Adjoint vector fields and differential operators on representation spaces
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ADJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS ONREPRESENTATION SPACES
DMITRI I. PANYUSHEVA
BSTRACT . Let G be a semisimple algebraic group with Lie algebra g . In 1979, J. Dixmierproved that any vector field annihilating all G -invariant polynomials on g lies in the k [ g ] -module generated by the ”adjoint vector fields”, i.e., vector fields ς of the form ς ( y )( x ) =[ x, y ] , x, y ∈ g . A substantial generalisation of Dixmier’s theorem was found by Levasseurand Stafford. They explicitly described the centraliser of k [ g ] G in the algebra of differentialoperators on g . On the level of vector fields, their result reduces to Dixmier’s theorem.The purpose of this paper is to explore similar problems in the general context of affinealgebraic groups and their rational representations. I NTRODUCTION
We work over an algebraically closed field k of characteristic zero. Throughout, G isa connected affine algebraic group with Lie algebra g . Suppose for a while that G issemisimple. In 1979, Jacques Dixmier proved a nice theorem on vector fields on g . Specif-ically, he showed that any vector field annihilating all G -invariant polynomials on g liesin the k [ g ] -module generated by the ”adjoint vector fields” [6, Theorem 2.1]. A substantialgeneralisation of Dixmier’s theorem was found by Levasseur and Stafford [13]. They ex-plicitly described the centraliser of k [ g ] G in the algebra of differential operators on g . Onthe level of vector fields, their result reduces to Dixmier’s theorem. The purpose of thispaper is to explore similar problems in the general context of affine algebraic groups andtheir rational representations.We show that Dixmier’s argument applies to the coadjoint representations of the so-called ’3-wonderful’ Lie algebras. Furthermore, the coadjoint representation can be re-placed with an arbitrary (finite-dimensional) representation, and this leads to three typesof interesting problems. Let now G be an arbitrary connected group. We say that g is if: (i) codim( g ∗ \ g ∗ reg ) > , (ii) k [ g ∗ ] G is a polynomial algebra of Krull dimen-sion ind g , and (iii) the sum of degrees of free homogeneous generators of k [ g ∗ ] G equals (dim g + ind g ) / . (Here g ∗ reg is the union of G -orbits of maximal dimension in g ∗ .)This definition intends to axiomatise good properties of reductive Lie algebras. Thereis also a method for generating new 3-wonderful algebras: if g is -wonderful and g ≃ g ∗ , Mathematics Subject Classification. then the semi-direct product g ⋉ g has the same properties. Below is a Dixmier-type resultfor the coadjoint representation of a 3-wonderful Lie algebra g . Theorem 0.1.
Let X be a polynomial vector field on g ∗ . Assume that X annihilates all of k [ g ∗ ] G .Then there is a polynomial mapping Y : g ∗ → g such that X ( ξ ) = Y ( ξ ) · ξ for any ξ ∈ g ∗ . The proof is essentially based on the fact that codim( g ∗ \ g ∗ reg ) > and certain vectorbundle on g ∗ reg appears to be trivial. A posteriori, this is related to some good propertiesof a homomorphism of k [ g ∗ ] -modules. Let Mor ( V, N ) denote the set of all polynomialmorphisms V → N , where V and N are k -vector spaces. It is a free graded k [ V ] -module ofrank dim N . Consider the homomorphism ˆ φ : Mor ( g ∗ , g ) → Mor ( g ∗ , g ∗ ) , ˆ φ ( F )( ξ ) := F ( ξ ) · ξ for ξ ∈ g ∗ . Then Theorem 0.1 merely says that Im ˆ φ equals the submodule of vector fieldsannihilating all of k [ g ∗ ] G . Furthermore, in the ”3-wonderful case” the kernel of ˆ φ appearsto be a free k [ g ∗ ] -module generated by G -equivariant morphisms.For an arbitrary G -module V , where g is not necessarily -wonderful, one can writeup three similar homomorphisms and consider similar problems. The most obviouspossibility is to replace g ∗ with V . This yields the homomorphism of k [ V ] -modules ˆ φ : Mor ( V, g ) → Mor ( V, V ) . Clearly, any vector field X ∈ Im ˆ φ annihilates all of k [ V ] G .The problem on the opposite inclusion is related to the structure of Ker ˆ φ , k [ V ] G , and V reg ,and we provide an appropriate analogue of Theorem 0.1. Two other possibilities are ˆ ψ : Mor ( V, V ∗ ) → Mor ( V, g ∗ ) and ˆ τ : Mor ( g , V ) → Mor ( g , V ) , where we also describe the respective images under similar conditions, and give someillustrations.Generalising the approach of [13], we regard the problem on Im ˆ φ as a special case of aproblem on differential operators on V . Let D ( V ) denote the ring of differential operatorson V with polynomial coefficients. As k [ V ] is identified with the differential operators oforder zero, one can consider the centraliser of k [ V ] G in D ( V ) , Cent D ( V ) ( k [ V ] G ) . Each x ∈ g gives rise to a linear operator on V and therefore a vector field. In this way, one obtainsthe Lie algebra homomorphism ς = ς V : g → { polynomial vector fields on V } ⊂ D ( V ) . The elements of ς ( g ) are called the adjoint vector fields (on V ). By the definition of ˆ φ , Im ˆ φ isthe k [ V ] -module generated by ς ( g ) . Clearly, C := Cent D ( V ) ( k [ V ] G ) contains k [ V ] and ς ( g ) ,and one may ask whether the subalgebra generated by k [ V ] and ς ( g ) , denoted A , is equalto C . On the level of vector fields, the equality A = C reduces to the assertion that any X annihilating all of k [ V ] G lies in Im ˆ φ , i.e., in k [ V ] ς ( g ) .For the adjoint representation of a semisimple Lie algebra, the equality A = C is provedin [13]. Adapting that method, we obtain a sufficient condition for A = C in a more DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 3 general framework. We assume that k [ V ] G is a polynomial algebra and Ker ˆ φ is a free k [ V ] -module, and impose determinantal constraints on the embedding Ker ˆ φ → Mor ( V, g ) .However, the reductivity of G is not assumed. (See Theorem 4.2 for precise formulations).The equality A = C and other results on C stem from assertions about certain G -stablesubvariety of V × g ∗ . Recall that for any G -module V , there is the moment map µ : V × V ∗ → g ∗ . Then κ : V × V ∗ → V × g ∗ is defined by letting κ ( v, ξ ) = ( v, µ ( v, ξ )) .Consider also the k [ V ] -module E = Im ˆ φ and its symmetric algebra, Sym k [ V ] ( E ) . Underappropriate constraints (alluded to above), we prove that • Sym k [ V ] ( E ) is a factorial domain of Krull dimension dim V + dim g − rk (Ker ˆ φ ) ; • Im κ = Spec ( Sym k [ V ] ( E )) and it is also a complete intersection in V × g ∗ ; • the generators of the ideal of Im κ are determined by a basis of Ker ˆ φ .From this, we deduce that gr A = gr C = Sym k [ V ] ( E ) , where gr ( . ) is the associated gradedring with respect to the filtration by the order of differential operators. Then the equality A = C follows. We also give a sufficient condition for C to be a free k [ V ] G -module (seeTheorem 4.9).Let g = g ⊕ g be a Z -graded semisimple Lie algebra. This grading (or the symmetricpair ( g , g ) ) is said to be N - regular , if g contains a regular nilpotent element of g . Our mainapplication concerns the isotropy representation ( G : g ) . We show that the hypotheses ofTheorems 4.2 and 4.9 are satisfied, modulo one exception, for ( G : g ) if ( g , g ) is N -regular.Hence A = C and C is a free k [ g ] G -module in these cases. Verification of all necessaryconditions requires a detailed information on the structure of the null-cone in the G -module g . We also provide other examples of the “ A = C phenomenon”; in particular,those for the coadjoint representation of non-reductive Lie algebras.The plan of the article is as follows. Section 1 contains preliminaries on group actionsand differential operators. In Section 2, we prove our analogue of Dixmier’s result for thecoadjoint representation of a -wonderful Lie algebra. Then we discuss, in Section 3, threegeneralisations to the case in which g ∗ is replaced with an arbitrary G -module. Section 4contains our results on the image of κ , Sym k [ V ] ( E ) , and the equality A = C . In Section 5,we consider applications to Z -gradings of semisimple Lie algebras and provide someother examples. In Section 6, we discuss possible connections between our results for Z -gradings and another generalisation of Dixmier’s result obtained in [12, 14].Some notation. If an algebraic group G acts on an irreducible affine variety X , then k [ X ] G is the algebra of G -invariant regular functions on X and k ( X ) G is the field of G -invariant rational functions. If k [ X ] G is finitely generated, then X//G := Spec k [ X ] G , andthe quotient morphism π X : X → X//G is the mapping associated with the embedding k [ X ] G ֒ → k [ X ] . We use dot ‘ · ’ to denote the action of (elements of) G and g on X . Forinstance, G · x is the orbit of x ∈ X . The stabiliser of x in g is denoted by g x . D. PANYUSHEV
All topological terms refer to the Zariski topology. The pairing of dual vector spaces isdenoted by h , i . If M is a subset of a vector space, then span ( M ) denotes the linear spanof M . Acknowledgements.
I am grateful to Thierry Levasseur for sending me his unpublished notesand useful e-mail exchange.
1. P
RELIMINARIES G be an affine algebraic group acting regularly on an irreducible algebraic va-riety X . We say that h ⊂ g is a generic stabiliser for the the action ( G : X ) if there existsa dense open subset Ω ⊂ X such that all stabilisers g x , x ∈ Ω , are G -conjugate to h . Thepoints of such an Ω are said to be generic . Generic stabilisers always exist if G is reductiveand X is smooth [21].Let X reg denote the set of all regular elements of X . That is, X reg := { x ∈ X | dim G · x > dim G · x ′ for all x ′ ∈ X } = { x ∈ X | dim g x dim g x ′ for all x ′ ∈ X } . As is well-known, X reg is a dense open subset of X . If we want to explicitly specify thegroup acting on X , we refer to G - regular elements. Definition 1. A G -variety X is said to have the codim– n property if codim X ( X \ X reg ) > n .We will mostly use this notion if X = V is a G -module. Example.
Let g be reductive and N ⊂ g the nilpotent cone. Then g (resp. N ) has thecodim– (resp. codim– ) property with respect to the adjoint representation [9].Recall that the index of g , denoted ind g , is the minimal dimension of stabilisers for theelements of the g -module g ∗ . That is, ind g = min ξ ∈ g ∗ dim g ξ = dim g η for any η ∈ g ∗ reg .1.2. For finite-dimensional k -vector spaces V and N , let Mor ( V, N ) denote the set of poly-nomial morphisms V → N . Clearly, Mor ( V, N ) ≃ k [ V ] ⊗ N and it is a free graded k [ V ] -module of rank dim N .If V and N are G -modules, then G acts on Mor ( V, N ) by the rule ( g ∗ F )( v ) = g · ( F ( g − · v )) .Then ( Mor ( V, N )) G =: Mor G ( V, N ) is the set of all polynomial G -equivariant morphisms V → N . It is a k [ V ] G -module, which is called the module of covariants of type N . If G isreductive, then the algebra k [ V ] G is finitely generated and each Mor G ( V, N ) is a finitelygenerated k [ V ] G -module.[All these constructions makes sense if V is replaced with any affine G -variety X .] DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 5 D ( V ) denote the algebra of differential operators on V , with polynomial coef-ficients. Recall that D ( V ) contains the symmetric algebra of V , S ( V ) , as the subalgebraof constant coefficient differential operators and k [ V ] as the subalgebra of differential op-erators of order zero. We always filter D ( V ) by the order of differential operators, hence gr n D ( V ) ≃ k [ V ] ⊗S n ( V ) and gr D ( V ) is isomorphic to k [ V ] ⊗S ( V ) = k [ V × V ∗ ] as algebras.Let Der ( k [ V ]) denotes the k [ V ] -module of all k -derivations of k [ V ] or, equivalently, themodule of polynomial vector fields on V . Then Der ( k [ V ]) ≃ Mor ( V, V ) . A vector field X can be regarded either as polynomial endomorphism of V or as linear endomorphism of k [ V ] . The respective notation is X ( v ) , v ∈ V and X { f } , f ∈ k [ V ] .2. A DJOINT VECTOR FIELDS AND WONDERFUL L IE ALGEBRAS
In this section, G is a connected algebraic group.Let V be a (finite-dimensional, rational) G -module. The differential of the G -action on k [ V ] yields a map ς = ς V : g → Der ( k [ V ]) ⊂ D ( V ) . Upon the identification Der ( k [ V ]) with Mor ( V, V ) , we see that ς ( e ) is just the linear operator on V corresponding to e ∈ g . Thevector fields on V of the form ς ( e ) are said to be the adjoint vector fields . For g semisimpleand V = g , Dixmier describes a relationship between the adjoint vector fields and vectorfields annihilating all of k [ g ] G [6, Theorem 2.1]. Below, we prove that this result naturallyextends to the coadjoint representations of certain non-reductive Lie algebras.In [9], Kostant established a number of fundamental properties of complex reductive Liealgebras. Motivated by these results, we give the following Definition 2.
