Order relations on induced and prime ideals in the enveloping algebra of a semisimple Lie algebra

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1 Journal of Algebra ) Order relations on induced and prime ideals in the enveloping algebra of a semisimple Lie algebra Nikolaos Papalexiou 1 Department of Mathematics, University of the Aegean, Karlovassi, Greece Received 25 May 2005 Available online 17 April 2006 Communicated by Susan Montgomery Abstract Let g be a semisimple Lie algebra and Ug) its enveloping algebra. Given an induced or prime ideal I in Ug) we formulate some inclusion relations for the set of primitive ideals containing I Elsevier Inc. All rights reserved. 0. Introduction Let g be a complex semisimple Lie algebra, h be a Cartan subalgebra of g and h the dual of h.letw be the associated Weyl group, R the root system associated to W and B a basis for R. Let Ug) be the enveloping algebra of g. For any λ h denote by Iλ) the annihilator of the simple highest weight module Lλ). In [11], A. Joseph attaches to any two-sided ideal I in Ug) a subvariety VI) of h, called the characteristic variety of I. The knowledge of VI) allows us to find all primitive ideals containing I. More precisely by [4, appendix], if I is a two-sided ideal of Ug) then we have: λ VI) I Iλ). Our object in the first two sections is to study the characteristic variety of an ideal in Ug) induced by a primitive ideal in the enveloping algebra of a Levi subalgebra of g. More precisely, address: papalexi@aegean.gr. 1 Partially supported by the Marie Curie Research Training Network Flags, Quivers and Invariant Theory in Lie Representation Theory grant No. MRTN-CT /$ see front matter 2006 Elsevier Inc. All rights reserved. doi: /j.jalgebra

2 2 N. Papalexiou / Journal of Algebra ) 1 15 let p be a parabolic subalgebra of g with Levi l, z the center of l, h =[l, l] h, R the root system of l and B a basis for R. For any ν h ) we fix the primitive ideal in Ul), J = Ann L ν). Given any λ z, we denote by I p J, λ) the annihilator in Ug) of the induced Ug)-module Ug) p L ν + λ). For any subset B of B,letD B be the set of all w W such that wb R +. In the first section, using Borho Jantzen s theory [3], we calculate explicitly the characteristic variety of I p J, λ) in case that [l, l] is isomorphic to sl2, C) Proposition 1.6). Furthermore, for l in general, using Duflo s [7, Proposition 12] and Borho Jantzen s results [3], we find an upper estimate for the characteristic variety of I p J, λ) Proposition 1.7) in the case where B R + ν+λ ={α R ν+λ ν + λ + ρ, ˇα 0}: Proposition 0.1. Let B B and p = p B the parabolic subalgebra defined by B. Let ν h ), J = Ann Ul) L ν) and λ z. Suppose that ν + λ is regular and B R ν+λ +. Denote by x the element in D B such that x ν + λ) = μ is dominant. Then, V I p J, λ) ) D β μ, α B β T α,x where T α,x denotes the set of β B such that xα,ϖ β 0. The main result of Section 2 will be the proof of the conjecture C1) posed by Borho and Joseph in [4, appendix] concerning the calculation of a lower estimate of the characteristic variety of an induced ideal. In our approach we project the induced ideal in Ul) and then we translate it using the translator functor of l-modules. In particular we prove Theorem 2.7): Theorem 0.2. Let ν h ), λ z such that J = Ann Ul) L ν) and I p J, λ) the corresponding induced ideal. Then, one has V I p J, λ) ) D B V J ) + λ ). Our aim in the third section will be the study of the characteristic variety for a prime ideal. The knowledge of this variety could enable us to determine the topology in the set Spec Ug) of prime ideals in Ug). By [4, Proposition 5.7] every prime ideal P can be expressed in the following way: P = I p J, Λ) := λ Λ I p J, λ), where Λ is an irreducible subset in z. At first, for every prime ideal P in Ug), using the results of the previous section, we prove an inclusion relation for VP ) Proposition 3.3): Proposition 0.3. Let P be a prime ideal such that P = I p J, Λ) with p a parabolic subalgebra with Levi l and J a primitive ideal in Ul). Then we have, V I p J, Λ) ) D B V J ) + Λ ). In the sequel, we calculate the Gelfand Kirillov dimension of a prime ideal Proposition 3.5) answering a question of Soergel in [13, 5.3]. At the end of this work we analyze the case of

