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On the Determinant of Shapovalov Form for Generalized Verma Modules
Journal of Algebra 215, 318᎐329 Ž1999. Article ID jabr.1998.7731, available online at http:rrwww.idealibrary.com on
On the Determinant of Shapovalov Form for Generalized Verma Modules Alexandre Khomenko and Volodymyr Mazorchuk Department of Mechanics and Mathematics, Kyi¨ Taras Sheychenko Uni¨ ersity, 64, Volodymyrska st., 252033, Kyi¨ , Ukraine E-mail: [email protected] Communicated by Wolfgang Soergel Received February 5, 1998
We define a generalization of the Shapovalov form for contragradient Lie algebras and compute its determinant for Generalized Verma modules induced from a well-embedded sl Ž2, ⺓. subalgebra. As a corollary we obtain a generalization of the BGG-theorem for Generalized Verma modules. 䊚 1999 Academic Press
1. INTRODUCTION The structure theory of Verma modules is a classical part in representation theory of Lie algebras. The first deep result in this direction was obtained in the original paper by I. Bernstein, I. Gelfand, and S. Gelfand wBGGx. This theorem Žwhich we will call the BGG-theorem. provides some criterion for the existence of a non-trivial homomorphism between two Verma modules over a complex semisimple finite-dimensional Lie algebra. The original proof by BGG uses some deep results on the structure of the Weyl group of the Lie algebra and refers to the Harish᎐Chandra theorem on central character of a Verma module. In eight years V. Kac and D. Kazhdan wKKx managed to generalize this result on Verma modules over an arbitrary contragradient complex Lie algebra with symmetrizable Cartan matrix. The most amazing thing is that their proof was quite elementary. The main tool in that proof was a special bilinear form defined on a Verma module by N. Shapovalov wSx. There are a lot of different generalizations of Verma modules. One of them, called ␣-stratified Generalized Verma modules ŽGVM., was studied intensively during the last years Žsee, for example, wCF, FM, KMx and 318 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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references therein .. For example, an analogue of the BGG-theorem for ␣-stratified GVM over a simple complex finite-dimensional Lie algebra was obtained in wFM, KMx. The technique used to prove this generalization is analogous to that of BGG. Certainly, it seems to be impossible to generalize this result using the BGG-method for infinite-dimensional algebras. Nevertheless, some information about the structure of GVMs w x over an affine Lie algebra of type AŽ1. 1 was obtained in F by introducing a generalization of the Shapovalov form. In the present paper we define a certain analogue of the Shapovalov form on the enveloping algebra of a contragradient Lie algebra and use this form to study the structure of GVMs induced from a well-embedded sl Ž2, ⺓. subalgebra. The family of GVMs considered in this paper is a bit bigger than one of ␣-stratified GVMs considered, for example, in wCFx. Nevertheless, the irreducibility criterion remains valid in this general case, but it seems to be a very easy generalization of the classical ␣-stratified case. Results, obtained in this paper, cover and generalize all known facts about the structure of GVMs wFM, KM, Fx. The structure of the paper is the following: In Section 2 we collect all necessary notations and preliminary results on GVMs. In Section 3 we define a generalization of the Shapovalov form, investigate its basic properties, and present a generalization of the determinant formula. In Section 4 we prove the determinant formula presented in Section 3. Finally, in Section 5 we obtain a criterion of irreducibility for a GVM and a generalization of the BGG theorem.
2. PRELIMINARIES Let ⺓ denote the complex field, ⺪ denote the set of integers, and ⺞ denote the set of all positive integers. All the notations that will be used in this paper without preliminary definition can be found in wMPx. For a Lie algebra ᑛ we will denote by UŽ ᑛ . its universal enveloping algebra. Let ᑡ be a complex contragradient Lie algebra Žor Chevalley algebra. associated with a complex Ž n = n.-matrix A s Ž a i j . Žsee wKKx.. We fix the standard triangular decomposition Ž ᑡq, ᑢ, Qq, . of ᑡ, where ᑢ is a Cartan subalgebra, Qq is the set of roots of ᑡq, and is an anti-involution on ᑡ Žsee wMP, KKx for details.. Let Q be the set of roots of the algebra ᑡ, i.e., Q s Qqjy Qq wMPx. For a root  let ᑡ  denote the corresponding root space. For the rest of the paper we fix a base of Qq and an element ␣ g satisfying the following conditions: the subalgebra ᑡ ␣ of ᑡ generated by ᑡ " ␣ should be isomorphic to sl Ž2, ⺓. and ᑡ should be an integrable Ži.e., direct sum of finite-dimensional modules. ᑡ ␣-module under the adjoint action. Let
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␣ ᑨ" s Ý  g Q q _ ␣ 4 ᑡ "  , ᑢ ␣ s h g ᑢ < ␣ Ž h. s 04 , ␣ s _ ␣ 4 . Then we ␣ ␣ have the following decomposition: ᑡ s ᑡ ␣ [ ᑨy [ ᑢ ␣ [ ᑨq . For ᑢ ␣ ␣ y␣ s ᑡ ␣ l ᑢ one obtains ᑡ ␣ s ᑡ [ ᑢ ␣ [ ᑡ . Fix some Weyl᎐Chevalley basis H␣ , X " ␣ in ᑡ ␣ . We also fix the dual elements H g ᑢ,  g Qq. Under the above choice of ␣ a simple reflection s␣ on ᑢ* is correctly defined and satisfies all the standard properties of a simple reflection. Let P denote the standard Kostant partition function with respect to and Pˆ denote the standard Kostant partition function with respect to s␣ Ž .. By a quasiroot we will mean any element q  g ᑢ*, where  g Qq and q is a positive rational number. A ᑡ-module V is said to be a weight module provided the action of ᑢ is diagonalizable on V. Any weight ᑡ-module V admits a weight-space decomposition V s [V, where runs through ᑢ* and V is the weight subspace corresponding to Žsee wDx.. For a weight module V by ch V we will denote its character wD, Sect. 7.5x. A weight ᑡ-module V is called ␣-stratified wCFx if the actions of X " ␣ are injective on V. An element ¨ / 0 of a weight ᑡ-module V will be called an ␣-highest weight vector ␣ provided ¨ g V for some g ⺓ and ᑨq ¨ s 0. Consider the standard quadratic Casimir operator c s Ž H␣ q 1. 2 q 4 Xy␣ X␣ in UŽ ᑡ ␣ .. For any pair a, b g ⺓ one can consider a ᑡ ␣-module N Ž a, b . uniquely defined by the following conditions: 䢇 䢇 䢇
b is the eigenvalue of c on N Ž a, b .; all weight spaces N Ž a, b .ay 2 k , k g ⺪ are one dimensional; all non-zero weight spaces of N Ž a, b . are exhausted by those listed
above; 䢇
N Ž a, b . is generated by N Ž a, b .a .
