +0 θ (?) shows that H 2 +0 θ (M) is a submodule o H 2 +0 θ (Ṽ 1(g)) = M(1), and thereore, so is M(1, µ(h1 hn 1) ). But this contradicts the simplicity o M(1). By Theorem 7.2, we have 2(n 2) (5) R V 1(g) = C[g ]/I W1, so XV 1(g) = Spec(C[g ]/I W1 ). Corollary 7.5. We have H (C[g ]/I W1 ) = C[z]. Proo. By Theorem 6.1(2), we know that R = H +0 H 2 (R (V θ 1(g)) V 1(g)). The assertion ollows since R = R +0 H 2 M(1) = C[z] by Theorem 7.4. (V θ 1(g)) As in Theorem 1.1, let l 1 be the standard Levi subalgebra o sl n generated by all simple roots except α 1, that is, l 1 = h + e αi, αi i 1. The center o l 1 is generated by ϖ 1. Thus, the Dixmier sheet closure S l1 by is given S l1 = G.C ϖ 1, see 2. The unique nilpotent orbit contained in S l1 G-orbit o θ. is Ind g l 1 (0) = O min, that is, the Lemma 7.6. We have V (I W1 ) N O min. Proo. First o all, we observe that O (22,1 n 4 ) it the smallest nilpotent orbit o g which dominates O min = O (2,1 n 2 ). By this, it means that i λ (2, 1 n 2 ) then λ (2 2, 1 n 4 ) where is the Chevalley order on the set P(n). Thereore, it is enough to show that V (I W1 ) does not contain O (22,1 n 4 ). Let O (2 2,1 n 4 ) that we embed into an sl 2 -triple (e, h, ) o g. Denote by g(h, i) the i-eigenspace o ad(h) or i Z and by + (h, i) the set o positive roots α + such that e α g(h, i). We have = α +(h,2) c α α with c α C. We

20 20 TOMOYUKI ARAKAWA AND ANNE MOREAU call the set o α + (h, 2) such that c α 0 the support o. Choose a Lagrangian subspace L g(h, 1) and set m := L i 2 g(h, i), J χ := x m C[g ](x χ(x)), with χ = ( ) g, as in 3. By Lemma 3.3, it is suicient to show that To see this, we shall use the vector C[g ] = I W1 + J χ. v 1 = e θ e θ1 e β e γ I W1. I n > 4, the weighted Dynkin diagram o the nilpotent orbit G. is So we can assume that h = ϖ 2 +ϖ n 2 and we can choose or the element β + γ. We see that θ, θ 1, β and γ all belong to + (h, 2). Since β and γ are in the support o, but not θ and θ 1, or some nonzero complex number c, we have e θ e θ1 e β e γ = c (mod J χ ). So v 1 = c (mod J χ ) and I W1 + J χ = C[g ], whence C[g ]/(I W1 + J χ ) = 0. For n = 4, the weighted Dynkin diagram o the nilpotent orbit G. is and we conclude similarly. Lemma 7.7. Let λ be a nonzero semisimple element o g. Then, λ V (I W1 ) i and only i λ G.C ϖ 1. Proo. Set or i, j {1,..., n 2}, with j i 2, and or i {2,..., n 2}, p i,j := h i h j, q i := h i (h i 1 + h i + h i+1 ). According to [AP08], the zero weight space o W 1 is generated by the elements p i,j and q i. Clearly, p i,j (ϖ 1 ) = 0 or j i 2 and q i (ϖ 1 ) = 0 or any i {2,..., n 2}. So ϖ 1 V (I W1 ), and whence G.C ϖ 1 V (I W1 ) since V (I W1 ) is a G-invariant cone. This proves the converse implication. For the irst implication, let λ be a nonzero semisimple element o g, and assume that λ V (I W1 ). Since V (I W1 ) is G-invariant, we can assume that λ h. Then write λ = i=1 λ i ϖ i, λ i C. Since p i,j (λ) = q i (λ) = 0 or all i, j, we get (6) (7) λ i λ j = 0, i, j {1,..., n 2}, j i 2, λ i (λ i 1 + λ i + λ i+1 ) = 0, i = 2,..., n 2. Since λ is nonzero, λ k 0 or some k {1,..., n 1}.