An algebraic Lie algebra g is said to be n - wonderful , if the following condi-tions are satisfied: (i) the coadjoint representation of g has the codim– n property. (ii) k [ g ∗ ] G is a polynomial algebra of Krull dimension l = ind g ; (iii) If f , . . . , f l are homogeneous algebraically independent generators of k [ g ∗ ] G , then P li =1 deg f i = (dim g + ind g ) / ; Remark 2.1.
We are only interested in n -wonderful algebras for n = 2 , . Let us point outsome connections between hypotheses of this definition, and their consequences.1. For any Lie algebra, trdeg k ( g ∗ ) G equals ind g and hence k [ g ∗ ] G contains at most ind g algebraically independent elements. Thus, condition (ii) also means that ( g , ad ∗ ) has suf-ficiently many polynomial invariants.2. If ( g , ad ∗ ) has the codim– property and f , . . . , f l ∈ k [ g ∗ ] G are algebraically inde-pendent, then P li =1 deg f i > (dim g + ind g ) / . Furthermore, if the equality holds, then f , . . . , f l freely generate k [ g ∗ ] G and(2.1) g ∗ reg = { ξ ∈ g ∗ | ( d f ) ξ , . . . , ( d f l ) ξ are linearly independent } , D. PANYUSHEV see [18, Theorem 1.2]. It follows that Eq. (2.1) holds for any -wonderful algebra. For g reductive, equality (2.1) is a celebrated result of Kostant [9, Theorem 0.1].3. The main result of [20] asserts that if g is -wonderful, then the Poisson commutativesubalgebra of k [ g ∗ ] obtained from k [ g ∗ ] G via the argument shift method is maximal for any ξ ∈ g ∗ reg .4. Any reductive Lie algebra is -wonderful. Several non-trivial examples of -wonderful algebras are discussed in [20, Section 4]. Theorem 2.2.
Let g be a -wonderful Lie algebra. Given a polynomial vector field X on g ∗ , thefollowing conditions are equivalent: (i) X annihilates all G -invariant polynomials on g ∗ ; (ii) X ( ξ ) ∈ g · ξ for any ξ ∈ g ∗ reg ; (iii) X ( ξ ) ∈ g · ξ for any ξ ∈ g ∗ ; (iv) There is a polynomial mapping Y ∈ Mor ( g ∗ , g ) such that X ( ξ ) = Y ( ξ ) · ξ for any ξ ∈ g ∗ .Proof. Recall that for f ∈ k [ g ∗ ] the polynomial X { f } ∈ k [ g ∗ ] is defined by(2.2) X { f } ( ξ ) = h X ( ξ ) , ( d f ) ξ i . It is therefore clear that (iv) ⇒ (iii) ⇒ (ii) ⇒ (i). It remains to prove the implication (i) ⇒ (iv).To this end, we need some preparations. Up to some obvious alterations, the rest of theproof is a repetition of the proof of Theorem 2.1 in [6].Set Ω = g ∗ reg . If ξ ∈ Ω , then ( d f ) ξ , . . . , ( d f l ) ξ form a basis for g ξ , in view of Definition 2and Eq. (2.1).Let E be the cotangent bundle of Ω , which is identified with E ≃ Ω × g . Let E ′ be thesub-bundle of E whose fibre of ξ is g ξ . The previous paragraph shows that the d f i ’s yielda trivialisation of E ′ . Let E ′′ be the sub-bundle of the tangent bundle of Ω whose fibre of ξ is g · ξ . Since the kernel of the surjective mapping ( x ∈ g ) ( x · ξ ∈ g · ξ ) is g ξ , one obtainsthe exact sequence of vector bundles → E ′ → E → E ′′ → and the exact sequence H (Ω , E ) → H (Ω , E ′′ ) → H (Ω , E ′ ) . Let O denote the structure sheaf of g ∗ . By [5, cor. 2.9, p.16], there exists an exact sequenceof cohomology groups H ( g ∗ , O ) → H (Ω , O | Ω ) → H g ∗ \ Ω ( g ∗ , O ) . DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 7
Here H ( g ∗ , O ) = 0 because g ∗ is affine, and it follows from the codim– property that H g ∗ \ Ω ( g ∗ , O ) = 0 [5, cor. 1.4, p.80]. Hence H (Ω , O | Ω ) = 0 . This fact and the trivilality of E ′ imply that H (Ω , E ′ ) = 0 . Thus, the homomorphism γ : H (Ω , E ) → H (Ω , E ′′ ) is onto.Suppose that X satisfies assumption (i). Then Eq. (2.2) and the linear independence ofthe differentials ( d f i ) ξ , ξ ∈ Ω , show that X also satisfies (ii). Therefore X | Ω is a section of E ′′ . The surjectivity of γ means that there exists a polynomial mapping Y : Ω → g suchthat X ( ξ ) = Y ( ξ ) · ξ for any ξ ∈ Ω . Since codim( g ∗ \ Ω) > , Y extends to a polynomialmapping Y : Ω → g , and the equality X ( ξ ) = Y ( ξ ) · ξ holds for all ξ ∈ g ∗ . (cid:3) Remark 2.3.