Since ᑢ s ᑢ ␣ [ ᑢ ␣ we can rewrite arbitrary g ᑢ* as s ␣ q ␣ , where ␣ g ᑢ ␣ and ␣ g ᑢ ␣ . Let a, b g ⺓ and g ᑢ* such that Ž H␣ . s ␣ Ž H␣ . s a. We can define a structure of an ᑢ-module on N Ž a, b . by setting h¨ s ␣ Ž h. ¨ for all h g ᑢ ␣ and all ¨ g N Ž a, b .. Further, we can ␣ ␣ Ž consider N Ž a, b . as a D s ᑢ q ᑡ ␣ [ ᑨq -module by setting ᑨq N a, b . s 0. The ᑡ-module M␣ Ž , b . s U Ž ᑡ .
m N Ž a, b . UŽ D .
is called the Generalized Verma module ŽGVM.. One can easily prove that M␣ Ž , b . is ␣-stratified if and only if b / Ž a q 1 q 2 l . 2 for all l g ⺪ Žsee also wCF, Theorem 2.1x.. An equivalent condition is that N Ž a, b . is irreducible. For M␣ Ž , b . we will denote by L␣ Ž , b . its unique irreducible
SHAPOVALOV FORM FOR GVM
321
quotient. Since ␣ is fixed we will omit it as an index in the subsequent notations of M␣ Ž , b . and L␣ Ž , b .. For a contragradient Lie algebra with a symmetrizable Cartan matrix let Ž⭈, ⭈ . denote the bilinear form on ᑡ wK, MPx. The corresponding bilinear form on ᑢ* will be also denoted by Ž⭈, ⭈ .. For a restricted weight ᑡ-module V we introduce the action of the Kac᎐Casimir operator ⍀ wKKx on V as follows: for ¨ g V , g ᑢ* let ⍀¨ s Ž q 2 , . ¨ q 2
Ý Ý eyŽ i. eŽ i. ¨ ,
 g Qq
i
where is an element in ᑢ* such that Ž , ␥ . s 1 for all ␥ g , the eŽ i. Ž i. y form a basis of ᑡ  and the ey . One can  form the dual basis of ᑡ easily check that the form Ž⭈, ⭈ . on ᑢ* is invariant under s␣ . 3. ␣-SHAPOVALOV FORM AND THE DETERMINANT FORMULA Set ᑢ Ž ␣ . s UŽ ᑢ . m ⺓w c x. Consider the following decomposition of UŽ ᑡ . wF, p. 88x.: ␣ ␣ U Ž ᑡ . s Ž ᑨy U Ž ᑡ . q U Ž ᑡ . ᑨq . [ ᑢ Ž ␣ . ⺓ w X␣ x X␣
[ ᑢ Ž ␣ . ⺓ w Xy␣ x Xy ␣ [ ᑢ Ž ␣ . . Let p be the projection of UŽ ᑡ . on ᑢ Ž ␣ . with respect to the above decomposition. We define the ␣-Shapovalov form Žor generalized Shapovalov form. F␣ on UŽ ᑡ . as a symmetric bilinear form with values in ᑢ Ž ␣ . as Žsee also wF, KK, MP, Sx. F␣ Ž x, y . s p Ž Ž x . y . ,
x, y g U Ž ᑡ . .
It is straightforward that the graded components UŽ ᑡ . , g ⺪ Q are orthogonal with respect to F␣ . Moreover, F␣ is contravariant, i.e., F␣ Ž zx, y . s F␣ Ž x, Ž z . y . for all x, y, z g UŽ ᑡ .. Consider a vector subspace ␣ ␣ M s U Ž ᑨy [ ᑡ ␣ . q U Ž ᑨy [ ᑡy␣ .
in UŽ ᑡ .. For g ⺪ Q we set M s M l UŽ ᑡ . . Clearly, each M is finite-dimensional. To calculate the dimension of M we have to introduce the notion of a Kostant ␣-function P␣ Žsee wMOx.. For ␥ s Ý  g a  g Q set ␣ Ž␥ . s Ý  g Ž _ ␣ 4. a  . Define the Kostant ␣-function P␣ : ᑢ* ª ⺞ j 04 as follows: for g ᑢ* set P␣ Ž . to
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be the maximum number of the decompositions
q n␣ s
Ý
 g Qq_ ␣ 4
n ␣ Ž  .
with non-negative integer coefficients, where n runs through all integers. It follows easily from the definition of P␣ that dim My s P␣ Ž .. For g ⺪ Q we denote by F␣ the restriction of F␣ on My . Let g ᑢ* and b g ⺓. Clearly, from the construction of N Ž a, b . it follows that GVM M Ž , b . is generated by M Ž , b . as a ᑡ-module. Let 0 / ¨ Ž , b. g M Ž , b . be a canonical generator of M Ž , b .. It is well known Žsee, for example, wCFx. that M ¨ Ž , b. s M Ž , b . since M Ž , b . is generated by ¨ Ž , b. . We can naturally identify ᑢ Ž ␣ . with the ring of polynomials on the ⺓-space Ž , b .< g ᑢ*, b g ⺓4 by setting c* s Ž0, 1.. Thus we can define the value F␣ ŽŽ , b .. of F␣ in the point Ž , b .. Now we can define a bilinear ⺓-valued form Fˆ␣ on M Ž , b . by setting Fˆ␣ Ž u1¨ Ž , b. , u 2¨ Ž , b. . s F␣ Ž u1 , u 2 . Ž Ž , b . . ,
u1 , u 2 g M .
One can easily obtain the following standard properties of Fˆ␣ : LEMMA 1. Ž1. The kernel of Fˆ␣ coincides with the unique maximal submodule in the module M Ž , b .. Ž2. Fˆ␣ is non-degenerate on M Ž , b . if and only if M Ž , b . is irreducible. Ž3. All weight subspaces of M Ž , b . are orthogonal with respect to Fˆ␣ . Proof. The proof is analogous to that for the classical Shapovalov form Žsee, for example, wMPx.. The main result of this paper is the following theorem which computes the determinant of F␣ : THEOREM 1. Let ᑡ be a contragradient Lie algebra with a symmetrisable Cartan matrix. Then for any g ᑢ* det F␣ s
⬁
Ł Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k . .
P Ž yk ␣ .
ks1
⬁
= Ł Ž Xya X␣ q Ž 1 y k . Ž H␣ q Ž H␣ . y Ž 1 y k . . . ks1
=
=
⬁
Ł
Ł
 g Qq_ ␣ 4 , ks1 s␣ Ž  .s 
ž
H q Ž H . y k
⬁
Ł
Ł
 , s␣ Ž  .< ks1  g Qq_ ␣ 4 , s␣ Ž  ./  4
žž
Ž ,  .
H q Ž H . y k
2
P␣ Ž yk  .
/
Ž ,  . 2
/
PˆŽ yk ␣ .
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= Hs␣ Ž  . q Ž Hs␣ Ž  . . y k
ž
Ž ,  . 2
/
P␣ Ž yk  .
q␣ Ž H . ␣ Ž Hs␣ Ž  . . Xy ␣ X␣
/
up to a non-zero constant factor, where all the roots  are taken with their multiplicities. We note that the product in the last factor of the above formula runs through all non-ordered pairs  , s␣ Ž  .4 such that  / s␣ Ž  .. 4. PROOF OF THE DETERMINANT FORMULA The proof of Theorem 1 follows the general line of the original proof in wKKx, although there are several differences and technical difficulties. To proceed we need the following lemmas. LEMMA 2. Up to a non-zero constant factor, det F␣ is a product of factors ha¨ ing one of the following forms: Ž1. Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k ..; Ž2. Ž Xya X␣ q Ž1 y k .Ž H␣ q Ž H␣ . y Ž1 y k ...; Ž3. s ; Ž4.
Ž ,  .
ž H q ŽH . y k ž ž H q ŽH . y k 



2
/ , where  is a quasiroot such that s Ž  . / ⭈ ž H q ŽH . y k / q
Ž ,  . 2
␣
s␣ Ž  .
s␣ Ž  .