3 N. Papalexiou / Journal of Algebra ) sl4) giving a negative answer for the Soergel s revised conjecture [13, Conjecture 5.1] see 1. An upper estimate of the characteristic variety of an induced ideal Let g be a complex semisimple Lie algebra, h be a Cartan subalgebra of g and h the dual of h. Letg = n + h n be the corresponding triangular decomposition of g. LetR be the root system of g relative to h and R + the set of positive roots of R. Letρ be the half-sum of positive roots. Denote by W the Weyl group of g. We consider the shifted by ρ action of W on h : w λ = wλ + ρ) ρ, for any w W and λ h.letug) be the enveloping algebra of g. For each μ h let Lμ) be the simple g-module of highest weight μ and let Iμ) be the annihilator of Lμ) in Ug). The object of this section is to find an upper estimate of the characteristic variety of an ideal in Ug) induced by a primitive ideal in the enveloping algebra of a Levi subalgebra of g. We shall denote by R μ the root system consisting of all α R such that μ + ρ, ˇα Z. We set R + μ ={α R μ μ + ρ, ˇα 0}. We shall say that μ is dominant if R + μ R +.LetW μ be the Weyl group of R μ.wealsosetd μ ={w W wr μ R + ) R + }, which identifies with the set of minimal right coset representatives of W μ in W. Let Prim μ Ug) be the set of primitive ideals of Ug) with central character μ h. Then, by [7], we know that the map w Ann Lw μ) defines a surjective map from W μ onto Prim μ Ug). Furthermore, Prim μ Ug) is ordered by inclusions and has Ug) ker χ μ as its unique minimal element. Suppose w 1,w 2 W μ. We shall write w 1 w 2 if and only if Iw 1 μ) Iw 2 μ). Givenw W μ, we denote by C μ w) or simply by Cw) if there is no confusion, the left cone over w in W μ consisting of all x W μ such that x w. Let P : Ug) Uh) = Sh) be the Harish-Chandra projection with kernel n Ug) + Ug)n +. Given now any two-sided ideal I of Ug) the characteristic variety of I is a variety in h defined by the following way: i=1 VI) = { λ h P I)λ) = 0 }. By Eq. ) in appendix of [4], it is known that the characteristic variety of an ideal I consists of all λ h such that I Iλ). Thus, if two primitive ideals have the same characteristic variety they are equal. Let I 1,...,I n two-sided ideals in Ug). Then by [11, Lemma 1v), vi)] we have, n ) n n ) n V I i = VI i ) and V I i = VI i ). i=1 Moreover, if I denotes the radical of an ideal I, then V I)= VI). These properties imply the lemma: Lemma 1.1. Let I be a semi-prime ideal in Ug) and let I 1,...,I n the minimal prime ideals containing I. Then VI) = i=1 n VI i ). i=1 i=1

4 4 N. Papalexiou / Journal of Algebra ) 1 15 We recall that see [11, Lemma 1ii)]) for any μ h, the characteristic variety of the primitive ideal Iμ)satisfies the following inclusions: {μ} V Iμ) ) W μ. It is clear that the characteristic variety of a primitive ideal is a finite set. More precisely, if μ h, is regular and dominant the characteristic variety of Iw μ) = Ann Lw μ), where w W μ,is given by the following formula see, for example, appendix of [4]): V Iw μ) ) = D μ Cw) μ. Given a primitive ideal Iw μ) in Ug) we denote by W Iw μ) the subset of W for which V Iw μ) ) = W Iw μ) μ. For any λ h we denote by Fλ) its facette and by ˆFλ) the upper closure of Fλ) see [10, 2.5]). Let Λ = λ + PR), where PR) denotes the lattice of integral weights. The following proposition is an immediate consequence of the translation principle. Proposition 1.2. Let μ Λ be a regular dominant weight and w W μ.ifw ν ˆFw μ) we consider the translator functor T ν μ :Prim w μ Ug) Prim w ν Ug) see [10, 4.12]). Then we have V T ν μ Iw μ)) = W Iw μ) ν. Proof. By the hypothesis we have that w ν is in the upper closure of the facette defined by w μ, hence we have V T ν μ Iw μ)) = V Iw ν) ) = W Iw ν) ν. Assume now that y is an element in W Iw ν).wehavey ν VI w ν)) if and only if Iw ν) Iy ν) [11, Lemma 1i)]. By the translation principle [10, Satz 5.8] Iw ν) Iy ν) is equivalent to the fact that Iw μ) Iy μ), which means that y W Iw μ). Then we have W Iw ν) ν = W Iw μ) ν. Let B be a basis of the root system R. For any subset B of B we set R = R ZB and R + = R R +.Letu + = α R + \R + g α and u = α R + \R + g α. Wesetn + B = α R + g α and n B = α R + g α. Then we define the parabolic subalgebra p = p B := n + h n B with nilradical u +. We also denote by l = l B = n + B h n B the corresponding Levi subalgebra of p and z its center. Then we have: g = u l u +, p = l u +, l =[l, l] z. Let h be a complement of z in h such that h =[l, l] h. It is obvious that R is a root system of l relative to h. Denote by W the corresponding Weyl group of l. For each μ h let L μ) be the simple l-module of highest weight μ. LetI μ) be the annihilator of L μ) in Ul). For