This theorem is a statement about the coadjoint representation of G . Thereare two key points in the proof. First, E ′ appears to be a trivial bundle and, second, codim( g ∗ \ Ω) > . Using this observation, we show in Section 3 that Theorem 2.2 admitsvarious generalisations to other representations of G . Remark 2.4.
We know that
Der ( k [ g ∗ ]) is a k [ g ∗ ] -module and ς ( g ) ⊂ Der ( k [ g ∗ ]) , where ς = ς g ∗ . Therefore, implication (i) ⇒ (iv) in Theorem 2.2 can be stated as follows:If X ∈ Der ( k [ g ∗ ]) annihilatesallof k [ g ∗ ] G , then X ∈ k [ g ∗ ] · ς ( g ) . Remark 2.5.
For an arbitrary Lie algebra g , define the homomorphism of k [ g ∗ ] -modules ˆ φ : Mor ( g ∗ , g ) → Mor ( g ∗ , g ∗ ) by ˆ φ ( F )( ξ ) = F ( ξ ) · ξ , ξ ∈ g ∗ . Then Im ˆ φ ⊂ { F : g ∗ → g ∗ | F ( ξ ) ∈ g · ξ ∀ ξ ∈ g ∗ } =: T . Since the elements of
Mor ( g ∗ , g ∗ ) is are just the vector field on g ∗ , the equivalence of con-ditions (iii) and (iv) in Theorem 2.2 reduces to the assertion that if g is -wonderful, then Im ˆ φ = T .Notice that Ker ˆ φ = { F : g ∗ → g | F ( ξ ) ∈ g ξ ∀ ξ ∈ g ∗ } . If Ker ˆ φ is a free k [ g ∗ ] -module(of rank l = ind g ) and F , . . . , F l is a basis, then E ′ is a trivial vector bundle over Ω ′ = { ξ ∈ g ∗ reg | F ( ξ ) , . . . , F l ( ξ ) are linearly independent } . For any f ∈ k [ g ∗ ] G , we have d f ∈ Ker ˆ φ ∩ Mor G ( g ∗ , g ) . If g is -wonderful, then [15, Theorem 1.9] applies to the coadjointrepresentation of g , and one concludes that Ker ˆ φ is freely generated by the differentialsd f , . . . , d f l . Then, using Eq. (2.1), we obtain Ω ′ = g ∗ reg . This argument shows that in somecases (actually, most interesting ones), the triviality of E ′ is closely related to the fact that Ker ˆ φ is a free k [ g ∗ ] -module generated by G -equivariant morphisms. Example . There is a procedure that generates new n -wonderful algebras from old ones(for n > ). Let q be a quadratic n -wonderful Lie algebra (”quadratic” means that q ∗ ≃ q as q -module) . Form the semi-direct product g = q ⋉ q (the second copy of q is anAbelian ideal of g ). Then g is again quadratic and n -wonderful. That g to be quadratic iselementary. Therefore we can deal with the adjoint representation of g . It then suffices toapply Theorem 7.1 in [17] to the case V = q . Roughly speaking, that theorem says that D. PANYUSHEV the passage q q ⋉ q doubles all data occurring in Definition 2. That is, dim g = 2 dim q , ind g = 2 ind q ; each basis invariant f i ∈ k [ q ] Q gives rise to two basis invariants in k [ g ] G ,and the degree for all three is the same. Finally, it is easily seen that q reg ⋉ q ⊂ g reg . Hencethe codim– n property is also preserved.In particular, one can start with any semisimple s and take s ⋉ s . This yields interestingexamples of -wonderful algebras. Notice that then this procedure can be iterated adinfinum.3. M ODULES OVER POLYNOMIAL RINGS ASSOCIATED WITH REPRESENTATIONS
Unless otherwise stated, G is an arbitrary connected algebraic group. Let V be a G -module. Associated with V , g , and g ∗ , there are at least three natural exact sequencesof modules over polynomial rings:(A) → Ker ˆ φ → Mor ( V, g ) ˆ φ → Mor ( V, V ) ,(B) → Ker ˆ ψ → Mor ( V, V ∗ ) ˆ ψ → Mor ( V, g ∗ ) ,(C) → Ker ˆ τ → Mor ( g , V ) ˆ τ → Mor ( g , V ) .The first two sequences consist of k [ V ] -modules, and the last one consists of k [ g ] -modules.Some of the properties of (A) and (B) have been studied in [15], whereas (B) and (C) havealso been considered in [17, Sect. 8]. Recall the definitions of ˆ φ, ˆ ψ, ˆ τ :(A) ˆ φ ( F )( v ) := F ( v ) · v , where v ∈ V ;(B) h ˆ ψ ( F )( v ) , x i := h x · v, F ( v ) i , where v ∈ V , x ∈ g , and h , i stands for the pairing ofelements of dual vector spaces. One can also exploit the moment mapping µ : V × V ∗ → g ∗ ,which is defined by h µ ( v, η ) , x i = h x · v, η i , where η ∈ V ∗ . Then ˆ ψ ( F )( v ) := µ ( v, F ( v )) .(C) ˆ τ ( F )( x ) := x · F ( x ) , where x ∈ g . Remark.