Ž ,  . 2
␣ Ž H . ␣ Ž Hs␣ Ž  . . Xy ␣ X␣ ,
/
where  is a quasiroot such that s␣ Ž  . /  . Proof. Consider a GVM M Ž , b . generated by a non-zero element
¨ Ž , b. g M Ž , b . . First we note that the module M Ž , b . is restricted wKKx and thus the action of ⍀ on it is well-defined. Applying ⍀ to ¨ Ž , b. one obtains ⍀ ¨ Ž , b. s ŽŽ q 2 , . q Ž b y ŽŽ , ␣ . q 1. 2 .r2. ¨ Ž , b. and thus ⍀
acts as ŽŽ q 2 , . q Ž b y Ž q , ␣ . 2 .r2. id on M Ž , b .. Consider the ᑡ ␣-module N Ž a, b . from the definition of M Ž , b .. Note that M Ž , b . can be reducible in two cases: if N Ž a, b . is reducible or if there exists an ␣-highest weight vector in some M Ž , b . with y f ⺪ ␣ . Suppose that N Ž a, b . is reducible. This is possible if and only if for some m g ⺞, X␣m Xym␣ ¨ Ž , b. s 0 or Xym␣ X␣m ¨ Ž , b. s 0 holds. By the direct calculations with UŽ ᑡ ␣ . we obtain m
Ł Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k . . ¨ Ž , b. s 0
ks1
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KHOMENKO AND MAZORCHUK
or m
Ł Ž Xya X␣ q Ž 1 y k . Ž H␣ q Ž H␣ . y Ž 1 y k . . . ¨ Ž , b. s 0.
ks1
Further, suppose that there exists an ␣-highest weight vector w in M Ž , b . for some g ᑢ* such that y f ⺪ ␣ . Then the eigenvalues of ⍀ on ¨ Ž , b. and w coincide and we obtain 2 Ž q 2 , . q Ž b y Ž Ž , ␣ . q 1 . . r2 2
s Ž q 2 , . q Ž b⬘ y Ž Ž , ␣ . q 1 . . r2
Ž 1.
for some b⬘ g ⺓. Clearly, the difference b⬘ y b polynomially depends on 'b after fixing ␣ and y Žsee wFMx.. Thus the formula above can be applied to the case Ž a q 1 q 2 n. 2 s b, n g ⺪. For such N Ž a, b . we get M Ž , b . to be an extension of two Verma modules Žwith respect to different bases in Q .. Now, using the fact that the action of ⍀ on a Verma module can be calculated at the highest weight vector, we obtain that b⬘ s b q 2'b Ž y , ␣ . q Ž y , ␣ . 2 Žhere 'b is a complex square root function which has two different values as soon as b / 0.. If Ž y , ␣ . s 0 the equality Ž1. reduces to Ž q 2 , . s Ž q 2 , . and we can use the same arguments as in the proof of wKK, Lemma 3.2x obtaining the factors Ž H q Ž H . y Ž  ,  .r2. Žhere  is not necessarily quasiroot.. If Ž y , ␣ . / 0 we can take two equalities of the form Ž1. corresponding to different values b1 and b 2 of 'b , transfer everything in the left-hand side, and multiply them. We obtain the following Žhere  s y .: 2 Ž 2 Ž q ,  . y Ž  ,  . y Ž q , ␣ . Ž  , ␣ . . y b Ž  , ␣ . 2 s 0.
The last equality can be rewritten in the form 2 Ž 2Ž q ,  . y Ž  ,  . y Ž q , ␣ . Ž  , ␣ . . y Ž  , ␣ . 2 Ž q , ␣ . 2
yŽ  , ␣ .
2
Ž b y Ž q , ␣ . 2 . s 0.
We note that
Ž q ,  . y Ž q , ␣ . Ž , ␣ . s Ž q ,  y Ž , ␣ . ␣ . s q ,  y
ž
s Ž q , s␣ Ž  . . .
2Ž  , ␣ .
Ž␣, ␣.
␣
/
SHAPOVALOV FORM FOR GVM
325
From this it follows that
Ž 2 Ž q ,  . y Ž  ,  . . Ž 2 Ž q , s␣ Ž  . . y Ž s␣ Ž  . , s␣ Ž  . . . q4
ž
1 4
Ž b y Ž q , ␣ . 2.
/Ž
␣ ,  . Ž ␣ , s␣ Ž  . . s 0.
Taking into account that 14 Ž b y Ž q , ␣ . 2 . is an eigenvalue of the operator Xy␣ X␣ , we obtain the factor of the form
Ž H q Ž H . y Ž  ,  . r2 . Ž Hs Ž  . q Ž Hs Ž  . . y Ž  ,  . r2 . ␣
␣
q ␣ Ž H . ␣ Ž Hs␣ Ž  . . Xy ␣ X␣ with the same arguments as in wKK, Lemma 3.2x. Now we only need to show that all  which appeared above are quasiroots. Suppose not. Thus we will have some factor of the determinant of F␣ corresponding to a non-quasiroot  . Calculating F␣ on a Verma submodule for some reducible N Ž a, b . we obtain a contradiction with wKK, Theorem 1x. The lemma is proved. By the PBW theorem we can define a new ␣-gradation on UŽ ᑡ . by setting the grade of X " ␣ and the grade of H␣ to be 0 and all the grades of other base elements in ᑡ to be 1. LEMMA 3. Up to a factor of grade zero the leading term of det F␣ with respect to the ␣-gradation is equal to ⬁
Ł
Ł HP Žyk  . . ␣
 g Qq_ ␣ 4 ks1
Proof. From the classical Shapovalov determinant formula wKKx it follows that the above formula is correct for det F␣yl ␣ , where l g ⺞ is big enough. to complete the proof it is sufficient to show that the leading term of det F␣ in the ␣-gradation does not depend on the shift on ␣ . Choose some PBW monomial base ¨ 1 , . . . , ¨ t in My and suppose that as soon as some ¨ i contains Xy ␣ this monomial should start with this Xy␣ . Consider the elements X␣ ¨ 1 , . . . , X␣ ¨ t and let W be a linear span of these elements. For 1 F i F t set ¨ˆi s X␣ ¨ i if ¨ i does not contain Xy ␣ and ¨ˆi s wi if ¨ i s Xy ␣ wi . Clearly, elements ¨ˆ1 , . . . , ¨ˆt form a basis of My q ␣ . Moreover, it follows from the definition of ¨ˆi that up to a factor of zero degree the leading term of det F␣y ␣ coincides with the leading term of the determinant of the form F␣ restricted to W Žwe will denote it by F␣ ŽW ... Since the base change from ¨ 1 , . . . , ¨ t to X␣ ¨ 1 , . . . , X␣ ¨ t is defined by the elements of zero grade it follows that det F␣ differs from det F␣ ŽW .
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KHOMENKO AND MAZORCHUK
by a factor of grade zero. This implies that the leading term of det F␣ in the ␣-gradation does not depend on the shift on ␣ . To proceed we have to define a Jantzen filtration on M Ž , b .. Choose z g ᑢ* such that Ž z,  . / 0 for all  g ⺪ Qq_ 0. Let t be an indetermi& ˜Ž , b. nate. By standard technique we can extend M Ž , b .& to the module M & over& the algebra U Ž ᑡ . s UŽ ᑡ . m ⺓w t x, where Ž , b . s Ž , b . q& t Ž z, 1. g ᑢ Ž ␣ . *s ᑢ Ž ␣ .* m ⺓w t x. Further we can&trivially extend on U Ž ᑡ . & and construct a bilinear form F . Using F one can define a bilinear & &␣ & ␣ t ˜ ⺓w t x-valued form F␣ on M Ž , b . . Setting M i to be equal to the set of all &
&
˜Ž , b . such that F␣tŽ ¨ , w . is divisible by t i for all w g elements ¨ in M & ˜Ž , b . we define a Jantzen filtration M &
&
&
˜Ž , b . s M 0 > M 1 > . . . M &
&
˜Ž , b . . The canonical epimorphism : M˜Ž , b . ª M Ž , b . Ž t ª 0. on M induces a filtration M Ž , b. s M 0 > M 1 > . . . of M Ž , b . which will be also called a Jantzen filtration. Proof of Theorem 1. We have only to calculate the degrees in det F␣ of the factors described in Lemma 2. For a quasiroot  , which is not proportional to ␣ , the proof of this fact is exactly the same as in wKK, Proof of Theorem 1x because of Lemma 3 and the remark that the functions P␣ Ž x y y ., y g ␣ H are linearly independent Žhere ␣ H is taken with respect to Ž⭈, ⭈ ... Thus we have only to calculate the degrees of the factors of the form 䢇 䢇
Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k ..; Ž Xya X␣ q Ž1 y k .Ž H␣ q Ž H␣ . y Ž1 y k ....