5 N. Papalexiou / Journal of Algebra ) any ν h ), we fix the primitive ideal in Ul), J = Ann L ν). Given any λ z, we denote by M p J, λ) or M p ν + λ) the induced Ug)-module: M p J, λ) = M p ν + λ) := Ug) p Ul)/J Cλ ) = Ug) p L ν + λ), where u + acts trivially on Ul)/J and C λ is the one-dimensional p-module of weight λ.wealso denote by I p J, λ) or I p ν + λ) the corresponding induced ideal: I p J, λ) = I p ν + λ) := Ann Ug) M p J, λ) = Ann Ug) M p ν + λ). We set W μ = W μ W.WealsosetD B ={w W wb R + }, which identifies with the set of minimal right coset representatives of W in W. For any subset S of R + we set D S = {w W ws R + }. Lemma 1.3. If S 1,S 2 R +, we have: i) D S1 D S2 = D S1 S 2 ; ii) if furthermore x W such that xs 1 R +, then D S1 x 1 = D xs1. Proof. i) We have D S1 ={w W ws 1 R + } and D S2 ={w W ws 2 R + }. Then obviously we have: D S1 D S2 = { w W ws1 S 2 ) R + } = DS1 S 2. ii) We see that w is an element of D S1 x 1 if and only if wxs 1 R + which is equivalent that w is an element of D xs1. Let B be a subset of B μ. We denote by w μ and w B the longest elements in W μ and W, respectively. The element w D μ := w μ w B is the longest element in D μ := D B W μ. Next we calculate the characteristic variety of some primitive ideals. Proposition 1.4. Let μ be a B-dominant regular weight and w W. Then, V Iw D μ w μ) ) = D B V I w μ) ). Proof. By [10, Satz 7.19] we have that { y Wμ Iw D μ w μ) Iy μ) } = { w 1 w 2 w1 D μ,w 2 W with I w μ) I w 2 μ) }. In addition, by Lemma 3.12 in [4] for all x D μ, y W μ one has Ixy μ) = Iy μ). Since B B μ, D μ D B W μ ) = D B. Hence, VI w D μ w μ)) = D B V I w μ)). Let α B and μ a B-regular dominant weight. Suppose that α R + μ. Denote by I αμ) the almost minimal primitive ideal Iw μ s α μ) in Ug). Then we have the following corollary. Corollary 1.5. The characteristic variety of I α μ) is equal to D α μ.

6 6 N. Papalexiou / Journal of Algebra ) 1 15 Proof. It is an immediate consequense of Proposition 1.4. The following proposition is a slight generalization of Satz 4.15 in [3]. We calculate the induced ideal in the case when the semisimple part of the Levi factor of the parabolic subalgebra is isomorphic to sl2, C). For any α B we denote by p α the parabolic subalgebra n + h g α with Levi l α = g α h g α. Proposition 1.6. Let α B and x D α. Let μ h be a dominant and B-regular weight. Suppose that xα R + μ. Let p α be the parabolic subalgebra defined by {α}. Weset Then we have T α,x = { β B xα,ϖ β 0 }. I pα x 1 μ ) = β T α,x I β μ). ) Furthermore, VI pα x 1 μ)) = β T α,x D β μ. Proof. Since x D α, xα R + which implies that xα = β B n β β, with n β N. Ifxα is of the form β 1 + β β r, with β 1,β 2,...,β r B, then by [3, Satz 4.15] we have I pα x 1 μ ) = r I βi μ). We will deduce our result using translation principle. Let x 1 D α such that x 1 α = β 1 + β 2 + +β r. Then by [3, Lemma 2.8], since α R +, we obtain x 1 μ i=1 T x 1 μ I x1 1 μ p α x 1 1 μ ) = I pα x 1 μ ). Thus, we get the formula ). Furthermore, since β T α,x and xα R +, β R μ +. By Corollary 1.5 we have that VI β μ)) is equal to D β μ. Thus, the formula of the characteristic variety follows from the properties of V ). Suppose that ν + λ is regular and B R ν+λ +. This implies that ν + λ is B -dominant. Thus, there exists x D B such that x ν + λ) is dominant. We prove now the basic result of this paragraph.

