On the Gel'fand-Kirillov conjecture for induced ideals in the semisimple case

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B ULLETIN DE LA S. M. F.

A ^NTHONY J ^OSEPH

On the Gel’fand-Kirillov conjecture for induced ideals in the semisimple case

Bulletin de la S. M. F., tome 107 (1979), p. 139-159

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107, 1979, p. 139-159.

ON THE GELTAND-KIRILLOV CONJECTURE FOR INDUCED IDEALS IN THE SEMISIMPLE CASE

BY

ANTHONY JOSEPH (*) [Tel-Aviv University]

RESUME. — Soient g une algebre de Lie semi-simple complexe, p une sous-algebre parabolique de 9, E un £/(p)-module simple de dimension finie, M Ie (7 (g)-module induit par E, et I = Ann M. La representation de U (9) dans M definit une injection \i/

de U (g)/I dans la partie ad g finie de Hom^ (M, M). Supposons que M est simple et v surjective. Alors I est un ideal primitif, done U (Q)/I admet un anneau de fractions Fract U(Q)II. Notons m == codim p, et ^m 1'algebre de Weyl d'mdice m sur C. On demontre que Fract U (g)/J = End E ® Fract ^rn, a isomorphisme pres et que Fract U(Q)II contient un sous-corps maximal commutatif qui est ad Q stable. Pour g simple ^ sl(n + 1) : n e N4', on montre que ce dernier resultat n'est pas toujours vrai pour un ideal primitif quelconque.

ABSTRACT. — Let 9 be a complex semisimple Lie algebra, p a parabolic subalgebra of g, E a finite dimensional U (p)-module, M the (7(g)-module induced from E, and / = Ann M. The representation of U (9) in M defines an injection \|/ of U (Q)/I into the ad g finite part of Horn/, (M, M). Suppose that M is simple and \y surjective.

Then I is a primitive ideal, and so U(cf)/I admits a ring of fractions Fract U(Q)II.

Set m = codim p and let ^rn denote the Weyl algebra of index m over C. It is shown that Fract U (g)/J = End E ® Fract J^n,, up to isomorphism and that Fract U (g)/J admits a maximal commutative subfield which is ad Q stable. For g simple ^ sl (n + 1) : n e N4", it is shown that this last result is not a general property of an arbitrary primitive ideal.

1. Introduction

1.1: Let A: be a commutative, algebraically closed field of characteristic zero, 9 a k — Lie algebra which is finite dimensional and algebraic. Let U (9) (resp. K(o)) denote the enveloping algebra (resp. field) of 9, and Prim £/(g)

(*) Texte recu Ie 2 juin 1978. Work supported by the Centre National de la Recherche Scientifique.

Anthony JOSEPH, Mathematics department, Tel-Aviv University, Ramat Aviv (Israel).

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(resp. Spec U (9)) the set of primitive (resp. prime) ideals of U (9).

Given I e Spec £/(g), then by Goldie's theorem U(^)/I admits a ring of fractions which for some /zeN⁴" is isomorphic to the matrix ring M^

over a skew field K. We call n the Goldie rank, and AT the Goldie field of I.

1.2: Given r , ^ e N , let ^ 5 denote the generalized Weyl algebra of order r and index s over /r. ja^ is isomorphic to the associative ^-algebra with identity and generators x^ 8/8 Xj : i = 1, 2, . . . , / - + s; j == 1, 2, . . . , r.

GEL'FAND and KIRILLOV conjectured ([12]-[14]) that K(o) is isomorphic to the generalized Weyl field Fract ^^, with dim 9 = 2 r+s, index 9 = ^.

This generalizes naturally to the conjecture that the Goldie field of any I e Spec U (9) is isomorphic to the Weyl field Fract ja^ 5 with Dim U(Q)/I == 2 r+s, Dim Cent (Fract U ( ^ ) / I ) = .y, where Dim denotes GEL'FAND-KIRILLOV dimension. This has been established for g solvable ([3], [25]) and for the minimal primitive ideals in 9 semisimple [6]. Yet outside g solvable the conjecture may be too strong. This is indicated by representation theory [14] and algebraic geometry. Indeed take r = sl (2, k), Q == r © m, where m is commutative, satisfies 3 < dim m < oo and is simple as an ad r module. Let 5" (m) denote the symmetric algebra over m, and C(g) the centre of K (9). Since C(g) = (Fract S (m)/, the most natural conscruction of the Weyl field for the zero ideal of U(o) requires Fract 5' (m) to be a pure transcendental extension of (Fract S (m)/;

but this is generally false (SERRE, unpublished). To avoid this difficulty one might demand that only some algebraic extension of the Goldie field be isomorphic to a Weyl field. This is established in [14] for the zero ideal in 9 semisimple, with the Weyl group being the Galois group of the extension. Yet it is technically easier and perhaps more natural to just restrict the conjecture to primitive ideals as suggested in [17]. Furthermore the orbital method for constructing induced ideals suggests the further conjecture [17] that 1 e Prim U(c^) is induced if, and only if, the Goldie field of I admits a maximal commutative subfield which is ad 9 stable. The analysis of [19] shows that this holds for 9 solvable.

1.3: Under certain minor technical restrictions, the main result of this paper (Theorem 4.3) establishes the first conjecture for induced primitive ideals in 9 semisimple. It obtains by refining the Goldie rank computation of CONZE-BERLINE and DUFLO [7] (Sect. 8), through the preparation theorem of [20]. A new feature derives from having to consider primitive ideals of Goldie rank > 1 and for this we derive a general ring theoretic result in Sect. 2. The second conjecture is discussed in Sect. 5.

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I should like to thank M. DUFLO for explaining to me what had been proved in [7].

Conventions. — Terms like noetherian mean left noetherian. The symbol # denotes the smash product defined in [25] (Remarks 2.9). It defines a skew polynomial extension in the sense of [3] (Sect. 4). An element of a ring is called regular if it is a non-zero divisor on both the left and the right. A left ideal of a ring is called essential if it intersects non-trivially with every non-zero left ideal of the ring. A ring R is said to be torsion-free if mx = 0, m e Z, x e R implies m = 0, or x = 0.

For g semisimple, a primitive ideal / is said to be induced if I Ann M;

M = ind (N, 0 ^ 9 , where a is a subalgebra of 9 (possibly g itself) and dim N < oo.