We will do it for the first kind of factors. One can apply analogous arguments for the second case. Consider a factor Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k .. for some fixed k g ⺞. Let N Ž a, b . be such that it has the unique submodule starting at the highest weight a y k ␣ . we note that in this case a f ⺪. One can easily choose g ᑢ* Ž Ž H␣ . s a. such that the GVM M Ž , b . has the unique non-trivial submodule N. Clearly, in the described case N is isomorphic to the Verma module M Ž y k ␣ .. From the definition of a Jantzen filtration we have M 0 s M Ž , b . and M 1 s N. Our goal is to prove that M 2 s 0. Since N is irreducible it follows that & & ˜Ž , b . and N˜ and either M 2 s N or M 2 s 0. Consider U Ž ᑡ . -modules M & ˜ Use the definition of F␣ to calculate let w be a canonical generator of N.
327
SHAPOVALOV FORM FOR GVM
&
F␣tŽ w, w .. By the direct application of sl Ž2.-theory we obtain that &
F␣t Ž w, w . s
k
Ł fk Ž t . ,
is1
where f k Ž t . g ⺓w t x such that f kX Ž0. / 0 satisfy the following condition: the differences between constant terms in f kq 1 and f k is equal to a y 2 k. Since a is not on integer it follows that the product in the formula above is divisible at most by t. But it is divisible by t since N˜ is a submodule. Thus the canonical generator of N belongs to M 1 and does not belong to M 2 . Hence M 2 s 0. Now we can claim that from the construction of the Jantzen filtration it follows immediately that det F␣ is divisible exactly by the P Ž y k ␣ .th power of Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k .. Žsee wKK, Proof of Theorem 1; MP, Sect. 6.6x.. This completes our proof.
5. STRUCTURE OF GVMs As in the classical case, the determinant formula for F␣ enables one to prove a generalization of the BGG-criterion for the embeddings of Verma modules Žsee wKK, Theorem 2; MP, Sect. 6.7x.. In this section we will formulate and prove an analogous result for GVMs induced from ᑡ ␣ . For , g ᑢ* and b1 , b 2 g ⺓ we set Ž , b1 . © Ž , b 2 . in one of the following cases: Ž1. b1 s b 2 and s y k ␣ for some k g ⺪; Ž2. b1 s b 2 " 2 b 2 Ž k  , ␣ . q Ž k  , ␣ . 2 for k g ⺞ and  g Qq_ ␣ 4 such that s y k  and
'
2 Ž q . Ž H . y k Ž  ,  . y Ž q . Ž H␣ . Ž  , ␣ . s " b 2 Ž  , ␣ .
'
Žhere an analytic branch of 'z function is fixed.. Denote by $ the transitive closure of the relation © on ᑢ* = ⺓. For each pair  / s␣ Ž  . of roots in Qq we fix some bijective map sign:  , s␣ Ž  . 4 ª "1 4 . We also set signŽ  . s 0 if Ž ␣ ,  . s 0 and fix some analytic branch of 'z function. For  g Qq_ ␣ 4 , k g ⺞, and b g ⺓ set f , k Ž b . s b q 2 signŽ  . b 2 Ž k  , ␣ . q Ž k  , ␣ . 2 . First of all it is worth nothing to formulate the following criterion of irreducibility of the module M Ž , b . which follows immediately from Theorem 1 and Lemma 1.
'
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THEOREM 2. M Ž , b . is irreducible if and only if the two following conditions are satisfied: Ž1. ŽŽ q , ␣ . q 2 k . 2 / b for all k g ⺪. Ž2. ŽŽ2Ž q ,  . y k Ž  ,  ..Ž2Ž q , s␣ Ž  .. y k Ž s␣ Ž  ., s␣ Ž  ... q Ž ␣ ,  .Ž ␣ , s␣ Ž  .. ⭈ Ž b y Ž q , ␣ . 2 .. / 0 for all  g Qq_ ␣ 4 and for all k g ⺞. Remark 1. The first condition of the above theorem is equivalent to the condition that the module N Ž a, b . Žsee definition of M Ž , b .. and thus the module M Ž , b . is ␣-stratified. Hence for ␣-stratified modules one needs to check only the second condition. The following theorem is a generalization of the BGG structure theorem for Verma modules Žsee wBGG, Theorem 2; KK, Theorem 2x.. THEOREM 3. The following statements are equi¨ alent: Ž1. LŽ , b1 . is a subquotient of M Ž , b 2 .; Ž2. M Ž , b1 . ; M Ž , b 2 .; Ž3. Ž , b1 . $ Ž , b 2 .. Proof. One can easily see that it is enough to prove that the first condition implies the third one. Other implications are easy. Using Theorem 1 all necessary steps can be done the same way as in wKK, Theorem 2x. We will only outline the basic statements. Consider the Jantzen filtration M Ž , b2 . s M 0 > M 1 > . . . defined in the previous section. Clearly &
ord F␣t Ž , b 2 . s
Ý dim Mi y , iG1
&
where ord denotes the maximal power of t dividing F␣t Ž , b 2 .. Further, it follows by direct calculation that
Ý ch M i s Ý ch M Ž y k ␣ . q Ý ch M ␣ Ž q k ␣ . iG1
k
q
k
Ý
Ž  , k.
ch M Ž y k  , f , k Ž b 2 . . ,
where the first sum is taken over positive integers k such that ŽŽ q , ␣ . y 2 k . 2 s b and M Ž . denotes the Verma module with respect to with the highest weight g ᑢ*, the second sum is taken over positive integers
SHAPOVALOV FORM FOR GVM
329
k such that ŽŽ q , ␣ . q 2 k . 2 s b, and M ␣ Ž . denotes the Verma module with respect to s␣ Ž . with the highest weight g ᑢ* and the last sum is taken over all pairs Ž  , k . g Qq= ⺞,  / ␣ such that Ž y k  , f , k Ž b 2 .. $ Ž , b 2 .. Now proof of the theorem follows by standard arguments using induction in Žsee wKK, Proof of Theorem 2x.. Remark 2. One can easily obtain that the equivalence Ž1. m Ž3. in Theorem 3 remains valid even for GVMs M Ž , b . that is not generated by M Ž , b . Žthis means that N Ž a, b . is not generated by N Ž a, b .a and we can forget about this condition on N Ž a, b ... This case can be reduced easily to that where N Ž a, b . is generated by N Ž a, b .a . ACKNOWLEDGMENTS The first author was partially supported by ISSEP, Grant GSU071130. The second author was partially supported by the grant of The Cabinet of Ukraine and by CRDF Grant UM1-327. We thank Professor V. Futorny for helpful discussions.
REFERENCES wBGGx I. Bernstein, I. Gelfand, and S. Gelfand, Structure of representations generated by a highest weight vector, Funct. Anal. Appl. 5 Ž1971., 1᎐9. wCFx A. J. Coleman and V. M. Futorny, Stratified L-modules, J. Algebra 163 Ž1994., 219᎐234. wDx J. Dixmier, ‘‘Algebres enveloppantes,’’ Gauthier᎐Villars, Paris, 1974. wFx Ž . V. M. Futorny, Irreducible non-dense AŽ1. 1 -modules, Pacific. J. Math. 172 1996 , 83᎐99. wFMx V. Futorny and V. Mazorchuk, Structure of ␣-stratified modules for finite-dimensional Lie algebras, I, J. Algebra 183 Ž1996., 456᎐482. wKx V. G. Kac, ‘‘Infinite-Dimensional Lie Algebras,’’ Prog. Math., Vol. 44, Birkhauser, ¨ Boston, 1983. wKKx V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebra, Ad¨ . Math. 34 Ž1979., 97᎐108. wKMx A. Khomenko and V. Mazorchuk, Generalized Verma modules over the Lie algebra of type G 2 , Comm. Algebra, in press. wMOx V. Mazorchuk and S. Ovsienko, Submodule structure of generalized Verma modules induced from generic Gelfand᎐Zetlin modules, Algebra Repr. Theory 1 Ž1998., 3᎐26. wMPx R. V. Moody and A. Pianzola, Lie algebras with triangular decomposition, in CMS Series of Monographs and Advanced Texts, Wiley, New York, 1995. wSx N. N. Shapovalov, On bilinear form on universal enveloping algebra of a complex semisimple Lie algebra, Funct. Anal. Appl. 6 Ž1972., 307᎐312.