7 N. Papalexiou / Journal of Algebra ) Proposition 1.7. Let B B and p = p B the parabolic subalgebra defined by B. Let ν h ) and λ z. Suppose that ν + λ is regular and B R ν+λ +. Denote by x the element in D B such that x ν + λ) = μ is dominant. Then, V I p ν + λ) ) D β μ. α B β T α,x Proof. Let I ν + λ) be the primitive ideal in Ul) of highest weight ν + λ. Denote by w B the longest element in W. By Proposition 12 in [7] we can write: I ν + λ) = α B I s α w B ν + λ) ). Thus, the induced ideal becomes, I p ν + λ) = Ann Ug) p Ul)/I ν + λ) ) / = Ann Ug) p Ul) I ) ) sα w B ν + λ) α B α B Ann Ug) p Ul)/I s α w B ν + λ) )) = α B I p sα w B ν + λ) ). It suffices now to compute the characteristic variety of I p s α w B ν + λ)), for any α B. We calculate first the characteristic variety of the induced ideal I pα s α w B ν + λ)). Letγ α = s α w B ) 1 α) B. We will show that I pα sα w B ν + λ) ) = I p sγα w B ν + λ) ). ) By [2, Lemma 10.4b)] we know that I pα sα w B ν + λ) ) = Ann [ Ind g p Ind l pα I s α w B ν + λ) ))], where I s α w B ν + λ)) is the primitive ideal in Ul α ) of highest weight s α w B ν + λ). Denote by I p α s α w B ν + λ)) the annihilator in Ul) of Ind l p α I s α w B ν + λ)). Since ν + λ is B -dominant, by Proposition 1.6 we have where D B γ α V I p α sα w B ν + λ) )) = D B γ α ν + λ), ={w W wγ α R + }. On the other hand, there exists a primitive ideal I s γα w B ν +λ)) in Ul) such that V I s γα w B ν +λ))) = D B γ α ν +λ). Thus, I p α s α w B ν + λ)) = I s γα w B ν + λ)), which implies Eq. ). Let now x D B be such that x ν + λ) = μ is B-dominant. Then, by Eq. ) above and Eq. ) of Proposition 1.6 we get I p sγα w B x 1 μ ) = β T α,x I β μ),

8 8 N. Papalexiou / Journal of Algebra ) 1 15 which implies that VI p s γα w B x 1 μ)) = β T α,x D β μ. Since {γ α α B }=B,wehave V I p ν + λ) ) α B V = α B V Ip sα w B x 1 μ )) Ip sγα w B x 1 μ )), and the theorem is proven. 2. A lower estimate of the characteristic variety of an induced ideal We keep the notation of Section 1. Our aim is to calculate a lower estimate of the characteristic variety of an induced ideal in Ug). Here we consider first the projection in Ul) of the induced ideal and then its translation principle relative to l. Let B be a subset of B μ. We recall that w D μ := w μ w B denotes the longest element in D μ = D B W μ. It is clear that w D μ B = w μ w B B = w μ B ) is a subset of B μ. Lemma 2.1. For any subset B of B μ we set B := w D μ B B μ. Denote by W and W the corresponding Weyl groups. Then, the map w w := w D μ w w 1 D μ induces a group isomorphism from W onto W. In addition w D = μ w 1 D μ. Proof. For any α B, w D μ s α w 1 D μ = s w D μ α. Since W is generated by its reflections, we have the first assertion of the lemma. Now, this isomorphism preserves Bruhat ordering, hence w D μ w B w 1 D μ = w B. Furthermore, wμ 2 = w2 B = 1, thus w D μ = w μw B = w μ w D μ w B w 1 D μ = w μ w μ w B w B w 1 D μ = w 1 D μ. Proposition 2.2. Let B B μ and B = w D μ B B μ and let R = R ZB and R = R ZB its corresponding roots systems. Then the map φ : α w D μ α from B to B induces an isomorphism of root systems R R. In addition it induces an isomorphism of Lie algebras φ : l B l B. Proof. Since φ : α w D μ α is a group isomorphism and α, β = w D μ α, w D μ β for any α, β B, φ extends uniquely to an isomorphism ˆφ : h h mapping R onto R and satisfying ˆφα), ˆφβ) = α, β for all α, β R. By isomorphism theorem [8, Theorem 14.2] there exists a unique isomorphism φ : l B l B extending φ. Let P : Ul) Uh) = Sh) be the relative Harish-Chandra projection with kernel n B Ul)+ Ul)n + B. We also define P l : Ug) Ul) the projection of Ug) onto the first summand