21 SHEETS AND ASSOCIATED VARIETIES OF AFFINE VERTEX ALGEBRAS 21 I k {1, n 1}, then by (6), λ j = 0 or j k 2. So by (7), either λ k 1 +λ k = 0 or λ k + λ k+1 = 0 since λ k 1 λ k+1 = 0 by (6). I k = 1, then by (6), λ j = 0 or j 3. So by (7), λ 2 (λ 1 + λ 2 ) = 0. I k = n 1, then by (6), λ j = 0 or j n 3. So by (7), λ n 2 (λ n 2 +λ n 1 ) = 0. We deduce that λ C ϖ 1 C ϖ n 1 C (ϖ i ϖ i+1 ) = 1 j n 1 1 j n 2 C (ε ε i 1 (n 1)ε i + ε i ε n ) All the above weights are conjugate to tϖ 1 or some t C under the Weyl group o (g, h) which is the group o permutations o {ε 1,..., ε n }, whence the expected implication. The ollowing assertion ollows immediately rom Lemma 2.1, Lemma 7.6 and Lemma 7.7. Proposition 7.8. We have V (I W1 ) = S l1. Remark 7.9. The zero locus o the Casimir element Ω in V (I W1 ) is O min. Indeed, by Lemma 2.1 the zeros locus o Ω in S l1 is contained in the nilpotent cone since Ω(ϖ 1 ) 0. The statement ollows since O min has codimension one in S l1. As a consequence, the zero locus o the ideal generated by CΩ W 1 S 2 (g) is O min. This latter act is known by [Ga82] using a dierent approach. Proposition The ideal I W1 is prime, and thereore it is the deining ideal o S l1. Proo. We apply Lemma 3.6 to the ideal I := I W1. First o all, l 1 and (e θ, h θ, θ ) satisy the conditions o Lemma 3.4 since z(l 1 ) = Cϖ 1 and g(h θ, i) = 0 or i > 2. It remains to veriy that the conditions (1),(2),(3),(4) o Lemma 3.6 are satisied. Condition (1) is satisied by Proposition 7.8. According to [Ga82, Corollary 2 and Theorem 1, Chap. V], the ideal I +I Ω is the deining ideal o O min. So condition (2) is satisied. The condition (3) is satisied too, by Corollary 7.5. At last, because Ω(ϖ 1 ) 0, condition (4) is satisied. In conclusion, by Lemma 3.6, I = I W1 is prime. Proo o Theorem 1.1 (1). The irst statement ollows immediately rom (4) and Proposition The second statement ollows rom the inclusions J S l1 = SS(V 1 (g)) red SS(V 1 (g)) J X V 1(g) = J S l1, where the irst equality ollows rom Corollary 4.2. Remark Let V k (g) -Mod g[t] be the ull subcategory o the category o V k (g)- modules consisting o objects that belong to KL k. Since H 2 +0 θ (V 1 (g)) = M(1), we have a unctor V 1 (g) -Mod g[t] M(1) -Mod, M H 2 +0 θ (M), where M(1) -Mod denotes the category o the modules over the Heisenberg vertex algebra M(1). The proo o Theorem 1.1 (1) and [AP14, Theorem 6.2] imply that this is a usion unctor.

22 22 TOMOYUKI ARAKAWA AND ANNE MOREAU Let p = l 1 p u be a parabolic subalgebra o g with nilradical p u. It is a maximal parabolic subalgebra o g. Let P be the connected parabolic subgroup o G with Lie algebra p. Set Y := G/(P, P ). As explained in 2, we have V (gr J Y ) = S l1, where J Y = ker ψ Y and ψ Y is the algebra homomorphism U(g) D Y. The variety Y is a quasi-aine variety, isomorphic to C n \{0}. It ollows that the natural embedding Y = C n \{0} C n induces an isomorphism D C n D Y. In this realization, the map ψ Y is described as ollows. ψ Y : U(g) D C n = z 1,..., z n,,..., z 1 z n e i,j z j (i j), z i h i z i + z i+1. z i z i+1 where e i,j is as beore the standard elementary matrix element and h i = e i,i e i+1,i+1. This has the ollowing chiralization: Let DC ch be the βγ-system o rank n, generated by ields β 1 (z),..., β n (z), γ 1 (z),..., γ n (z), satisying n OPE γ i (z)β j (w) δ ij z w, γ i(z)γ j (w) 0, β i (z)β j (w) 0, see e.g. [Kac98]. We have R D ch C = C[T C n ], A(D ch n C n) = D C n, where D C n denotes the Weyl algebra o rank n, which is identiied with the algebra o dierential operators on the aine space C n. In particular, DC ch is a chiralization n o D C n. Lemma The ollowing gives a vertex algebra homomorphism. ˆψ Y : V 1 (g) D ch C n, e i,j (z) : β j (z)γ i (z) : (i j), h i (z) : β i (z)γ i (z) : + : β i+1 (z)γ i+1 (z) :. Note that the map ˆψ Y induces an algebra homomorphism U(g) = A(V 1 (g)) A(D ch C n) = D C n, which is identical to ψ Y. Theorem 7.13 ([AP14]). The map ˆψ Y actors through the vertex algebra embedding (8) By Theorem 7.13, ˆψ Y V 1 (g) D ch C n. induces a homomorphism A(V 1 (g)) A(D ch C n) = D C n. Theorem (1) The ideal gr J Y C[g ] is prime and hence it is the deining ideal o S l1. (2) The natural homomorphism R V 1(g) gr A(V 1 (g)) is an isomorphism.