On the Determinant of Shapovalov Form for Generalized Verma Modules Alexandre Khomenko and Volodymyr Mazorchuk Department of Mechanics and Mathematics, Kyi¨ Taras Sheychenko Uni¨ ersity, 64, Volodymyrska st., 252033, Kyi¨ , Ukraine E-mail: [email protected] Communicated by Wolfgang Soergel Received February 5, 1998
We define a generalization of the Shapovalov form for contragradient Lie algebras and compute its determinant for Generalized Verma modules induced from a well-embedded sl Ž2, ⺓. subalgebra. As a corollary we obtain a generalization of the BGG-theorem for Generalized Verma modules. 䊚 1999 Academic Press
1. INTRODUCTION The structure theory of Verma modules is a classical part in representation theory of Lie algebras. The first deep result in this direction was obtained in the original paper by I. Bernstein, I. Gelfand, and S. Gelfand wBGGx. This theorem Žwhich we will call the BGG-theorem. provides some criterion for the existence of a non-trivial homomorphism between two Verma modules over a complex semisimple finite-dimensional Lie algebra. The original proof by BGG uses some deep results on the structure of the Weyl group of the Lie algebra and refers to the Harish᎐Chandra theorem on central character of a Verma module. In eight years V. Kac and D. Kazhdan wKKx managed to generalize this result on Verma modules over an arbitrary contragradient complex Lie algebra with symmetrizable Cartan matrix. The most amazing thing is that their proof was quite elementary. The main tool in that proof was a special bilinear form defined on a Verma module by N. Shapovalov wSx. There are a lot of different generalizations of Verma modules. One of them, called ␣-stratified Generalized Verma modules ŽGVM., was studied intensively during the last years Žsee, for example, wCF, FM, KMx and 318 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
SHAPOVALOV FORM FOR GVM
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references therein .. For example, an analogue of the BGG-theorem for ␣-stratified GVM over a simple complex finite-dimensional Lie algebra was obtained in wFM, KMx. The technique used to prove this generalization is analogous to that of BGG. Certainly, it seems to be impossible to generalize this result using the BGG-method for infinite-dimensional algebras. Nevertheless, some information about the structure of GVMs w x over an affine Lie algebra of type AŽ1. 1 was obtained in F by introducing a generalization of the Shapovalov form. In the present paper we define a certain analogue of the Shapovalov form on the enveloping algebra of a contragradient Lie algebra and use this form to study the structure of GVMs induced from a well-embedded sl Ž2, ⺓. subalgebra. The family of GVMs considered in this paper is a bit bigger than one of ␣-stratified GVMs considered, for example, in wCFx. Nevertheless, the irreducibility criterion remains valid in this general case, but it seems to be a very easy generalization of the classical ␣-stratified case. Results, obtained in this paper, cover and generalize all known facts about the structure of GVMs wFM, KM, Fx. The structure of the paper is the following: In Section 2 we collect all necessary notations and preliminary results on GVMs. In Section 3 we define a generalization of the Shapovalov form, investigate its basic properties, and present a generalization of the determinant formula. In Section 4 we prove the determinant formula presented in Section 3. Finally, in Section 5 we obtain a criterion of irreducibility for a GVM and a generalization of the BGG theorem.
2. PRELIMINARIES Let ⺓ denote the complex field, ⺪ denote the set of integers, and ⺞ denote the set of all positive integers. All the notations that will be used in this paper without preliminary definition can be found in wMPx. For a Lie algebra ᑛ we will denote by UŽ ᑛ . its universal enveloping algebra. Let ᑡ be a complex contragradient Lie algebra Žor Chevalley algebra. associated with a complex Ž n = n.-matrix A s Ž a i j . Žsee wKKx.. We fix the standard triangular decomposition Ž ᑡq, ᑢ, Qq, . of ᑡ, where ᑢ is a Cartan subalgebra, Qq is the set of roots of ᑡq, and is an anti-involution on ᑡ Žsee wMP, KKx for details.. Let Q be the set of roots of the algebra ᑡ, i.e., Q s Qqjy Qq wMPx. For a root  let ᑡ  denote the corresponding root space. For the rest of the paper we fix a base of Qq and an element ␣ g satisfying the following conditions: the subalgebra ᑡ ␣ of ᑡ generated by ᑡ " ␣ should be isomorphic to sl Ž2, ⺓. and ᑡ should be an integrable Ži.e., direct sum of finite-dimensional modules. ᑡ ␣-module under the adjoint action. Let
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␣ ᑨ" s Ý  g Q q _ ␣ 4 ᑡ "  , ᑢ ␣ s h g ᑢ < ␣ Ž h. s 04 , ␣ s _ ␣ 4 . Then we ␣ ␣ have the following decomposition: ᑡ s ᑡ ␣ [ ᑨy [ ᑢ ␣ [ ᑨq . For ᑢ ␣ ␣ y␣ s ᑡ ␣ l ᑢ one obtains ᑡ ␣ s ᑡ [ ᑢ ␣ [ ᑡ . Fix some Weyl᎐Chevalley basis H␣ , X " ␣ in ᑡ ␣ . We also fix the dual elements H g ᑢ,  g Qq. Under the above choice of ␣ a simple reflection s␣ on ᑢ* is correctly defined and satisfies all the standard properties of a simple reflection. Let P denote the standard Kostant partition function with respect to and Pˆ denote the standard Kostant partition function with respect to s␣ Ž .. By a quasiroot we will mean any element q  g ᑢ*, where  g Qq and q is a positive rational number. A ᑡ-module V is said to be a weight module provided the action of ᑢ is diagonalizable on V. Any weight ᑡ-module V admits a weight-space decomposition V s [V, where runs through ᑢ* and V is the weight subspace corresponding to Žsee wDx.. For a weight module V by ch V we will denote its character wD, Sect. 7.5x. A weight ᑡ-module V is called ␣-stratified wCFx if the actions of X " ␣ are injective on V. An element ¨ / 0 of a weight ᑡ-module V will be called an ␣-highest weight vector ␣ provided ¨ g V for some g ⺓ and ᑨq ¨ s 0. Consider the standard quadratic Casimir operator c s Ž H␣ q 1. 2 q 4 Xy␣ X␣ in UŽ ᑡ ␣ .. For any pair a, b g ⺓ one can consider a ᑡ ␣-module N Ž a, b . uniquely defined by the following conditions: 䢇 䢇 䢇
b is the eigenvalue of c on N Ž a, b .; all weight spaces N Ž a, b .ay 2 k , k g ⺪ are one dimensional; all non-zero weight spaces of N Ž a, b . are exhausted by those listed
above; 䢇
N Ž a, b . is generated by N Ž a, b .a .
Since ᑢ s ᑢ ␣ [ ᑢ ␣ we can rewrite arbitrary g ᑢ* as s ␣ q ␣ , where ␣ g ᑢ ␣ and ␣ g ᑢ ␣ . Let a, b g ⺓ and g ᑢ* such that Ž H␣ . s ␣ Ž H␣ . s a. We can define a structure of an ᑢ-module on N Ž a, b . by setting h¨ s ␣ Ž h. ¨ for all h g ᑢ ␣ and all ¨ g N Ž a, b .. Further, we can ␣ ␣ Ž consider N Ž a, b . as a D s ᑢ q ᑡ ␣ [ ᑨq -module by setting ᑨq N a, b . s 0. The ᑡ-module M␣ Ž , b . s U Ž ᑡ .
m N Ž a, b . UŽ D .