2. A stepping-up theorem

2.1: Let A be a torsion-free ring, X a locally nilpotent derivation of A, and set Ax == { aeA; X a = 0 }. From A being prime Goldie, it does not follow that Ax is prime Goldie. For example, take A = M^, X = ad e with e upper triangular. Yet we show that the converse does hold and we analyse the structure of A with respect to Ax.

Given Fa subspace of A, set Vx = V r\ Ax. Call V, X-stable if XV c: K Clearly Vx ^ 0, for any non-zero ^-stable subspace K Let S (resp. T) denote the set of regular elements of A (resp. A^.

LEMMA. - T == Sx.

T ^ Sx trivially. Conversely given t e T, a e A such that ta = 0 (or at = 0), choose n e N such that X" aeAX-{Q}. Then t (X" a) = 0 (or (JT ^) t = 0) and so X" a = 0, contradicting the choice of n.

2.2: From now on we assume that Ax is a prime Goldie ring.

LEMMA. - Given A + A^x, then B : = [ a e A ; X a e T ] ^ 0.

Set £/ = { M e ^4; M = 1^, for some v e A } . Then L^ is a two-sided ideal of Ax. If A ^ ^, then ^x ^ 0. Since y^ is prime Goldie,

Ux n T ^ 0 and so j8 7^ 0.

2.3: From now on we assume that A 7^ ^.

LEMMA. — For each pair beB, t e T, there exists a pair b' e B, t ' e T such that t1b = V t.

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Set s = Xbe T. Since T is an Ore set for Ax, there exists a pair s ' e T, a e A^x such that as = s ' t. Then a e T and

X(bs't-sab) = s s ' t - s a s == 0.

Hence there exists s " e T, such that

stf(bstt-sab)eAxt^At.

This gives 6' e A such that ^" 5'afi = 6' r. Then (JT&') t = s" sas = s " s s ' t.

Now f e T so by 2.1, we obtain A" V = s " s s ' e T. Hence b ' E B and setting ^/ = s" sa, we obtain t1 e T and t ' b = b' t, as required.

2.4: Given C a subring of ^4 containing Ax and for which T is an Ore subset, we denote by Cj. = { t ~1 c; t e T, c e C } the localization of C at T.

Let D be the subring of A generated by A^x and B. By 2.1 and 2.3, T is an Ore subset for Z).

LEMMA:

(i) For each ae A there exists s e T such that sa e D;

(ii) T ^ an Ore subset for A and Ay = Dy

We recall our assumption that A + Ax. Then by 2.2, there exists x e DT such that X x = 1. Since A is torsion-free, one has

Xn: == (nrr^eDr for each neN^

and this element satisfies X" x^ = 1. For each m e N, set 4 n = { ^ e ^ j r ^ a ^ O } .

Consider oe^\^_i. Choose t e T , such that b : = = t x ^ e D . Then Z" (A (Z" ^-^) = 0 and so b (Z" d)-ta e A,.,. Yet b (X" a)eD and so taeD+A^-i. Since ^4o = ^x c A we obtain (i) by induction on n, and (ii) follows from (i).

Remark. — The above rather indirect proof of (ii) is necessitated by the fact that we do not know that A is Goldie.

2.5: Set R = Fract Ax. Obviously (A^)x = R. We have seen that there exists xeAj. such that Xx = 1. From X\_x, R~\ = [1, R~\ = 0, we obtain [x, R~\ c: R. Define the skew polynomial extension (R # Q [jc])

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of R by Q [x] to be the Q-algebra with underlying vector space jR ® Q [x^ (where ® = ®o) and multiplication

(a # ^)(& # X^s) == a& # x^+ra [x, &] # x¹-⁴-⁸-¹.

Since Ax is a prime Goldie ring, there exists n e N4' and a skew field K such that 1? = Mn ® K, up to isomorphism. By [20] (Lemma 6.3), we may adjust ad x by an inner derivation of Mn 0 K to obtain

{R # Q [x]) = M, ® (K # Q [x]),

up to an isomorphism. Furthermore K # Q [x] is an Ore domain and we set K ' = Fract (K # Q [x]).

THEOREM:

(i) Ay = (R # Q [x]), Mp /o an isomorphism', (ii) ^4 ^ a prime Goldie ring;

(iii) Fract A = M^ ® K\ up to an isomorphism^

(iv) Ay is the largest X-stable subring of Fract A on which X is locally nilpotent.

Recall that X is a locally nilpotent derivation of A^^x, X x = 1, and (^r)^ = ^. By [18] (2.2 (i), Taylor's lemma), this gives (i). We have seen that Fract (R # Q [.x]) = M^ 00 7C/. Through the converse of Goldie's theorem, this gives (ii) and hence (iii). (iv) obtains from [18]

(2.6 (iv)) applied to (iii).

2.6: Let k be a commutative field of characteristic zero, A an associative A:-algebra, and m a finite dimensional k-Lie algebra of locally nilpotent derivations of A. By [18] (2.2), m is a nilpotent Lie algebra. Set A^m = Q {A^X; Xem}. The following immediate consequence of 2.5 is needed later on.

THEOREM. — Suppose A^ is a prime Goldie ring. Then A is a prime Goldie ring and Fract A^ = (Fract ^)m.

Remark. - Set R = Fract ^m. Then R = M^ ® K, for some n e N4' and some skew field K. By 2.5, Fract A == M^ 00 K ' , for some skew field K ' which is the quotient field of some skew polynomial extension Q of K.

Furthermore each Xe m is a locally nilpotent derivation of Q and Q^ == K.

Given Q^ c: Cent Q (or K c: Cent K ' ) it follows by [18] (3.2), that Q is a generalized Weyl algebra over K. This generalizes [18], (3.2), to the prime situation.

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3. A theorem of Conze-Berline and Duflo

3.1: Let 9 denote a complex, semisimple Lie algebra with triangular decomposition 9 = n @ 1) © n~ ([8], 1.10.14). Let u—>u denote the principle antiautomorphism of U (cf). Let R (resp. ^+) denote the set of non-zero (resp. positive) roots for t) in 9. Let u —> ^ denote the antiauto- morphism of order 2 of U (9) defined by ^ == X_^ for all a e R, and tH = H for all He I). Set b = n © I), and given p =? b a subalgebra of 9, we write

p = = m © r , $ = [r, r], ^ = t ) n s and m~ =tm, where m (resp. r) is the nilradical (resp. reductive part) of p. Let R⁺ (resp. R^) denote the positive (resp. negative) roots for I) in r. Let W (resp. W^) denote the Weyl group for (9,1)) (resp. ($, I)i)), and w^ the unique element of W^ taking R^ into J^~, Let p (resp. pg) denote the half sum of roots of R+ (resp. R^^R^). One has pg e l)j-. Set cr, = p- p,. Let ^ (resp. ^+) denote the set of integral (resp. dominant integral) weights defined with respect to $. Given ^i e ^+, let E^ denote the simple, finite dimensional (7(s)-module with highest weight ^. Given Xel)* such that Xi : = X|^e^4', let E^ denote the simple £/(p)-module whose restriction to U ($) coincides with E^ and whose restriction to 1) is X, and let e^ denote its highest weight vector. Set

M,(?i)=ind'(£,,pt9) ([8], 5.1.1).