Ker ˆ φ is a G -stable submodule of Mor ( V, g ) ; and likewise for Ker ˆ ψ and Ker ˆ τ .Note that, for V = g ∗ , the sequences (A) and (B) coincide, and we obtain the situationof Remark 2.5. Also, the sequences (A) and (C) coincide if V = g . Below we formu-late Dixmier-type statements, which characterise the images of ˆ φ, ˆ ψ , and ˆ τ under similar(rather restrictive) assumptions. Case (A) . Here
Ker ˆ φ = { F : V → g | F ( v ) ∈ g v ∀ v ∈ V } and Im ˆ φ ⊂ { F : V → V | F ( v ) ∈ g · v ∀ v ∈ V } . Set Ω φ = V reg . Consider three vector bundles on Ω φ : E ′ φ = { ( v, x ) | x · v = 0 } = { ( v, x )) | x ∈ g v } ⊂ Ω φ × g ,E φ = Ω φ × g , E ′′ φ = { ( v, x · v ) | v ∈ Ω φ , x ∈ g } ⊂ Ω φ × V and the corresponding exact sequence → E ′ φ → E φ → E ′′ φ → . Arguing as in the proofof Theorem 2.2, one obtains DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 9
Proposition 3.1.
Suppose E ′ φ is a trivial vector bundle and codim( V \ Ω φ ) > . Then Im ˆ φ = { F : V → V | F ( v ) ∈ g · v ∀ v ∈ V } . In other words, if F ( v ) ∈ g · v for all v ∈ Ω φ , then there is F : V → g such that F ( v ) = F ( v ) · v forall v ∈ V . This is not a complete analogue of Theorem 2.2, since we obtain only equivalence of thefollowing three conditions on the vector field F : V → V : (ii) F ( v ) ∈ g · v for any v ∈ V reg ; (iii) F ( v ) ∈ g · v for any v ∈ V ; (iv) There is an F ∈ Mor ( V, g ) such that F ( v ) = F ( v ) · ξ for any ξ ∈ V .In order to add condition (i) F annihilates all of k [ V ] G to this list, one has to impose some constraints on k [ V ] G . For instance, it suffices to requirethat the quotient field of k [ V ] G equals to k ( V ) G and that dim( span { d f v | f ∈ k [ V ] G } ) =trdeg k ( V ) G for any v ∈ V reg . (Cf. the proof of Theorem 2.2). Actually, these two condi-tions are not too restrictive. These are always satisfied if G is semisimple and k [ V ] G is apolynomial (free) algebra (see [8]).The problem of triviality for E ′ φ is connected with the question of whether Ker ˆ φ is a free k [ V ] -module. This seems to be related to the property that a generic stabiliser for ( g : V ) isabelian. In the following sections, we study case (A) more carefully, prove a more generalresult, and provide some examples. Case (B) . Here
Ker ˆ ψ = { F : V → V ∗ | µ ( v, F ( v )) = 0 ∀ v ∈ V } and Im ˆ ψ ⊂ { F : V → g ∗ | F ( v ) ∈ µ ( v, V ∗ ) ∀ v ∈ V } . Again, we take Ω ψ = V reg . Consider three vector bundles on Ω ψ : E ′ ψ = { ( v, ξ ) | µ ( v, ξ ) = 0 } = { ( v, ξ ) | ξ ∈ ( g · v ) ⊥ } ⊂ Ω ψ × V ∗ ,E ψ = Ω ψ × V ∗ , E ′′ ψ = { ( v, µ ( v, ξ )) | v ∈ Ω ψ , ξ ∈ V ∗ } ⊂ Ω ψ × g ∗ and the corresponding exact sequence → E ′ ψ → E ψ → E ′′ ψ → . Arguing as in the proofof Theorem 2.2, one obtains Proposition 3.2.
Suppose E ′ ψ is a trivial vector bundle and codim( V \ Ω ψ ) > . Then Im ˆ ψ = { F : V → g ∗ | F ( v ) ∈ µ ( v, V ∗ ) ∀ v ∈ V } . In other words, if F ( v ) ∈ µ ( v, V ∗ ) for all v ∈ Ω ψ , then there is F : V → V ∗ such that F ( v ) = µ ( v, F ( v )) for all v ∈ V . The hypotheses of Proposition 3.2 are satisfied in the following situation.
Theorem 3.3.
Suppose g is semisimple, k [ V ] G is polynomial, and codim( V \ V reg ) > . Then Ker ˆ ψ is a free k [ V ] -module and Proposition 3.2 applies.Proof. Let f , . . . , f l ∈ k [ V ] G be the basis invariants. By [15, 1.9 & 1.10], Ker ˆ ψ is afree k [ V ] -module generated by d f , . . . , d f l . By [8, Korollar 1], V reg ⊂ { v ∈ V | ( d f ) v , . . . , ( d f l ) v are linearly independent } . It follows that E ′ ψ is a trivial bundle on V reg . (cid:3) It follows from [20, Remark 4.5] that, under the assumptions of Theorem 3.3, a genericstabiliser for ( G : V ) has to be non-trivial, i.e., max dim G · v < dim V . Case (C) . For x ∈ g , we set V x = { v ∈ V | x · v = v } . Here Ker ˆ τ = { F : g → V | F ( x ) ∈ V x ∀ x ∈ g } and Im ˆ τ ⊂ { F : g → V | F ( x ) ∈ x · V ∀ x ∈ g } . Set Ω τ = { x ∈ g | dim V x is minimal } . Consider three vector bundles on Ω τ : E ′ τ = { ( x, v ) | x · v = 0 } = { ( x, v ) | v ∈ V x } ⊂ Ω τ × V, E τ = Ω τ × V,E ′′ τ = { ( x, x · v ) | x ∈ Ω τ , v ∈ V } ⊂ Ω τ × V and the corresponding exact sequence → E ′ τ → E τ → E ′′ τ → . Arguing as in the proofof Theorem 2.2, one obtains Proposition 3.4.
Suppose E ′ τ is a trivial vector bundle and codim( g \ Ω τ ) > . Then Im ˆ τ = { F : g → V | F ( v ) ∈ x · V ∀ x ∈ g } . In other words, if F ( x ) ∈ x · V for all x ∈ Ω τ , then there is F : g → V such that F ( x ) = x · F ( x ) for all x ∈ q . It is remarkable that if G is reductive, then Ker ˆ τ is always a free k [ g ] -module [17, Theo-rem 8.6]. There is also a special case in which all the assumptions of Proposition 3.4 aresatisfied. Theorem 3.5.