is called the Generalized Verma module ŽGVM.. One can easily prove that M␣ Ž , b . is ␣-stratified if and only if b / Ž a q 1 q 2 l . 2 for all l g ⺪ Žsee also wCF, Theorem 2.1x.. An equivalent condition is that N Ž a, b . is irreducible. For M␣ Ž , b . we will denote by L␣ Ž , b . its unique irreducible
SHAPOVALOV FORM FOR GVM
321
quotient. Since ␣ is fixed we will omit it as an index in the subsequent notations of M␣ Ž , b . and L␣ Ž , b .. For a contragradient Lie algebra with a symmetrizable Cartan matrix let Ž⭈, ⭈ . denote the bilinear form on ᑡ wK, MPx. The corresponding bilinear form on ᑢ* will be also denoted by Ž⭈, ⭈ .. For a restricted weight ᑡ-module V we introduce the action of the Kac᎐Casimir operator ⍀ wKKx on V as follows: for ¨ g V , g ᑢ* let ⍀¨ s Ž q 2 , . ¨ q 2
Ý Ý eyŽ i. eŽ i. ¨ ,
 g Qq
i
where is an element in ᑢ* such that Ž , ␥ . s 1 for all ␥ g , the eŽ i. Ž i. y form a basis of ᑡ  and the ey . One can  form the dual basis of ᑡ easily check that the form Ž⭈, ⭈ . on ᑢ* is invariant under s␣ . 3. ␣-SHAPOVALOV FORM AND THE DETERMINANT FORMULA Set ᑢ Ž ␣ . s UŽ ᑢ . m ⺓w c x. Consider the following decomposition of UŽ ᑡ . wF, p. 88x.: ␣ ␣ U Ž ᑡ . s Ž ᑨy U Ž ᑡ . q U Ž ᑡ . ᑨq . [ ᑢ Ž ␣ . ⺓ w X␣ x X␣
[ ᑢ Ž ␣ . ⺓ w Xy␣ x Xy ␣ [ ᑢ Ž ␣ . . Let p be the projection of UŽ ᑡ . on ᑢ Ž ␣ . with respect to the above decomposition. We define the ␣-Shapovalov form Žor generalized Shapovalov form. F␣ on UŽ ᑡ . as a symmetric bilinear form with values in ᑢ Ž ␣ . as Žsee also wF, KK, MP, Sx. F␣ Ž x, y . s p Ž Ž x . y . ,
x, y g U Ž ᑡ . .
It is straightforward that the graded components UŽ ᑡ . , g ⺪ Q are orthogonal with respect to F␣ . Moreover, F␣ is contravariant, i.e., F␣ Ž zx, y . s F␣ Ž x, Ž z . y . for all x, y, z g UŽ ᑡ .. Consider a vector subspace ␣ ␣ M s U Ž ᑨy [ ᑡ ␣ . q U Ž ᑨy [ ᑡy␣ .
in UŽ ᑡ .. For g ⺪ Q we set M s M l UŽ ᑡ . . Clearly, each M is finite-dimensional. To calculate the dimension of M we have to introduce the notion of a Kostant ␣-function P␣ Žsee wMOx.. For ␥ s Ý  g a  g Q set ␣ Ž␥ . s Ý  g Ž _ ␣ 4. a  . Define the Kostant ␣-function P␣ : ᑢ* ª ⺞ j 04 as follows: for g ᑢ* set P␣ Ž . to
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be the maximum number of the decompositions
q n␣ s
Ý
 g Qq_ ␣ 4
n ␣ Ž  .
with non-negative integer coefficients, where n runs through all integers. It follows easily from the definition of P␣ that dim My s P␣ Ž .. For g ⺪ Q we denote by F␣ the restriction of F␣ on My . Let g ᑢ* and b g ⺓. Clearly, from the construction of N Ž a, b . it follows that GVM M Ž , b . is generated by M Ž , b . as a ᑡ-module. Let 0 / ¨ Ž , b. g M Ž , b . be a canonical generator of M Ž , b .. It is well known Žsee, for example, wCFx. that M ¨ Ž , b. s M Ž , b . since M Ž , b . is generated by ¨ Ž , b. . We can naturally identify ᑢ Ž ␣ . with the ring of polynomials on the ⺓-space Ž , b .< g ᑢ*, b g ⺓4 by setting c* s Ž0, 1.. Thus we can define the value F␣ ŽŽ , b .. of F␣ in the point Ž , b .. Now we can define a bilinear ⺓-valued form Fˆ␣ on M Ž , b . by setting Fˆ␣ Ž u1¨ Ž , b. , u 2¨ Ž , b. . s F␣ Ž u1 , u 2 . Ž Ž , b . . ,
u1 , u 2 g M .
One can easily obtain the following standard properties of Fˆ␣ : LEMMA 1. Ž1. The kernel of Fˆ␣ coincides with the unique maximal submodule in the module M Ž , b .. Ž2. Fˆ␣ is non-degenerate on M Ž , b . if and only if M Ž , b . is irreducible. Ž3. All weight subspaces of M Ž , b . are orthogonal with respect to Fˆ␣ . Proof. The proof is analogous to that for the classical Shapovalov form Žsee, for example, wMPx.. The main result of this paper is the following theorem which computes the determinant of F␣ : THEOREM 1. Let ᑡ be a contragradient Lie algebra with a symmetrisable Cartan matrix. Then for any g ᑢ* det F␣ s
⬁
Ł Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k . .
P Ž yk ␣ .
ks1
⬁
= Ł Ž Xya X␣ q Ž 1 y k . Ž H␣ q Ž H␣ . y Ž 1 y k . . . ks1
=
=
⬁
Ł
Ł
 g Qq_ ␣ 4 , ks1 s␣ Ž  .s 
ž
H q Ž H . y k
⬁
Ł
Ł
 , s␣ Ž  .< ks1  g Qq_ ␣ 4 , s␣ Ž  ./  4
žž
Ž ,  .
H q Ž H . y k
2
P␣ Ž yk  .
/
Ž ,  . 2
/
PˆŽ yk ␣ .
323
SHAPOVALOV FORM FOR GVM
= Hs␣ Ž  . q Ž Hs␣ Ž  . . y k
ž
Ž ,  . 2
/
P␣ Ž yk  .
q␣ Ž H . ␣ Ž Hs␣ Ž  . . Xy ␣ X␣
/
up to a non-zero constant factor, where all the roots  are taken with their multiplicities. We note that the product in the last factor of the above formula runs through all non-ordered pairs  , s␣ Ž  .4 such that  / s␣ Ž  .. 4. PROOF OF THE DETERMINANT FORMULA The proof of Theorem 1 follows the general line of the original proof in wKKx, although there are several differences and technical difficulties. To proceed we need the following lemmas. LEMMA 2. Up to a non-zero constant factor, det F␣ is a product of factors ha¨ ing one of the following forms: Ž1. Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k ..; Ž2. Ž Xya X␣ q Ž1 y k .Ž H␣ q Ž H␣ . y Ž1 y k ...; Ž3. s ; Ž4.
Ž ,  .
ž H q ŽH . y k ž ž H q ŽH . y k 



2
/ , where  is a quasiroot such that s Ž  . / ⭈ ž H q ŽH . y k / q
Ž ,  . 2
␣
s␣ Ž  .
s␣ Ž  .
Ž ,  . 2
␣ Ž H . ␣ Ž Hs␣ Ž  . . Xy ␣ X␣ ,
/
where  is a quasiroot such that s␣ Ž  . /  . Proof. Consider a GVM M Ž , b . generated by a non-zero element
¨ Ž , b. g M Ž , b . . First we note that the module M Ž , b . is restricted wKKx and thus the action of ⍀ on it is well-defined. Applying ⍀ to ¨ Ž , b. one obtains ⍀ ¨ Ž , b. s ŽŽ q 2 , . q Ž b y ŽŽ , ␣ . q 1. 2 .r2. ¨ Ž , b. and thus ⍀
acts as ŽŽ q 2 , . q Ž b y Ž q , ␣ . 2 .r2. id on M Ž , b .. Consider the ᑡ ␣-module N Ž a, b . from the definition of M Ž , b .. Note that M Ž , b . can be reducible in two cases: if N Ž a, b . is reducible or if there exists an ␣-highest weight vector in some M Ž , b . with y f ⺪ ␣ . Suppose that N Ž a, b . is reducible. This is possible if and only if for some m g ⺞, X␣m Xym␣ ¨ Ž , b. s 0 or Xym␣ X␣m ¨ Ž , b. s 0 holds. By the direct calculations with UŽ ᑡ ␣ . we obtain m
Ł Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k . . ¨ Ž , b. s 0
ks1
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KHOMENKO AND MAZORCHUK
or m
Ł Ž Xya X␣ q Ž 1 y k . Ž H␣ q Ž H␣ . y Ž 1 y k . . . ¨ Ž , b. s 0.