One has

M,W= (7(9)®(;(p)^-p,= t7(nn®£^= U(n-)e^

up to isomorphisms. In particular. My (^) is generated as a £/ (n'~)-module by a highest weight vector of weight ^+<7g—p. Set 4+03 = Ann Mp (^).

Identify £/ : = U (9) ® £/ (9) canonically with U (9 © 9), and consider Home (My (^-), Mp (^^/)) as a (/-module through

( ( a ® & ) . r ) m = C a T b ) m , for all

a, be U (9), meM^(X), reHomc(M^(X), My(X')).

Define the embedding j : 9->9 © 9, through j(X) = (A",-^), for all JTe 9. Set I = 7 (9) and let L (My (^), My (K')) denote the £/-submodule of Home (My (k). My (^/)) of I finite elements.

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Consider (7 (9) as a (7-module through

(a®b).u =taub, for all a, b, u e l/(9).

The representation of U (9) in My (K) induces an injective homomorphism ^F of ^(9)/4+o, into L(My d), My (^)). By [7] (2.12, 4.7, 5.5 and 6.3), we have the following proposition.

PROPOSITION. - Suppose that 2 ((X+o,), a)/(a, a) ^ N4', /or a//

^eR+\R^, then:

(i) Mp (^) f5' irreducible;

(ii) x? ^ surjective.

Remarks. — Clearly^x? is surjective if, and only if, ^ (1) is a cyclic vector for L (My (K), My (^)). Given p $ b, it can happen that My (K) is irreducible, yet x? is not surjective [7] (6.5). On the other hand, it can also happen that 7^+^ = 4.+^, given ^ + 0 g e ^(Pi'+a,), even though X ^ ^/. Furthermore, if—w^(k+<jy) lies in the closure of che positive Weyl chamber, then the hypothesis of the lemma is satisfied. For integral K these induced ideals are just the primitive ideals minimal for given

"T-invariant" [4] (2.17 d). An implicit conjecture in [7] is that these are no further induced ideals; but this is false [24] (3.7). (i) is a special case of [12] (Satz 3).

3.2: We assume from now on that My (X) is irreducible (in which case 4+CTg ls primitive) and that ^F is surjective, so we can identify

^(9)/4+J, with L (My (X), My (k)). Set A == U(Q)/I^^ and let S denote the set of regular elements of A. The following theorem is due to CONZE-BERLINE and DUFLO [7].

THEOREM. - For each b e (Horn (My (X), My (^)))7 (lt) there exists s e S such that a : = sb e A. Furthermore a ^ 0 if b ^ 0.

Remarks. — The theorem is not explicitly stated in [7]; but DUFLO pointed out to me that it follows from their analysis. In detail, given l^e^ such that [i |^ = 0, then the identity map on U(n~) induces by passage to the quotient, a linear isomorphism 9p of My (K) into My (^—\i) Given b as above, then by [7] (5.9), one can choose p, such that 9^ b e L (My (\), My (K - ^)) and by [7] (8.4 and 4.8), such that My (k - ^i) is irreducible. Then by [7] (5.8), L (My (k-[Ji), My (k)) is non-zero and by [7] (8.5), there exists (p^ e L (My (^~[i). My (X)) such that (p^ 9^

is a regular element of L (My (k). My (^)). The first part obtains on setting a = (p^ 9^ b, s = (p^ 6^. The last part is explicitly proved in [23] (5.9).

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4. Main theorem

4.1: Retain the notation of 3.1, and set M = My (X), E = E^p . Let T : £/(?)—)• End £' define the irreducible representation of U (r/ acting in E. Given ae U(m~), be U(o), define

^ ® T (fc) G (Horn (M, M)V^(m) through

(ra®^(b))(^®e)=ua®^ft(b)e, for all ueU(m~),eeE.

A straightforward computation gives the following lemma.

LEMMA:

(i) the map a 00 T (b) i-> ^ ® T (&) extends linearly to an isomorphism of U (m~) ® End E onto (Horn (M, M))³ <^m);

(ii)/or ^// A^er, one has,

-OCZ).(r, ® r(fc)) = (r^ ^ ® T(fc))+(r, ® T [Z, &]);

(iii) (Horn (M, M))-^{7 (m)} ^ y(s) finite and so generated by the action ofj (s) on (Horn (M, M))-⁷ ("^

4.2: Set 5 = (Hom(M, M)Y'^\ By 4.1 (1), B is a prime, noetherian ring and Fract B = K(m~) ® End £, up to an isomorphism. Assume the hypotheses and notation of 3.2, and in particular identify A : == U(^)/I^y

with L (M, M).^as

PROPOSITION. - A^ admits a ring of fractions, and Fract ^^m" = Fract B.

Let us recall the basic structure theorem for £/(g) given in [20] (3.4).

Define the subalgebra c(m) (or, simply, c) of b as in [20] (2.6), and set c~ =^tc. Let m^ denote the nilradical of c~. Then c~ == m^ © I with I a subalgebra of t) and n~ => nto =» m~. Furthermore by [20]

(2.6 (iii)), we have Z(mo) c ^(c-)¹"" = Z(m-). By [20] (2.4), there exists z e Z ( m o ) such the localized algebra U(^\ is defined and takes the form

U(^ = (U^r° #z(mo) U(c-)\ =((U^r°^z^-) U(m,)) # U(l))^

Here the smash product # is defined through the adjoint action of c"

in £/(g). Suppose /e Spec U(^) satifies I n Z (nto) = 0. Then through the methods of [3] (Sect. 4) applied to the above formula (as show in [20],

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Sect. 6) it follows that the localized algebra (^(cQAO^0 is defined and is prime noetherian. Take / = 4+og- Since the restriction of ind" (r; p f 9) to £/(m~) identifies with its left regular representation, it follows that I r\ U(m~) = 0. Hence the localized algebra A^° is defined and is prime noetherian. By 2.6 applied to rrio'/m", it follows that A^~ is defined and is prime, Goldie. By 2.6 applied to m~, we have

(FractA)"¹' = (FractA^" = FractA^" = FractA¹"".