Let g be reductive, t ⊂ g a Cartan subalgebra, and e ∈ g a regular nilpotentelement. Suppose that ( ✸ ) dim V e = dim V t .Then Ω τ ⊃ g reg and if F ( x ) ∈ x · V for all x ∈ g reg , then there is F : g → V such that F ( x ) = x · F ( x ) for all x ∈ g .Proof. It easily follows from assumption ( ✸ ) that dim V z = dim V t for any regularsemisimple z ∈ g . Therefore the minimal value of dim V x is the dimension of the zero-weight space of V , which is positive. That is, the open subset Ω τ contains the regularsemisimple and nilpotent elements. It follows that Ω τ ⊃ g reg . By [9], codim( g \ g reg ) = 3 .For triviality E ′ τ , it is enough to prove that Ker ˆ τ is a free k [ g ] -module, and the latter hasbeen done in [17, Theorem 8.6]. (cid:3) DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 11
Remark 3.6.
The above equality dim V z = dim V t (with z semisimple) means that eachnonzero weight of V (with respect to t ) is a multiple of a root. Using this observation,one easily obtains the complete list of irreducible representations of simple Lie algebrassatisfying assumption ( ✸ ) . Here it is: • the adjoint representation of g ; • ( B n , ϕ ) , ( B n , ϕ ) , ( C n , ϕ ) , ( F , ϕ ) , ( G , ϕ ) , ( G , ϕ ) ; • ( A , mϕ ) for any m ∈ N .Actually, each of the cases (A), or (B), or (C) deserves a special thorough treatment. Inthe following sections, we concentrate on case (A), partly in view of its connections withdifferential operators. Another reason is that similar properties of sequences (B) and (C)for representations of reductive groups have been studied in [17, Section 8].4. D IFFERENTIAL OPERATORS AND INVARIANT POLYNOMIALS
In Section 3, three possibilities to generalise Dixmier’s results have been discussed. Theseare related to three sequences of modules over polynomial rings. It seems that case (A) isthe most interesting one, because the problem can further be transferred to the setting ofdifferential operators on V .The discussion of case (A) in Section 3 shows that if a G -module V satisfies certainexplicit conditions, then a vector field F : V → V annihilates all of k [ V ] G if and only ifthere is F ∈ Mor ( V, g ) such that F ( v ) = F ( v ) · v for all v ∈ V . In other words, ( ♦ ) F { f } = 0 for any f ∈ k [ V ] G if and only if F ∈ k [ V ] ς ( g ) .(cf. Remark 2.4). Let us restate ( ♦ ) using the algebra of differential operators D ( V ) .Let C = Cent D ( V ) ( k [ V ] G ) denote the centraliser of k [ V ] G in D ( V ) . Clearly, C contains k [ V ] and ς ( g ) . Let A be the subalgebra of C generated by k [ V ] and ς ( g ) . Note that avector field F and a polynomial f ∈ k [ g ] G commute as differential operators if and onlyif F { f } = 0 . Therefore assertion ( ♦ ) can also be interpreted as the coincidence of A and C on the level of vector fields.Motivated by Dixmier’s result [6, Theorem 2.1] and a question by Barlet, Levasseur andStafford proved that A = C for the adjoint representation of a semisimple Lie algebra g [13]. In this section, we prove such an equality in a more general setting.We assume below that the G -module V has the property that k [ V ] G is finitely generatedand the quotient field of k [ V ] G equals k ( V ) G . The latter is equivalent to that a generic fibreof π V : V → V //G contains a dense G -orbit.We work with the sequence of graded k [ V ] -modules → Ker ˆ φ → Mor ( V, g ) ˆ φ → Mor ( V, V ) . Here rk ˆ φ = max v ∈ V dim g · v [15, Prop. 1.7] and therefore rk (Ker ˆ φ ) = min v ∈ V dim g v .Set m = min v ∈ V dim g v = dim g − dim V + dim V //G . Assume that
Ker ˆ φ is a free (graded) k [ V ] -module, and let F , . . . , F m be a homogeneous basis for Ker ˆ φ . Write E for the k [ V ] -module Im ˆ φ . Then we obtain the exact sequence(4.1) → m M i =1 k [ V ] F i ˆ β → Mor ( V, g ) ˆ φ → E → . Using the morphisms F i : V → g , we define a variety Y as follows: Y = { ( v, η ) ∈ V × g ∗ | h F i ( v ) , η i = 0 , i = 1 , . . . , m } . Recall that
Ker ˆ φ is a G -stable submodule of Mor ( V, g ) . Therefore for any g ∈ G there exist u ( g )1 , . . . , u ( g ) m ∈ k [ V ] such that g ∗ F i = P j u ( g ) j F j . This readily implies that Y is G -stable.Choose a basis e , . . . , e n for g . Using this basis, we identify Mor ( V, g ) = k [ V ] ⊗ g with k [ V ] n . Then we can write F j ( v ) = P ni =1 F ij ( v ) e i , where F ij ∈ k [ V ] . If we regard (4.1) as asequence → k [ V ] m ˆ β → k [ V ] n ˆ φ → E → , then ˆ β becomes an n × m -matrix with entries F ij . Let I t ( ˆ β ) be the ideal of k [ V ] generatedby t × t minors of ˆ β . Following [7], consider the series of determinantal conditions for thepresentation of E : ( F d ) ht I t ( ˆ β ) > m − t + 1 + d for t m .The ideals I t ( ˆ β ) are independent of the presentation of E . These are Fitting ideals of E ,see e.g. [24, 1.1]. Let Sym k [ V ] ( E ) denote the symmetric algebra of the k [ V ] -module E .Then Sym k [ V ] ( E ) = L ∞ n =0 Sym k [ V ] ( E ) n and each Sym k [ V ] ( E ) n is a finitely generated graded k [ V ] -module. Conditions ( F d ) are very useful in the study of properties of the symmetricalgebras of modules. Utility of these conditions in Representation and Invariant theoryhas been demonstrated in [15, 13, 17]. Theorem 4.1.