ks1
Further, suppose that there exists an ␣-highest weight vector w in M Ž , b . for some g ᑢ* such that y f ⺪ ␣ . Then the eigenvalues of ⍀ on ¨ Ž , b. and w coincide and we obtain 2 Ž q 2 , . q Ž b y Ž Ž , ␣ . q 1 . . r2 2
s Ž q 2 , . q Ž b⬘ y Ž Ž , ␣ . q 1 . . r2
Ž 1.
for some b⬘ g ⺓. Clearly, the difference b⬘ y b polynomially depends on 'b after fixing ␣ and y Žsee wFMx.. Thus the formula above can be applied to the case Ž a q 1 q 2 n. 2 s b, n g ⺪. For such N Ž a, b . we get M Ž , b . to be an extension of two Verma modules Žwith respect to different bases in Q .. Now, using the fact that the action of ⍀ on a Verma module can be calculated at the highest weight vector, we obtain that b⬘ s b q 2'b Ž y , ␣ . q Ž y , ␣ . 2 Žhere 'b is a complex square root function which has two different values as soon as b / 0.. If Ž y , ␣ . s 0 the equality Ž1. reduces to Ž q 2 , . s Ž q 2 , . and we can use the same arguments as in the proof of wKK, Lemma 3.2x obtaining the factors Ž H q Ž H . y Ž  ,  .r2. Žhere  is not necessarily quasiroot.. If Ž y , ␣ . / 0 we can take two equalities of the form Ž1. corresponding to different values b1 and b 2 of 'b , transfer everything in the left-hand side, and multiply them. We obtain the following Žhere  s y .: 2 Ž 2 Ž q ,  . y Ž  ,  . y Ž q , ␣ . Ž  , ␣ . . y b Ž  , ␣ . 2 s 0.
The last equality can be rewritten in the form 2 Ž 2Ž q ,  . y Ž  ,  . y Ž q , ␣ . Ž  , ␣ . . y Ž  , ␣ . 2 Ž q , ␣ . 2
yŽ  , ␣ .
2
Ž b y Ž q , ␣ . 2 . s 0.
We note that
Ž q ,  . y Ž q , ␣ . Ž , ␣ . s Ž q ,  y Ž , ␣ . ␣ . s q ,  y
ž
s Ž q , s␣ Ž  . . .
2Ž  , ␣ .
Ž␣, ␣.
␣
/
SHAPOVALOV FORM FOR GVM
325
From this it follows that
Ž 2 Ž q ,  . y Ž  ,  . . Ž 2 Ž q , s␣ Ž  . . y Ž s␣ Ž  . , s␣ Ž  . . . q4
ž
1 4
Ž b y Ž q , ␣ . 2.
/Ž
␣ ,  . Ž ␣ , s␣ Ž  . . s 0.
Taking into account that 14 Ž b y Ž q , ␣ . 2 . is an eigenvalue of the operator Xy␣ X␣ , we obtain the factor of the form
Ž H q Ž H . y Ž  ,  . r2 . Ž Hs Ž  . q Ž Hs Ž  . . y Ž  ,  . r2 . ␣
␣
q ␣ Ž H . ␣ Ž Hs␣ Ž  . . Xy ␣ X␣ with the same arguments as in wKK, Lemma 3.2x. Now we only need to show that all  which appeared above are quasiroots. Suppose not. Thus we will have some factor of the determinant of F␣ corresponding to a non-quasiroot  . Calculating F␣ on a Verma submodule for some reducible N Ž a, b . we obtain a contradiction with wKK, Theorem 1x. The lemma is proved. By the PBW theorem we can define a new ␣-gradation on UŽ ᑡ . by setting the grade of X " ␣ and the grade of H␣ to be 0 and all the grades of other base elements in ᑡ to be 1. LEMMA 3. Up to a factor of grade zero the leading term of det F␣ with respect to the ␣-gradation is equal to ⬁
Ł
Ł HP Žyk  . . ␣
 g Qq_ ␣ 4 ks1
Proof. From the classical Shapovalov determinant formula wKKx it follows that the above formula is correct for det F␣yl ␣ , where l g ⺞ is big enough. to complete the proof it is sufficient to show that the leading term of det F␣ in the ␣-gradation does not depend on the shift on ␣ . Choose some PBW monomial base ¨ 1 , . . . , ¨ t in My and suppose that as soon as some ¨ i contains Xy ␣ this monomial should start with this Xy␣ . Consider the elements X␣ ¨ 1 , . . . , X␣ ¨ t and let W be a linear span of these elements. For 1 F i F t set ¨ˆi s X␣ ¨ i if ¨ i does not contain Xy ␣ and ¨ˆi s wi if ¨ i s Xy ␣ wi . Clearly, elements ¨ˆ1 , . . . , ¨ˆt form a basis of My q ␣ . Moreover, it follows from the definition of ¨ˆi that up to a factor of zero degree the leading term of det F␣y ␣ coincides with the leading term of the determinant of the form F␣ restricted to W Žwe will denote it by F␣ ŽW ... Since the base change from ¨ 1 , . . . , ¨ t to X␣ ¨ 1 , . . . , X␣ ¨ t is defined by the elements of zero grade it follows that det F␣ differs from det F␣ ŽW .
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by a factor of grade zero. This implies that the leading term of det F␣ in the ␣-gradation does not depend on the shift on ␣ . To proceed we have to define a Jantzen filtration on M Ž , b .. Choose z g ᑢ* such that Ž z,  . / 0 for all  g ⺪ Qq_ 0. Let t be an indetermi& ˜Ž , b. nate. By standard technique we can extend M Ž , b .& to the module M & over& the algebra U Ž ᑡ . s UŽ ᑡ . m ⺓w t x, where Ž , b . s Ž , b . q& t Ž z, 1. g ᑢ Ž ␣ . *s ᑢ Ž ␣ .* m ⺓w t x. Further we can&trivially extend on U Ž ᑡ . & and construct a bilinear form F . Using F one can define a bilinear & &␣ & ␣ t ˜ ⺓w t x-valued form F␣ on M Ž , b . . Setting M i to be equal to the set of all &
&
˜Ž , b . such that F␣tŽ ¨ , w . is divisible by t i for all w g elements ¨ in M & ˜Ž , b . we define a Jantzen filtration M &
&
&
˜Ž , b . s M 0 > M 1 > . . . M &
&
˜Ž , b . . The canonical epimorphism : M˜Ž , b . ª M Ž , b . Ž t ª 0. on M induces a filtration M Ž , b. s M 0 > M 1 > . . . of M Ž , b . which will be also called a Jantzen filtration. Proof of Theorem 1. We have only to calculate the degrees in det F␣ of the factors described in Lemma 2. For a quasiroot  , which is not proportional to ␣ , the proof of this fact is exactly the same as in wKK, Proof of Theorem 1x because of Lemma 3 and the remark that the functions P␣ Ž x y y ., y g ␣ H are linearly independent Žhere ␣ H is taken with respect to Ž⭈, ⭈ ... Thus we have only to calculate the degrees of the factors of the form 䢇 䢇
Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k ..; Ž Xya X␣ q Ž1 y k .Ž H␣ q Ž H␣ . y Ž1 y k ....
We will do it for the first kind of factors. One can apply analogous arguments for the second case. Consider a factor Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k .. for some fixed k g ⺞. Let N Ž a, b . be such that it has the unique submodule starting at the highest weight a y k ␣ . we note that in this case a f ⺪. One can easily choose g ᑢ* Ž Ž H␣ . s a. such that the GVM M Ž , b . has the unique non-trivial submodule N. Clearly, in the described case N is isomorphic to the Verma module M Ž y k ␣ .. From the definition of a Jantzen filtration we have M 0 s M Ž , b . and M 1 s N. Our goal is to prove that M 2 s 0. Since N is irreducible it follows that & & ˜Ž , b . and N˜ and either M 2 s N or M 2 s 0. Consider U Ž ᑡ . -modules M & ˜ Use the definition of F␣ to calculate let w be a canonical generator of N.