On the other hand by 3.2 and 4.1 (iii), we have FractA => B and so

(Fract A/"" -=> FractB =3 FractAm",

which with the previous equality gives the required assertion.

Remark. - From /eSpec £/(g), it does not follow that (^(g)//)" is prime. For example take / of finite codimension > 1.

4.3 (Notations 3.1, 3.2, 4.1): Set

I = l^^, n = dim £, 2 m = 2 codim p = Dim U (g)//

([I], 2.3 (&)). Recall that / = Ann M.

THEOREM. - Suppose that M = ind" (E, p T 9) is simple, and that the representation of U (9) in M sets up a bijection of U (g)/7 onto the l-finite part L (M, M) of Home (M, M) (cf. (3.1)). Then

(i) Fract U ( ^ ) I I = M^ ® Fract ^, ^p ^ ^ isomorphism;

(ii) 6W^ may choose

^ = C [xi, X2, ..., x^, 819x^ 918x2, . . . , 3/<9xJ 5'o ^^ C [xi, x^ . . . , x^] LV ad ^'stable.

Define c, m~, I and z as in 4.2. As in 4.2, we obtain from [20] (3.4), that A^ is defined and takes the form

A, = (A-- #z(m-) ^(c~)). = (((A"" (2) z(m-) ^Ono)) # ^(0)z.

We remark that (A111" (x)z(m-) ^o)) = (-4mu ®z(mo) ^^o)). ^ that

the smash product is defined through the adjoint action of I in U (9). This

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coincides with the action of j (I) in U (9) considered as a (7-module. Let S denote the set of regular elements of y4"1. Then z e S and

S-^A = (FractA-- ®z(m-) UW) # U(l),

=((iC(m-)®End£)(x)^-)[/(mo))# C7(I), by 4.2,

= End£ ® ((IC(m-)® z^-) l7(mo)) # 17 (I)),

up to an isomorphism. The last step was obtained by adjusting each ad X : Xe I by an inner derivation of End E (x) ((K(m~) (§)^ ^-^ U(m~)). This is achieved by dropping the second term in the right hand side of 4.1 (ii).

It follows that up to an isomorphism

Fract((X(m-)(x) ;^-) U (mo)) # 17 (I)) = Fract C/(m- ® c~)/J, where J is the two-sided ideal generated by the semi-invariants

y (x) 1-1 (x) y :yeZ(m~).

Obviously J is a prime ideal and the Lie algebra m~ © c~ is solvable and algebraic. Noting that

Dim l/(9)/J = Dim U(m ~ © c~ )fJ = dim m + dime - Dim Z(m~ ) = 2 dim m", by [20] (2.6 (ii)); we obtain (i) from [25] (Cor. 6.4 (1)) or from [3] (6.8).

(ii) obtains on adapting [19] to prime ideals or as follows. Observe that in the identification of U (c0//with L (M, M), the restriction to U (m~) defines the left regular representation of U(m~) and furthermore from FractA"1" we get a second copy of U(m~) which (cf. 4.1) identifies with the right regular representation of U(m~). Then since

Dim((L/(m-)(2) z^-) U(mo)) # U (I)) = 2dimm- = 2m = Dim^, it follows from [20] (5.2) that ja^ may be chosen so that Q is represented by first order differential operators on C [x^, x^ . . . , x^] (with a zero order part depending also on r). Hence (ii).

Remarks. — The proof of the theorem is entirely algebraic and consequently the base field can be any algebraically closed commutative field of characteristic zero. Yet excepting the case p = b ([6], Sect. 6) the proof of 3.2 (ii) is partly analytical. Part (ii) asserts that the Goldie field of / admits a maximal commutative subfield which is ^-stable. Yet excepting sl(2) and sl(3) not all primitive ideals are induced ([4], [6]).

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Thus it still remains to show that for such non-induced ideals the Goldie field does not admit a maximal commutative subfield which is ^-stable.

This is discussed in Sect. 5.

4.4: We round off the discussion by showing that an induced ideal is primitive only if it is induced from a parabolic subalgebra. (We remark that this condition is not also a sufficient one, even if the module from which one induces is simple and finite dimensional ([4], [21]).)

Let F(g) denote the invariants of the symmetric algebra S (9) and let Y+ be the subspace of ^(9) spanned by homogeneous invariants of positive degree. Identify g with 9* through the Killing form and recall [8] (8.1.3 (ii)), that the zero variety of 5' (9) 7+ is the cone ^ of nilpotent elements of 9.

LEMMA. — Let a be a subalgebra of 9. If Y+ c S (9) a, then a is a parabolic subalgebra of g.

We have a¹ c ^f and [a, a¹] c= a¹. Thus if we can show that a¹ <= a, it will follow that a1 is a nilpotent subalgebra and hence that a is parabolic.

Let a^ denote the algebraic hull of a and m the nilradical of Oi, and 1)^ a Cartan subalgebra for the reductive part r of di. By [8] (1.10.16), there exists a triangular decomposition

g = n © I ) © n ~ with n =» m and t)=>I)i.

If Xe r is an t)i weight vector of non-zero weight, then Xe n © n~ and so 1)^ n I) c a^ n t) c: a¹ n I) c: J^ n I) == 0.

Thus I)i = t) and since [di, a1] c a1, it follows that a1 is spanned by root vectors. Thus a = a11 is also spanned by root vectors (and in particular di = a). Suppose a1 $ a. Then there exists a e R, such that X^ e a1, 2^ ^ a. This gives X_y e a1, which contradicts the fact that Xy^ +Ar_a ^e/T.

4.5: Let a be a subalgebra of 9 and W a £/(a)-module. Set M = ind (W, a f 9), / = Ann M.

PROPOSITION. — T/'/ePrim £/(g), then a is a parabolic subalgebra.

Let gr be the gradation functor for the canonical filtration of U (9).

If 7 e Prim U (9), then (/ n Z (g)) e Max Z (g) and so gr (/ ) n Y (9) = Y+.