9 N. Papalexiou / Journal of Algebra ) of Ug) = Ul) u Ug) + Ug)u + ). It is obvious that P = P P l. For any two-sided ideal J in Ul) we denote by V J ) its relative characteristic variety defined by the following way: V J ) = { λ h P J )λ) } = 0. We shall now calculate the relation between the characteristic varieties of primitive ideals in Ul B ) and in Ul B ). Proposition 2.3. Suppose that μ is a B-dominant regular weight. Let w W and w = w D μ w w 1 D μ W. Then, one has V I w μ) ) = w D μ V I w w 1 D μ μ)). Proof. Proposition 2.2 implies that the following diagram is commutative: Ul B ) φ Ul B ) P P Uh) ˆφ Uh) Consequently, since φ 1 I w μ)) = I w 1 D μ w μ), we obtain P I w μ) ) = ˆφP φ 1 I w μ) ) = ˆφP I w w 1 D μ μ)) which implies, using Proposition 2.2, that for any λ h P I w μ) ) λ) = ˆφP I w w 1 D μ μ)) λ) = P I w w 1 D μ μ)) w 1 D μ λ). Thus, V I w μ) ) = { λ h P I w μ) ) λ) = 0 } = { λ h P I w w 1 D μ μ)) w 1 D μ λ) = 0 } = w D μ { λ h P I w w 1 D μ μ)) λ) = 0 } = w D μ V I w w 1 D μ μ)). Proposition 2.4. Assume that μ is a B-dominant regular weight. For any w W, a) the induced ideal I pb w μ) is primitive. More precisely, I pb w μ) = Iw D μ w μ)

10 10 N. Papalexiou / Journal of Algebra ) 1 15 and V I pb w μ) ) = D B V I w μ) ). b) The induced ideal I pb w w 1 D μ μ) is primitive. More precisely, and V I pb I pb w w 1 D μ μ) = Iw D μ w μ) w w 1 D μ μ)) = D B V I w μ) ) = D B V I w w 1 D μ μ)). Proof. a) Let w W and w := w D μ w w 1 D μ W. By Lemma 7.16 in [10], Lw w 1 D μ μ) = Ug) UpB ) L w w 1 D μ μ), where L w w 1 D μ μ) denotes the simple highest weight l B - module. Hence I pb w w 1 D μ μ) = I w w 1 D μ μ). 2.1) By Satz in [10] and Lemma 2.1, since the induced ideal is primitive I pb w μ) = I pwd μ B w D μ w μ) = I pb Consequently, by Eq. 2.1) and Lemma 2.1, we also have w w 1 D μ μ). I pb w μ) = I w w 1 D μ μ) = Iw D μ w μ). Proposition 1.4 now implies that VI pb w μ)) = D B V I w μ)). b) By assertion a) it follows that I pb w μ) is a primitive ideal equal to Iw D μ w μ) = Iw w 1 D μ μ). On the other hand, [10, Satz 15.26] implies that I p B w μ) = I pb w D μ w μ) = I pb w w 1 D μ μ). Hence I p B w w 1 D μ μ) is a primitive ideal equal to Iw w 1 D μ μ). Finally, by assertion a), Lemma 1.3ii) and Proposition 2.3 one has V I pb w w 1 D μ μ)) = V I w w 1 D μ μ)) = V Iw D μ w μ) ) = D B V I w μ) ) = D B w 1 D μ w D μ V I w w 1 D μ μ)) = D B V I w w 1 D μ μ)). For any Ug)-module M we write P l M) its restriction as Ul)-module. We denote by T ν μ the translator functor of l-modules.