Suppose that
Ker ˆ φ is a free module and condition ( F ) is satisfied by E . Then (i) Sym k [ V ] ( E ) is a factorial domain of Krull dimension dim V + dim g − m . (ii) Y is an irreducible factorial complete intersection, and k [ Y ] = Sym k [ V ] E . (iii) Y = Im ( κ ) , where κ : V × V ∗ → V × g ∗ is defined by κ ( v, ξ ) = ( v, µ ( v, ξ )) . Here v ∈ V, ξ ∈ V ∗ and µ : V × V ∗ → g ∗ is the moment mapping. (iv) Let p : Y → V be the (surjective) projection. If I is a prime ideal of k [ V ] with ht ( I ) > ,then ht ( I k [ Y ]) > .Proof. (i) The exact sequence (4.1) shows that E has projective dimension at most one.Therefore part (i) follows from [2, Prop. 3 & 6]. DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 13 (ii) The universal property of symmetric algebras implies that
Sym k [ V ] ( E ) is the quo-tient of Sym k [ V ] ( k [ V ] ⊗ g ) = k [ V × g ∗ ] modulo the ideal ”generated by the image of ˆ β ”. Moreprecisely, each F i determines the polynomial b F i ∈ k [ V × g ∗ ] by the rule b F i ( v, η ) = h F i ( v ) , η i , η ∈ g ∗ , and the ideal in question is generated by b F , . . . , b F m . Hence Sym k [ V ] ( E ) = k [ Y ] ,and the other assertions follow from (i).(iii) Clearly, Im ( κ ) is an irreducible subvariety of V × g ∗ . Taking the (surjective) pro-jection to V and looking at the dimension of the generic fibre, one finds that dim Im ( κ ) =dim V + max(dim g · v ) = dim V + dim g − m . Since F i ( v ) · v = 0 for all v ∈ V , we have b F i ( v, µ ( v, ξ )) = h F i ( v ) , µ ( v, ξ ) i = h F i ( v ) · v, ξ i = 0 . Hence each b F i vanishes on Im κ and Im ( κ ) ⊂ Y . Since both varieties have the samedimension and are irreducible, they are equal.(iv) According to [7], Remarks on pp. 664–5, this property is equivalent to condition ( F ) . See also [24, Remark 1.3.9] (cid:3) As a by-product of this theorem, we obtain the following description of
Sym k [ V ] ( E ) = k [ Y ] . Consider the linear map i : g → V ∗ ⊗ V which is induced by the moment map µ : V × V ∗ → g ∗ . (The map i is injective if and only if the representation g → gl ( V ) isfaithful.) In this way, we obtain certain copy of g sitting in V ∗ ⊗ V ⊂ k [ V × V ∗ ] . Then k [ Y ] is isomorphic to the subalgebra of k [ V × V ∗ ] generated by k [ V ] and i ( g ) . Theorem 4.2.
Suppose that k [ V ] G is a polynomial algebra freely generated by f , . . . , f l ; Ker ˆ φ is a free k [ V ] -module; V has the codim– property and { ( d f i ) v } are linearly independent for any v ∈ V reg ; condition ( F ) is satisfied for E = Im ˆ φ .Let A and C be given the filtration induced from D ( V ) . Then gr C = gr A ≃ Sym k [ V ] ( E ) .Proof. The proof of Levasseur and Stafford for the adjoint representation of a semisimple g [13, Section 3] carries over mutatismutandis to this more general situation. The followingis very close to their original proof. Lemma 4.3.
For v ∈ V reg , let R denote the local ring of V at v . Then there exists a basis ofderivations { ∂ i | i = 1 , . . . , s } of Der R such that ∂ i ( f j ) = δ i,j for all i, j l and R ς ( g ) = L si = l +1 R ∂ i . [Here s = dim V .] Proof.
There is a one-to-one correspondence between the bases for the R -module Der R and the k -bases for the tangent space T v ( V ) . Since { ( d f i ) v , i = 1 , . . . , l } are linearly in-dependent, the annihilator of span { ( d f i ) v , i = 1 , . . . , l } in T v ( V ) is g · v . Choose a basis ( e , . . . , e s ) in V such that h e i , ( d f j ) v i = δ i,j , i, j l , and span { e l +1 , . . . , e s } = g · v . Thenthe corresponding basis for Der R will work. (Cf. the proof of Lemma 3.1 in [13]). (cid:3) Since gr C ⊂ gr D ( V ) = k [ V × V ∗ ] , certainly gr A ⊂ gr C are domains. Next, gr A containsthe subalgebra generated by k [ V ] and gr ( ς ( g )) . It is easily seen that gr ( ς ( g )) = i ( g ) . Itfollows from Theorem 4.1 that, on the geometric level, we obtain the chain of dominantmorphisms(4.2) V × V ∗ → Spec (gr C ) → Spec (gr A ) → Im κ = Y ⊂ V × g ∗ , where the resulting map V × V ∗ → Im κ is just κ . Lemma 4.4.