Yet / c: £/(g) Ann W and so gr (/) <= S(^) a, in virtue of the Poincare- Birkhoff-Witt theorem. This gives 7+ c= S (9) a and so a is parabolic by 4.4.

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5. The subfield criterion for induced ideals

Here we show that the Goldie field of a certain non-induced primitive ideal constructed in [16] does not admit a maximal commutative subfield which is ad 9 stable. These ideas are inspired by the work of NGHI^M ([26], [27]); but since we require results valid outside solvable Lie algebras our analysis has to be somewhat different. It is motivated by [9].

5.1: Let 9 be a finite dimensional fc-Lie algebra. Extend the gradation functor gr for the canonical filtration of U(^) to K(^) through

gr^^gr^grW; a, be U (9), a^O

(cf. [22], Chap. II or [13], Sect. 3). Through the identification gr (U fe)) = S (9); gr defines a Poisson bracket { , } onR (9) : = Fract S (9) ([22], 2.1,2.2). Given K a subfield of ^(9), we set K' = Fract gr (K).

A subfield K ' of R (9) is called strongly commutative if { a, b} = 0, for all a,beK\

Now assume that K is a commutative ad 9 stable subfield of K (9). Then K' is a strongly commutative ad 9 stable subfield of ^(9). Let ^ ( K ) (resp. Q ( K ' ) ) denote 9 K+K (resp. ^ K ' - ^ - K ' ) considered as a subspace of K (9) (resp. R (9)) over K (resp. K'). The set { X }^ ^ : X, e 9, is said to be a cobase for 9 (K) over K (resp. 9 ( K ' ) over K ' ) if with the identity adjoined it becomes a basis for 9 {K) (resp. for 9 {K')). Obviously 9 (K) and 9 (K') admit cobases though these may not coincide. Observe further that 9 (K) (resp. 9 (K')) is closed under commutation (resp. Poisson bracket) and so are Lie algebras (The Lie algebra 9 (K) and the notion of a cobase for 9 ( K ) are due to NGHIEM [26], 1.2.3.) Recall that for a commutative field, Dim coincides with transcendence degree (over k). Let dim denote dim^.

LEMMA:

(i) DimK' ^ Dim K;

(ii) 1 +dim 9 ^ dim^ 9 (K') +Dim K'\

(iii) dim,,, 9 ( K ' ) ^ dim^ 9 (K);

(i) is elementary. Let {^}^i be a cobasis for 9 ( K ' ) over K\ One has dim^/ 9 OQ = /+!. Since {^},Li generates ^(9) over K ' and Dim R (9) = dim 9, this gives (ii).

Let { Xi }?^ ^ be a basis for 9. Let 33 denote the set of all cobases for 9 (K) over K formed from subsets of { X^ }^. The set 25 is trivially non-empty. Given b e S, let b (K/) denote the linear span of { b, 1 } over K\ and set /(&) = { ie {1, 2, . . . , n }; X, e b ( K ' ) }. Choose 6 e 93

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maximizing card I(b). For (iii), it suffices to show that card I ( b ) = n.

We can assume that b = { X^ }^^ and set XQ = 1. Given card I(b) < n, choose r e { w +1, ..., n ] , r ^ I(b\ Since b is a cobase for 9 (^) over K, there exist jc. e K such that

(5.1) ^=£'=0^.

Yet ^^^(J^') by hypothesis, so leading terms on the right-hand side of (5.1) must cancel. This defines a non-empty set J c: { 0, 1, 2, . . . , m } such that

(5.2) 0=£,^(grx,)X,; grx,eX'-{0}.

Since K ' is a field, card J > 1. Choose 7 * e 7 \ { 0 } and set b' = { b\Xj } u {X,}. Since Xj + 0, we obtain V e 93 from (5.1). Since JT, e b' and JT, e b' (K') by (5.2), it follows that card / (6') ^ 1 +card / (&), contradicting the choice of 6. Hence O¹¹)-

5.2: In addition to the hypothesis of the above lemma assume that K contains its commutant in 9 (K). Set dim^ 9 (K) == w+1, and let 8^- denote the Kronecker delta.

LEMMA. — There exist x^ e K, yj e 9 (K), i,j = 1, 2, . . . , m, such that (0 [^.^j] == 8»^

00 bi^j]^^;

(ni) { ^ » ^o' ^o = 1. is ^ basis for 9 (K).

Obviously m = 0, if and only if, K = K((^). Suppose m > 0, and let / be the largest non-negative integer such that x^e K, yj e 9 (K), i,j = 1, 2, ...,/, can be chosen to satisfy (i). Then {1, y^ y^ ' . • » Y i } are linearly independent over K and so / < m. Given y e ^ ( K ) , then a^: = [x;, ^] e ^T, and we set

/^-I^i^^-

Then [x,, /] = 0. Suppose a : = \x, y ' ~ \ i=- 0, for some x e K. Set bi = [^ ^], ^ = ^+1. yi+i = ^~¹ y. ^' = Yi-bi yi+i' Then

^^ y'^ i = l, 2, . . . , / + ! } satisfy (i) contradicting the maximality of /.

Hence [X, y] = 0, and so y ' e ^ since ^ contains its commutant in 9 (K).

It follows that { 1, j^i, ^2, . . . , Yi} spans 9 (K) over ^. Hence I ^ m, which with the opposite inequality established above proves (i) and (iii).

Through the Jacobi identity [x^, [^y, ^]] = 0. Yet [^y, y^\ eg (K), and so (ii) follows from (i) and (iii).

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5.3: Retain the hypotheses and notation of 5.2. By 5.2 (iii), the y, generate over K a subalgebra S of ^(9) containing U(Q). Consider ad x,, i = 1, 2, . . . , m, as spanning a commutative Lie algebra n of locally nilpotent derivations of S. By 5.2 (i), 5'" = K. By [18] (2.6 (iv)), S admits a left quotient field Fract S which must obviously coincide with ^(g). By [18] (2.6 (vi)), ^ contains its commutant in ^(9) and in particular €(9) c: K. Set K ' = Fract gr (A:).

Given L a skew field over ^ define Tdeg^L (or simply TdegL) as follows. Let V denote a finite dimensional A: subspace of L containing the identity and given b e L, m e N, let (b VY denote the k subspace ofL spanned by { ^ ^ • • • ^m; a, e b V }. Define

TdegL=sup, inf^olim.^10^1111^^.

logm PROPOSITION:

(i) w+Tdeg K < Tdeg Fract S = dim 9;

(ii) dim^ 9 (A:) +Dim K = dim 9 +1;

(iii) DimA:= Dim A:';

(iv) Dim K ^ 1/2 (dim 9 4-index 9), m7A equality if Dim C (9) = index 9, so in particular if 9 is algebraic.