327
SHAPOVALOV FORM FOR GVM
&
F␣tŽ w, w .. By the direct application of sl Ž2.-theory we obtain that &
F␣t Ž w, w . s
k
Ł fk Ž t . ,
is1
where f k Ž t . g ⺓w t x such that f kX Ž0. / 0 satisfy the following condition: the differences between constant terms in f kq 1 and f k is equal to a y 2 k. Since a is not on integer it follows that the product in the formula above is divisible at most by t. But it is divisible by t since N˜ is a submodule. Thus the canonical generator of N belongs to M 1 and does not belong to M 2 . Hence M 2 s 0. Now we can claim that from the construction of the Jantzen filtration it follows immediately that det F␣ is divisible exactly by the P Ž y k ␣ .th power of Ž Xya X␣ q k Ž H␣ q Ž H␣ . y k .. Žsee wKK, Proof of Theorem 1; MP, Sect. 6.6x.. This completes our proof.
5. STRUCTURE OF GVMs As in the classical case, the determinant formula for F␣ enables one to prove a generalization of the BGG-criterion for the embeddings of Verma modules Žsee wKK, Theorem 2; MP, Sect. 6.7x.. In this section we will formulate and prove an analogous result for GVMs induced from ᑡ ␣ . For , g ᑢ* and b1 , b 2 g ⺓ we set Ž , b1 . © Ž , b 2 . in one of the following cases: Ž1. b1 s b 2 and s y k ␣ for some k g ⺪; Ž2. b1 s b 2 " 2 b 2 Ž k  , ␣ . q Ž k  , ␣ . 2 for k g ⺞ and  g Qq_ ␣ 4 such that s y k  and
'
2 Ž q . Ž H . y k Ž  ,  . y Ž q . Ž H␣ . Ž  , ␣ . s " b 2 Ž  , ␣ .
'
Žhere an analytic branch of 'z function is fixed.. Denote by $ the transitive closure of the relation © on ᑢ* = ⺓. For each pair  / s␣ Ž  . of roots in Qq we fix some bijective map sign:  , s␣ Ž  . 4 ª "1 4 . We also set signŽ  . s 0 if Ž ␣ ,  . s 0 and fix some analytic branch of 'z function. For  g Qq_ ␣ 4 , k g ⺞, and b g ⺓ set f , k Ž b . s b q 2 signŽ  . b 2 Ž k  , ␣ . q Ž k  , ␣ . 2 . First of all it is worth nothing to formulate the following criterion of irreducibility of the module M Ž , b . which follows immediately from Theorem 1 and Lemma 1.
'
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THEOREM 2. M Ž , b . is irreducible if and only if the two following conditions are satisfied: Ž1. ŽŽ q , ␣ . q 2 k . 2 / b for all k g ⺪. Ž2. ŽŽ2Ž q ,  . y k Ž  ,  ..Ž2Ž q , s␣ Ž  .. y k Ž s␣ Ž  ., s␣ Ž  ... q Ž ␣ ,  .Ž ␣ , s␣ Ž  .. ⭈ Ž b y Ž q , ␣ . 2 .. / 0 for all  g Qq_ ␣ 4 and for all k g ⺞. Remark 1. The first condition of the above theorem is equivalent to the condition that the module N Ž a, b . Žsee definition of M Ž , b .. and thus the module M Ž , b . is ␣-stratified. Hence for ␣-stratified modules one needs to check only the second condition. The following theorem is a generalization of the BGG structure theorem for Verma modules Žsee wBGG, Theorem 2; KK, Theorem 2x.. THEOREM 3. The following statements are equi¨ alent: Ž1. LŽ , b1 . is a subquotient of M Ž , b 2 .; Ž2. M Ž , b1 . ; M Ž , b 2 .; Ž3. Ž , b1 . $ Ž , b 2 .. Proof. One can easily see that it is enough to prove that the first condition implies the third one. Other implications are easy. Using Theorem 1 all necessary steps can be done the same way as in wKK, Theorem 2x. We will only outline the basic statements. Consider the Jantzen filtration M Ž , b2 . s M 0 > M 1 > . . . defined in the previous section. Clearly &
ord F␣t Ž , b 2 . s
Ý dim Mi y , iG1
&
where ord denotes the maximal power of t dividing F␣t Ž , b 2 .. Further, it follows by direct calculation that
Ý ch M i s Ý ch M Ž y k ␣ . q Ý ch M ␣ Ž q k ␣ . iG1
k
q
k
Ý
Ž  , k.
ch M Ž y k  , f , k Ž b 2 . . ,
where the first sum is taken over positive integers k such that ŽŽ q , ␣ . y 2 k . 2 s b and M Ž . denotes the Verma module with respect to with the highest weight g ᑢ*, the second sum is taken over positive integers
SHAPOVALOV FORM FOR GVM
329
k such that ŽŽ q , ␣ . q 2 k . 2 s b, and M ␣ Ž . denotes the Verma module with respect to s␣ Ž . with the highest weight g ᑢ* and the last sum is taken over all pairs Ž  , k . g Qq= ⺞,  / ␣ such that Ž y k  , f , k Ž b 2 .. $ Ž , b 2 .. Now proof of the theorem follows by standard arguments using induction in Žsee wKK, Proof of Theorem 2x.. Remark 2. One can easily obtain that the equivalence Ž1. m Ž3. in Theorem 3 remains valid even for GVMs M Ž , b . that is not generated by M Ž , b . Žthis means that N Ž a, b . is not generated by N Ž a, b .a and we can forget about this condition on N Ž a, b ... This case can be reduced easily to that where N Ž a, b . is generated by N Ž a, b .a . ACKNOWLEDGMENTS The first author was partially supported by ISSEP, Grant GSU071130. The second author was partially supported by the grant of The Cabinet of Ukraine and by CRDF Grant UM1-327. We thank Professor V. Futorny for helpful discussions.
REFERENCES wBGGx I. Bernstein, I. Gelfand, and S. Gelfand, Structure of representations generated by a highest weight vector, Funct. Anal. Appl. 5 Ž1971., 1᎐9. wCFx A. J. Coleman and V. M. Futorny, Stratified L-modules, J. Algebra 163 Ž1994., 219᎐234. wDx J. Dixmier, ‘‘Algebres enveloppantes,’’ Gauthier᎐Villars, Paris, 1974. wFx Ž . V. M. Futorny, Irreducible non-dense AŽ1. 1 -modules, Pacific. J. Math. 172 1996 , 83᎐99. wFMx V. Futorny and V. Mazorchuk, Structure of ␣-stratified modules for finite-dimensional Lie algebras, I, J. Algebra 183 Ž1996., 456᎐482. wKx V. G. Kac, ‘‘Infinite-Dimensional Lie Algebras,’’ Prog. Math., Vol. 44, Birkhauser, ¨ Boston, 1983. wKKx V. G. Kac and D. A. Kazhdan, Structure of representations with highest weight of infinite-dimensional Lie algebra, Ad¨ . Math. 34 Ž1979., 97᎐108. wKMx A. Khomenko and V. Mazorchuk, Generalized Verma modules over the Lie algebra of type G 2 , Comm. Algebra, in press. wMOx V. Mazorchuk and S. Ovsienko, Submodule structure of generalized Verma modules induced from generic Gelfand᎐Zetlin modules, Algebra Repr. Theory 1 Ž1998., 3᎐26. wMPx R. V. Moody and A. Pianzola, Lie algebras with triangular decomposition, in CMS Series of Monographs and Advanced Texts, Wiley, New York, 1995. wSx N. N. Shapovalov, On bilinear form on universal enveloping algebra of a complex semisimple Lie algebra, Funct. Anal. Appl. 6 Ž1972., 307᎐312.