11 N. Papalexiou / Journal of Algebra ) Proposition 2.5. Take μ h arbitrary in some facette F and let ν h in the upper closure of F. Then T ν μ P l Ind g p L μ) = P l Ind g p T ν μ L μ). Proof. Let E be a simple finite-dimensional Ul)-module with extremal weight ν μ. T μ νp l Ind g p L μ) is defined to be the direct summand of E P l Ind g p L μ) with central character defined by ν and since E P l Ind g p L μ) is isomorphic to P l Ind g pe L μ)), wehave T ν μ P l Ind g p L μ) = P l Ind g p E L μ) ) ν = P l Ind g p T ν μ L μ). Lemma 2.6. For each two-sided ideal I in Ug), Proof. Since P = P P l,wehave VI) = V P l I) ). VI) = { λ h } P I)λ) = 0 = { λ h P P l I)λ) = 0 } = V P l I) ). We will prove now the basic result of this paragraph. We will calculate a lower estimate of the characteristic variety of an induced ideal. This proves conjecture C1) in [4, appendix]. Theorem 2.7. Let ν h ), λ z such that J = Ann Ul) L ν) and I p J, λ) the corresponding induced ideal. Then, one has V I p J, λ) ) D B V J ) + λ ). Proof. Let μ h dominant in the Weyl orbit of ν + λ. There exists w W such that w μ = ν + λ. Since W = D B W, there exists w W and x D B such that w = w x 1. By Lemma 7.16 in [10], the induced module Ug) pb L w w 1 D μ μ) is simple, hence P l Ann Ug) Ind g p L w w 1 D μ μ) = Ann Ul) P l Ind g p L w w 1 D μ μ). 2.2) The elements w D μ and x are in D B,sow 1 D μ μ and x 1 μ are B -dominant. Thus, we may consider the translator functor T x 1 μ w 1 D μ μ: T x 1 μ w 1 D μ μ P l Ann Ug) Ind g p L w w 1 D μ μ) = = T x 1 μ w 1 D μ μ Ann Ul) P l Ind g p L w w 1 D μ μ) Ann Ul) T x 1 μ w 1 D μ μp l Ind g p L w w 1 D μ μ).

12 12 N. Papalexiou / Journal of Algebra ) 1 15 By Proposition 2.5 the last is equal to Ann Ul) P l Ind g p L w x 1 μ) which contains P l Ind g p I w x 1 μ). Thus using Lemma 2.6 one gets V I pb w x 1 μ )) = V P l Ind g p I w x 1 μ )) By Eq. 2.2), Propositions 1.2 and 2.4b) V T x 1 μ w 1 D μ μp l Ann Ug) Ind g p L w w 1 D μ μ)). V T x 1 μ w 1 D μ μp l Ann Ug) Ind g p L w w 1 D μ μ)) = V T x 1 μ Ann w 1 D μ μ Ul) P l Ind g p L w w 1 D μ μ)) = D B V I w x 1 μ )). Thus, finally, summarizing one has V I pb w x 1 μ )) D B V I w x 1 μ )) = D B V I ν + λ) ) = D B V J ) + λ ). Remark 2.8. The proposition above implies obviously a better result for the characteristic variety of I p J, λ): V I p J, λ) ) 3. Characteristic variety and prime spectrum ξ D B V J )+λ) V Iξ) ). The object of this section is to describe the characteristic variety of a prime ideal in Ug).According to [4, Corollary 5.7], every prime ideal can be expressed in an induced form convenient for studying its characteristic variety using the results of Section 2. First, following W. Soergel in [13], given any subset Ω in h we define the semi-prime ideal I Ω in Ug) by the following way: I Ω = λ Ω Iλ). If Ω is irreducible, I Ω is prime [6, Proposition 3.2.5]. In addition the map Ω I Ω from the set of irreducible subsets in h to Spec Ug) is surjective. Hence, if P is a prime ideal in Ug), there exists an irreducible subset Ω in h such that P = I Ω. Next by W. Borho and A. Joseph in [4] there exist a parabolic subalgebra p of g with Levi subalgebra l, center z and ν h ), and an irreducible subset Λ of z such that I Ω = I p J, Λ) := λ Λ I p J, λ),