The inequality in (i) follows from [18] (2.6 (iv), 4.2 (ii)) applied to 5.2.

(Note that in [18], Dim is denoted by Dim1 and Tdeg by Dim). The equality in (i) follows from the identity Fract 5' = A: (9) and [18] (4.3).

One has Dim K == Tdeg K, since K is commutative and then (ii) and (iii) follow from (i) and 5.1. Since K ' is strongly commutative, we obtain DimK' ^ l/2(dim9+index9) by [22] (2.7). By 5.2 (i), the ^,, z = l , 2 , . . . , w , are algebraically independent over €(9) and so m +Dim C (9) ^ Dim K := dim 9-w, by (ii). Hence

m ^ 1/2 (dim 9-Dim C (9)).

Then by (ii), we have

Dim K == dim 9 - m ^ 1/2 (dim 9 + Dim C (9)), which gives (iii).

Remarks. - Unlike NGHIEM we do not assume that K is generated by its intersection with U(o). This case is simpler; for example (iii) would be a consequence of [22] (2.5), which does not required K to be ad 9 stable. However this restriction is inappropriate outside 9 solvable.

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Again for 9 solvable one can show that any commutative ad 9 stable subfield of K (9) can be embedded in a commutative ad 9 stable subfield K which contains its commutant in Q(K). This generalizes [26] (III. 4);

but the argument given in [26] needs a little modification. Finally it is noted above that a commutative ad 9 subfield K of K (9) which contains its commutant in 9 (K) necessarily contains its commutant in ^(9); i. e. it is maximal commutative. This generalizes [26] (III. 7).

5.4: Now let 9 be a simple Lie algebra. It is known that 9* (identified with 9 through the Killing form) admits a unique nilpotent orbit OQ of minimal non-zero dimension. In [16], we constructed a completely prime, primitive ideal Jo for which OQ u {0 } is the zero variety ^ (gr Jo) of grJo. The ideal Jo can be described as follows. First there exists a solvable algebraic Lie subalgebra r of 9 satisfying dim r = dim OQ , index r = 0, and containing the highest root eigenvector E (see [16], Sect. 4). Secondly there exists an algebra homomorphism 0: U (9) —> U (r)^

satisfying 0 |y = / d (There is only one such homomorphism if 9 ^ sl (n+1): n eN⁴-, [16], Theorem 4.3.) Set Jo = ker 0. Then

^(grJo) = OQ u { 0 } ([16], Prop. 10.2). Furthermore <D defines, by passage to the quotient and localization at E, a bijection <3>:

(U^W^U(X)E

whose restriction to (7(r)^ is the identity. In [16] (Sect. 8), it was shown that for 9 ^ sl (n +1), n e N⁴', the ideal Jo is not induced from any proper subalgebra of 9. This was obtained through a dimensionality estimate which gave little insight into why Jo is not induced. This motivates the following theorem.

THEOREM. - Suppose 9 ^ sl(n+l), /zeN⁴'. Then Fract U(^)/Jo does not admit a maximal commutative subfield which is ad 9 stable.

Let K be such a subfield. Through 0, the subfield K may be considered as a subfield of K(x). Set K = Fract gr (K). From K\ we construct for certain/e OQ, a polarization of 9 in/. Yet for 9 + sl(n+l), n e N4', ([16], Prop. 3.5), no/e OQ is polarizable and this contradiction will prove the theorem.

5.5: Define a map O' : ^—>R(x), through ^ (X) = grO(Z), for all X e 9. Recall that deg extends to R (r) through deg a~1 b = deg b - deg a.

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LEMMA:

(i) ^ ' ( X ) = X,for all X ex;

(ii) deg 0' (X) = 1, for all Xe 9;

(iii) { <T (1-), 0' (7) } = <D' [X, 7], for all X, Ye 9;

(iv) (/, 0' (2-)) = (/; ^),/or ^/ ife 9 ^ alife OQ satisfying f (E) ^ 0.

(i) is immediate, (ii) obtains from the explicit formula [16] (Theo- rem 5.3), for 0, or as an easy consequence of (i) and the fact that R (r)¹ reduces to scalars. (iii) follows from (ii). For each A" eg, there exists

^ e N , such that ^(d) (X)-X) e^o. By (ii), E8 (fS>\X)-X)eff JQ.

Hence (iv).

5.6: Identify g with 9* through the Killing form, and set/= ^E (nota- tion 3.1). Then/e OQ, f(E) ^ 0 and r identifies with the tangent space to OQ at / ([16], Sect. 4). Consider 0 = { g e O^g (E) ^ 0 } in r* by restriction. Then (always in the Zariski topology), we have the folloning lemma.

LEMMA. — 0 is a non-empty open subset ofx*.

5.7: Define a linear map (p : g ® R (r) —> R (r) through (p (X ® x) = 0' (f) jc.

Let {^ }^ ^ be a basis for 9 such that { X, }^ ^ is a basis for r and for each fe R (g), let S^f denote its partial derivative with respect to X^

Define a linear map d : R (r) —> 9 ® 7? (r), through

^ = £?= i (^. ® {8i a)) = Er= i (X, ® (3, a)).

Define an antisymmetric bilinear two-form B on 9 ® ^ (r) through

^(^•,^) == (p [x,^]. Let F denote the linear span { d a ' . a e K ' } over R (r), and let F1 denote the orthogonal complement of V with respect to B.

Set N = ker B.

LEMMA : (i) Fez ^;

(ii) [F1, F1] c= F1; (iii) N n F = 0;

(iv) dim^) F = l/2dimr;

(v) rank2? = dimr;

(vi) dim^^ V1 = dim 9-1/2 dimr;

(vii) B(V1, V1) = 0.

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Given a, be K\ then

B(da, db) = S^=i (p([X,, Z,] ® (5,a)(a, b))

= E^ i \X,, Z,] (3, a) (8, b), by 5.6 (i)

= { a, b ] = 0, since K' is strongly commutative.

Hence (i).

Given ^ , ^ e g ® ^ ( r ) , aeR(x\ set a,={0(Z,),a} and

^ = £?= i (^i ® ^), ^ = D= i (^ ® ^), b

= £?=i j^n c = Z?=i ^^F

Then using 5.6 (i), (ii), a straightforward computation gives (5.3) B([x, y], da) = B(x, 6)-B(^, c).