13 N. Papalexiou / Journal of Algebra ) where J = Ann L ν) is a primitive ideal in Ul). First we have a crucial lemma which is due to A. Joseph. Lemma 3.1. For any irreducible subset Λ of z we have V λ Λ ) I p J, λ) = V I p J, λ) ). λ Λ Proof. By [3, Lemma 3.9a)] we have I := λ Λ I pj, λ) = λ Λ I pj, λ). Thus, by [11, Lemma 1v)] we take the inclusion V λ Λ ) I p J, λ) V I p J, λ) ). λ Λ Suppose now that ν VI), then I Iν). By [4, Theorem 5.1] there exists λ 0 Λ such that I I p J, λ 0 ), then ν VI ν)) VI p J, λ 0 )), hence the inclusion is proven. Let α B and p α the parabolic subalgebra defined by B ={α} with Levi factor l α.let ν h ) be a dominant regular relative to B ={α}, Λ an irreducible subset of z and J = Ann Ulα ) L ν) a primitive ideal in Ul α ). Denote by C the fundamental chamber. Let w 1,w 2,...,w s W such that w i ν + Λ for i = 1, 2,...,s.WesetΛ i = w i C ν + Λ. It is obvious that s i=1 Λ i = Λ. We recall that for any w W, T α,w ={β B wα,ϖ β 0}. Now using the results of Proposition 1.6 of Section 1 we calculate explicitly the characteristic variety of a prime ideal P = I pα J, Λ) induced by a parabolic subalgebra which Levi factor is isomorphic to sl2, C) and a primitive ideal J = Ann L ν) in Ul) such that α R + ν+λ for all λ Λ. Proposition 3.2. Let Λ an irreducible subset of z. Suppose that α R ν+λ + for any λ Λ. Then, V I pα J, Λ) ) s = i=1 λ Λ i β T α,wi D β w i ν + λ). Proof. It follows by Proposition 1.6 and Lemma 3.1 above. The following proposition gives a partial answer to the conjecture C2) in [4, appendix]. Proposition 3.3. Let P be a prime ideal such that P = I p J, Λ) with p a parabolic subalgebra with Levi l and J a primitive ideal in Ul). Then we have, V I p J, Λ) ) D B V J ) + Λ ). 3.1) Proof. It follows by Theorem 2.7 and Lemma 3.1. Remark 3.4. Inclusion 3.1) is in general strict and conjecture C2) is not true see [12, example]).

14 14 N. Papalexiou / Journal of Algebra ) 1 15 The following proposition calculates the Gelfand Kirillov dimension of Ug)/P for a prime ideal P in Ug) and answers Problem 3 in [13]. Proposition 3.5. Let P be a prime ideal in Ug). There exists a parabolic subalgebra p of g with Levi l, a rigid primitive ideal J of Ul) and an irreducible subset Λ of the dual of the center z of l such that P = λ Λ I pj, λ). Then, the Gelfand Kirillov dimension of Ug)/P is equal to Dim Ug)/P = 2dimg/p + Dim Ul)/J ) + dim Λ. Proof. Let RΛ) be the ring of regular functions on Λ as a p-module by identifying z with p/[p, p]. Then, the prime ideal I p J, Λ) is the annihilator of the induced module Uu ) L ν) Sz )/RΛ). Thus, Dim Ug)/P = Dim Ug)/I p J, Λ) = 2Dim Uu ) L ν) Sz )/RΛ) ) = 2dimg/p + Dim Ul)/J ) + dim Λ. Thus we have the proposition. We give now a main example: 3.1. The case of sl4) Let R ={α 1,α 2,α 3,α 1 + α 2,α 2 + α 3,α 1 + α 2 + α 3 } be the set of positive roots of the root system of sl4) with base B ={α 1,α 2,α 3 } and Weyl group W.Letϖ α1,ϖ α2,ϖ α3 the fundamental weights corresponding to α 1,α 2,α 3.TakeB ={α 1,α 3 } and p B the corresponding parabolic with Levi l B and center z. LetW ={1,s 1,s 3,s 1 s 3 } be the Weyl group of l B. It is obvious that D B ={1,s 2,s 1 s 2,s 3 s 2,s 1 s 3 s 2,s 2 s 1 s 3 s 2 }.Takeν = α 1 + α 3 and λ z = Cϖ α2. We consider the irreducible subset in h, Ω = ν + z and the corresponding prime ideal I Ω. For any λ z 0 = Cϖ α 2 \ Zϖ α2, VI pb ν + λ)) = D B V J ) + λ). Let z 1 = Zϖ α 2 and Ω 1 = ν + z 1.We decompose Ω 1 = 6 i=1 Ω1 i such that: By Lemma 3.1, one has that Ω 1 1 ={λ Ω 1 λ + ρ C}, Ω 1 2 ={λ Ω 1 λ + ρ s 2 C}, Ω 1 3 ={λ Ω 1 λ + ρ s 2 s 1 C}, Ω 1 4 ={λ Ω 1 λ + ρ s 2 s 3 C}, Ω 1 5 ={λ Ω 1 λ + ρ s 2 s 3 s 1 C}, Ω 1 6 ={λ Ω 1 λ + ρ s 2 s 3 s 1 s 2 C}. VI Ω ) = λ z V IpB ν + λ) )).