Now K is ad 9 stable, that is [0 (X), K~] ^ K and so { ^ (X), K ' } c K \ for all A^e 9. Then ^ e ^/, implies that b, c e V and so (ii) follows from

(5.3).

Given x e V n N, write

x = ]^= i x^ d0(: x^eR ((), ^ e K\

By definition of TV, we have, for all i = 1, 2, . . . , m,

(5.4) 0=B(X,,x)=^i E7-i[^^(W.

Now index r = 0 and so det [X,, Xj ] ^ 0. Substitution in (5.4) gives x == 0, Hence (iii).

One has dim^ V == Dim K ' = 1/2 dim r, by 5.3 (iii), (iv) (the first equality is elementary). Hence (iv).

For all X, Ye 9, we have B(X® 1, V ® 1) = 0' [X, V] c S^E.

Given /er*, such that f(E) -^ 0, define a two-form B^ on 9 through Bf (X, Y) == (/, 0' [^ r]). Then there exists a non-empty open set Q' c: r* such that rank B = rank ^ for all /e 0'. With ft as defined in 5.6, choose /e Q n Q'. Then

(/, 0'[JG Y]) == (/, [X, F]), for all X, YCQ, by 5.5 (iv).

Hence rank B = dim ^o == dim. r, which gives (v). By (iii) and (iv), dim^ (^ V = dim^ (,) (9 ® jR (r)) - dim^ (,) 7 = dim 9 -1/2 dim r,

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which gives (vi). Finally we have V+N c: F1, by (i) and by (iii)-(v), that Dim^(F+AO == dim^^ V1. Hence (vii).

5.8: Set V^ = V1 n (9 ® 5' (r)). Given /e r*,

^=E?=i(^®x,)e9®5(r), set

<^ / > =D=A(^ /) and pf = {<x, />; xe7i}.

By 5.7 (ii), py is a subalgebra of 9. Since V^ generates V over R (r), there exists a non-empty open set fi." c r* such that dim pf = dim^ — V1, for all/eQ'7. Choose /e Q n 0.". Then by 5.5 (iv), we have for all x , y e Pi, that

a[<^/x<^/>])=a^(^^)).

Then by 5.7 (vii), py is subordinate to /. By 5.7 (vi), dim pj- = dim 9 —1/2 dim r = dim g —1/2 dim OQ . Hence pj is a polarization of 9 in /. This proves the theorem.

Remark. — Is a maximal commutative subfield of K (r) which is ad r stable, necessarily a pure transcendental extension of k9 (cf. [19], Theorem 2.3).

5.9: The above construction of a polarization differs from that given by NGHIEM in [27] and which does not use the d map. NGHIEM construction requires that K be generated by its intersection with U (9) (or some image of ?/(g)). This is fine for 9 solvable, but too restrictive for g semisimple as the following lemma shows.

LEMMA. — Take 9 semisimple and 1 e Prim U(c^). Let K be a commu- tative ad 9 stable subfield of Fract U(o)/L Then ( U ( ^ ) / I ) n K reduces to scalars.

After DUFLO [11] (Theorem 1), there exists ^ e ()* such that / = AnnL (^+p), where L(X+p) is the unique simple (7 (cO-module with highest weight vector e of weight K. Let n : U (g) —> U (g)/7, be the natural projection. It is clear that A '. = (U (cQ/7) n Kis an ad 9 module.

Then by [8] (4.2.5), A reduces to scalars or there exists H e l ) * — { 0 }, and a weight vector ^-ê ^(cO"" for which n(a. ) e ^ — { 0 } . We have A-n : = a-êK^^-> rbr otherwise a-ê/. Since Z ( X + p ) has a unique

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weight vector a_a e U (9) of weight a, a simple, for which n (flt-oc) e A — { 0 }, and ^_a ^ 7^ 0. Let $„ denote the 57(2) subalgebra generated by Xy and X_^. We can assume that a-^ is contained in a simple 5a submodule P^

of U(o) with TT(T^) c= ^4 and having lowest weight vector ^-^, ^ e N4' . Then ^-a = (adi^Y"1 a-tv.^ u? to a scalar, and so a_^e^ ^ 0. Let m be the nilradical of b ® k X,^ and P2 : ^fe)-^ (7(^+t)) the projection defined by the decomposition U(^) == £ / ( < ^ + t ) ) ® C m U (9) + U (9) m), and 9 : ^(Q)^—^ £/(t)), the Harish-Chandra homomorphism [8] (7.4.3).

Then P" (^-J = A^ o, for some ae U^+f))^ satisfying [$„, a] == 0 and P" (^^ = ^ ^, where ^ : = (ad^ X,) a.,, e F,. Now

aê^=Pâ.ê^=X.^(a)e^

Hence 9 (a) (^) ^ 0, A^ ZLa ^ ^ O? through the characterization of Ann e^

noted in [11] (1, Modules de Vermd). Yet

[^, a _ J ^ == ^a-,^, = P^JP'OI.J^ = (O^C^^ZL^, ^ 0, which contradicts the commutativity of A.

5.10: If 9 = sl(n^-\), ^ e N⁴' ; then (PQ admits polarization and the inducing procedure gives a family J^ (^k e k) of completely prime, primitive ideals [16] (3.5). These all take the form J^ = ker 0^, for some homo- morphism 0^ : U (9) —> U (r)^ (notation 5.4). Furthermore for each X e k, Fract £/(c0/.4 identifies with K(x) and admits a maximal commutative subfield K^ which is ad 9 stable.

K^ can be chosen to be generated by its intersection with U(x)E. Indeed taking m = 1/2 dim r, one knows that U (x) is isomorphic to the subalgebra of the Weyl algebra k ^ x^ . . . , ^, y^ y^ . . . , ^J, ^, = B/Bx; with generators

^ ( f = 1, 2, ..., m),

x,_, 0 = 1, 2, ..., m-1), (^^-l^E^'i1^^)-

Then 0 (9 © fe) is spanned by ^, x< ^., x, (^= i x^. ^ + X + 2), f,y = 1, 2, . . . , m, and we may choose K = k (x^ x^, . . . , x^).

Let us examine the polarization construction in detail for sl(2).