15 N. Papalexiou / Journal of Algebra ) Thus, we find VI Ω ) = D B Ω {s 1 s 2,s 3 s 2,s 2 s 1,s 2 s 3,s 1 s 3 s 2,s 3 s 2 s 1,s 1 s 2 s 3 } s 2 Ω1 2 s 1s 3 s 2 Ω1 5 ) {s 3 s 2 s 1,s 1 s 2 s 3,s 2 s 1 s 3 } s 1 s 2 Ω1 3 s 3s 2 Ω1 4 ). Take μ s 2 Ω1 2. Then Is 2s 1 s 3 s 2 μ) I Ω. Set λ 1 = s 2 s 1 s 3 s 2 μ. Then we have λ 1 VI Ω ) but λ 1 / ξ D B Ω VI ξ)). Thus Soergel s revised conjecture see is not true in general. Acknowledgments Part of this work was effected during the visit of the author at the Mathematics Department of the Weizmann Institute in Israel. I thank A. Joseph for his advice during the preparation of this work and W. Soergel for a helpful conversation. References [1] D. Barbasch, D. Vogan, Primitive ideals and orbital integrals in complex exceptional groups, J. Algebra 80 2) 1983) [2] W. Borho, P. Gabriel, R. Rentschler, Primideale in Einhüllenden auflösbarer Lie-Algebren Beschreibung durch Bahnenräume), Lecture Notes in Math., vol. 357, Springer-Verlag, Berlin, [3] W. Borho, J.C. Jantzen, Über primitive Ideals in der Einhüllenden einer halbeinfachen Lie-Algebra, Invent. Math. 39 1) 1977) [4] W. Borho, A. Joseph, Sheets and topology of primitive spectra for semisimple Lie algebras, J. Algebra 244 1) 2001) [5] N. Bourbaki, Éléments de mathématique, Masson, Paris, 1981 Groupes et algèbres de Lie. Chapitres 4, 5 et 6 [Lie groups and Lie algebras. Chapters 4, 5 and 6]). [6] J. Dixmier, Algèbres enveloppantes, Gauthier Villars, Paris, 1974, Cahiers Scientifiques, Fasc. XXXVII. [7] M. Duflo, Sur la classification des idéaux primitifs dans l algèbre enveloppante d une algèbre de Lie semi-simple, Ann. of Math. 2) 105 1) 1977) [8] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Grad. Texts in Math., vol. 9, Springer- Verlag, New York, 1978 second printing, revised). [9] J.C. Jantzen, Modulen mit einem höchsten Gewicht, Lecture Notes in Math., vol. 750, Springer-Verlag, Berlin, [10] J.C. Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete 3) [Results in Mathematics and Related Areas 3)], vol. 3, Springer-Verlag, Berlin, [11] A. Joseph, A characteristic variety for the primitive spectrum of a semisimple Lie algebra, in: Non-Commutative Harmonic Analysis, Actes Colloq., Marseille-Luminy, 1976, in: Lecture Notes in Math., vol. 587, Springer-Verlag, Berlin, 1977, pp [12] N. Papalexiou, On the prime spectrum of the enveloping algebra and characteristic varieties, J. Algebra Appl., in press. [13] W. Soergel, The prime spectrum of the enveloping algebra of a reductive Lie algebra, Math. Z ) 1990)