Set x = jq, y = y^. The canonical basis [E, H, F } for sl (2) is represented above through E = y, H= —yx—'k, F=yx2+2x^ and satisfies the relations [_E, F] = -2 ^, [Jf, £] = E, [Jf, F] = -F. Then

J , = £/(g)(^-^(^+l)+X(X-l)),

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and we set I = gr (7Q = S (9) (EF-H²). We have

r = k H Q k E , K^ = fe^-^H+^i)), X' = fe(E-¹^).

Set

u = H ® E - E ® H , v=E(SH²-2(H(SEH)+¥®E², considered as elements of 9 ® S (r). Then (notation 5.7), V is spanned by M, N by u, and F'¹ by { u, v }. Furthermore [u, v] = (1 (x) £') i?.

Taking/= '£', p^. has basis { 77, F}, that is p^ = 'b (notation 3.1). It is well-known that the J^ are induced from the one-dimensional represen- tation of T) defined in the obvious fashion by the character 7^ \ H\-^\.

5.11: The analysis of 5.5-5.8 applies in principle to any / e Prim U (9), though in practice certain technical difficulties arise. Yet just as the results of [16] (as summarized in 5.4) are generalized in [20] (6.10, 6.14, 6.15), so the results of 5.5-5.8 admit a corresponding generalization.

The details are left to the reader.

REFERENCES

[1] BORHO (W.). — Berechnung der Gel'fand-Kirillov-Dimension bei induzierten Darstellungen, Math. Annalen, t. 225, 1977, p. 177-194.

[2] BORHO (W.). — Primitive vollprime Ideale in der Einhiillenden von so (5, C), J . of Algebra, t. 43, 1976, p. 619-654.

[3] BORHO (W.), GABRIEL, (P.) und RENTSCHLER (R.). — Primideale in Einhullenden ausflosbarer Lie-algebren. — Berlin, Springer-Verlag, 1973 (Lecture Notes in Mathematics, 357).

[4] BORHO (W.) und JANTZEN (J. C.). — Uber primitive Ideale in der Einhfillenden einer halbeinfachen Lie-Algebra, Invent. Math., Berlin, t. 39, 1977, p. 1-53.

[5] BORHO (W.) und KRAFT (H.). — Uber die Gel'fand-Kirillov-Dimension, Math.

Annalen, t. 220, 1976, p. 1-24.

[6] CONZE (N.). — Algebres d'operateurs differentiels et quotients des algebres envelop- pantes. Bull. Soc. math. Fr., t. 102, 1974, p. 379-415.

[7] CONZE-BERLINE (N.) et DUFLO (M.). — Sur les representations induites des groupes semi-simples complexes. Compos. Math., Groningen, t. 34, 1977, p. 307-336.

[8] DDCMIER (J.). — Algebres enveloppantes, Paris, Gauthier-Villars, 1974 (Cahiers scientifiques, 37).

[9] DDMER (J.), DUFLO (M.) et VERGNE (M.). — Sur la representation coadjointe d'une algebre de Lie, Compos. Math., Groningen, t. 29, 1974, p. 309-323.

[10] DDCMIER (J.). — Polarisations dans les algebres de Lie, II, Bull. Soc. math. France, t. 104, 1976, p. 145-164.

[11] DUFLO (M.). — Sur la classification des ideaux primitifs dans 1'algebre enveloppante d'une algebre de Lie semi-simple. Annals of Math. Series 2,1.105,1977, p. 107-120.

[12] GEL'FAND (I. M.) and KIRILLOV (A. A.). — On fields connected with the enveloping algebras of Lie algebras [in Russian], Dokl. Nauk S.S.S.R., 1.167,1966, p. 503-505.

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[13] GEI/FAND (I. M.) and KIRILLOV (A. A.). — Sur les corps lies aux algebres envelop- pantes des algebres de Lie. — Paris, Institut des hautes Etudes scientifiques, 1966 (Publications mathematiques, 31, p. 5-19).

[14] GEL'FAND (I. M.) and KIRILLOV (A. A.). — The structure of the Lie field connected with a semisimple decomposable Lie algebra [in Russian], Funkcional Anal.

i Prilozen., t. 3, 1969, p. 726.

[15] JANTZEN (J. C.). — Kontravariante Formen auf induzierten Darstellungen halbeinfacher Lie-Algebren, Math. Annalen, t. 226, 1977, p. 53-65.

[16] JOSEPH (A.). — The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. sclent. EC. Norm. Sup., 4° serie, t. 9, 1976, p. 1-30.

[17] JOSEPH (A.). — The algebraic method in representation theory (enveloping algebras),.

"Group theoretical methods in physics" [4, 1975, Nijmegen], p. 95-109. — Berlin, Springer-Verlag, 1976 (Lecture Notes in Physics, 50).

[18] JOSEPH (A.). A generalization of the GeFfand-Kirillov conjecture, Amer. J. of Math., t. 99, 1977, p. 1151-1165.

[19] JOSEPH (A.).— Proof of the GeFfand-Kirillov conjecture for solvable Lie algebras, Proc. Amer. math. Soc., t. 45, 1974, p. 1-10.

[20] JOSEPH (A.). — A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. of Algebra, t. 48, 1977, p. 241-289.

[21] JOSEPH (A.). — Primitive ideals in the enveloping algebras of sl (3) and sp (4), 1976 (unpublished).

[22] JOSEPH (A.). — Sur les algebres de Weyl, Lecture notes, 1974, (unpublished).

[23] JOSEPH (A.). — Towards the Jantzen conjecture. Compos. Math., Groningen (in the press).

[24] JOSEPH (A.). — GeFfand-Kirillov dimension for the annihilators of simple quotients of Verma modules. J. London math. Soc., t. 18, 1978, p. 50-60.

[25] MCCONNELL (J. C.). — Representations of solvable Lie algebras and the GeFfand-Kirillov conjecture, Proc. London math. Soc., Series 3, t. 29, 1974, p. 453-484.

[26] NGHIEM (X. H.). — Sur certains sous-corps commutatifs du corps enveloppant d'une algebre de Lie resoluble. Bull. Sc. math., 26 serie, t. 96, 1972, p. 111-128, [27] NGHIEM (X. H.). — Sur certaines representations d'une algebre de Lie resoluble

complexe. Bull. Sc. r/.ath., 2e serie, t. 97, 1973, p. 105-128.

BULLETIN DE LA SOCIETE MATHEMA.TIQUE DE FRANCE

On the Gel'fand-Kirillov conjecture for induced ideals in the semisimple case

B ULLETIN DE LA S. M. F.

A NTHONY J OSEPH