The above mapping ϕ : Spec (gr C ) → Im κ is birational.Proof. We prove a more precise assertion that, for any v ∈ V reg , there is the equality oflocal rings (gr C ) v = (gr A ) v = k [ Y ] v .Recall that R = k [ V ] v . For any ring R containg k [ V ] , its localisation with respect to themultiplicative subset { f ∈ k [ V ] | f ( v ) = 0 } is denoted by R v . Then D ( V ) v = R { ∂ , . . . , ∂ s } is the non-commutative algebra generated by derivations constructed in Lemma 4.3 and C v = { D ∈ D ( V ) v | [ D, f i ] = 0 , i = 1 , . . . , l } . The last formula and Lemma 4.3 readily imply that C v = A v = R { ∂ l +1 , . . . , ∂ s } . Let ¯ ∂ i denotes the image of ∂ i in gr D ( V ) . Lemma 4.3 also shows that R · i ( g ) = s M i = l +1 R ¯ ∂ i . Thus, gr C v = gr A v = gr R { ∂ l +1 , . . . , ∂ s } = R [ ¯ ∂ l +1 , . . . , ¯ ∂ s ] v = R [ i ( g )] v = k [ Y ] v . (cid:3) Finally, we prove that ϕ : Spec (gr C ) → Im κ is an isomorphism. We already know that ϕ is birational and that Im κ is normal (Theorem 4.1). By Richardson’s lemma, see e.g. [3,3.2 Lemme 1], it suffices to verify that Im ϕ contains an open subset whose complementis of codimension > . Thanks to Eq. (4.2), this reduces to the same question for κ : V × V ∗ → Im κ ≃ Y .It is easily seen that if ( v, ξ ) ∈ Im κ = Y and v ∈ V reg , then ( v, ξ ) ∈ Im κ . Invoking theprojection p : Y → V shows that p − ( V reg ) is on open subset lying in Im κ . Since V has thecodim– property, we conclude, using Theorem 4.1(iv), that the complement of p − ( V reg ) is of codimension > .This completes the proof of Theorem 4.2. (cid:3) Corollary 4.5. (i) A = C ; Moreover, C is an Auslander-Gorenstein, Cohen-Macaulay, Noether-ian domain and a maximal order; (ii) the centre of C is k [ V ] G . DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 15 (See the proof of Corollary 3.3 in [13].)Levasseur and Stafford also prove that, in their situation, both C and gr C = Sym k [ V ] ( E ) are free modules over k [ g ] G , see [13, Corollary 3.4]. We return to this question below.There is a particular case of Theorem 4.2, where the assumptions simplify considerably. Proposition 4.6.
Suppose that k [ V ] G is a polynomial algebra freely generated by f , . . . , f l ; V has the codim– property and { ( d f i ) v } are linearly independent for any v ∈ V reg ; there is v ∈ V such that g v = 0 .Then Y = Im κ = V × g ∗ and A = C .Proof. Indeed, here
Ker ˆ φ = 0 and condition ( F ) becomes vacuous. (cid:3) Verification of condition ( F ) is the most difficult part in possible applications of Theo-rem 4.2. In the rest of the section, we describe a geometrical approach to it (cf. similarapproach in [17, § Ker ˆ φ is a free module (of rank m ).Using the basis morphisms F i : V → g , define the stratification of V as follows: X i = { v ∈ V | dim span { F ( v ) , . . . , F m ( v ) } m − i } . Then X i +1 ⊂ X i and X = V . As the ideal I t ( ˆ β ) defines X m − t +1 , condition ( F ) preciselymeans that dim X i dim V − i − for any i > . In particular, codim X > . In case ofthe coadjoint representation of a -wonderful Lie algebra, this becomes just the codim– condition on the set of non-regular points, which is used in the proof of Theorem 2.2.Consider the quotient map π V : V → V //G ≃ A l . As usual, we say that π − V ( π V (0)) =: N = N V is the null-cone of (the G -module) V . Set X i ( N ) := X i ∩ N . Sometimes, the studyof { X i } can be reduced to that of { X i ( N ) } .Recall that Ker ˆ φ is a G -stable submodule of Mor ( V, g ) . Therefore if F , . . . , F m is a basisof Ker ˆ φ , then { g ∗ F i } i is another basis for any g ∈ G . It is not clear a priori that the F i ’s should be G -equivariant. Consequently, subvarieties X i are not necessarily G -stable.However, in all known examples the freeness of Ker ˆ φ does mean that there is a basisconsisting of G -equivariant morphisms. (Cf. Remark 2.5 and Theorem 5.1 below). Forthis reason, we wish to assume that F i ∈ Mor G ( V, g ) . Proposition 4.7.
Under the first two assumptions of Theorem 4.2, suppose that a generic fibre of π V is a (closed) G -orbit, N contains finitely many G -orbits, and each F i lies in Mor G ( V, g ) . If ( ♣ ) codim N X i ( N ) > i + 1 for any i > , then ( F ) is satisfied.An equivalent form of condition ( ♣ ) is that N reg ⊂ X ( N ) and, for any v ∈ N \ N reg , dim( span { F i ( v ) | i = 1 , . . . , m } ) + codim N G · v > m + 1 . Proof.
The finiteness assumption guarantees us that dim N = dim V − dim V //G and N reg ⊂ V reg . Furthermore, all the fibres of π V are of dimension dim V − dim V //G , see e.g. [3, Prop. 6in p.146] (the reductivity of G is not needed in this place). By Chevalley’s theorem, π V isan open map. Consequently, it is onto. By the assumption, each X i is G -stable. Let π i,V bethe restriction of π V to X i . Then X i ( N ) = π − i,V ( π i,V (0)) . Therefore dim X i dim X i ( N ) + dim π V ( X i ) dim N − ( i + 1) + (dim V //G −
1) = dim V − ( i + 2) . Here we used the fact that π V ( X i ) is a proper subvariety of V //G for i > . Indeed, V \ X is a dense open subset of V and there is a dense open subset Ξ ⊂ ( V \ X ) such that if G · v is a fibre of π V for any v ∈ Ξ . The second part is an easy reformulation of condition ( ♣ ) ,which uses the finiteness for G -orbits in N . (cid:3) Recall that v ∈ V reg if and only if dim g v = m . Since codim N G · v = dim g v − m , yet anotherform of the above conditions is dim( span { F i ( v ) | i = 1 , . . . , m } ) + dim g v = 2 m for any v ∈ N reg , (4.3) dim( span { F i ( v ) | i = 1 , . . . , m } ) + dim g v > m + 1 for any v ∈ N \ N reg . (4.4) Remark 4.8. If Ker ˆ φ is a free k [ V ] -module generated by G -equivariant morphisms, thena generic stabiliser for ( G : V ) is commutative. Indeed, the G -equivariance implies that F i ( v ) lies in the centre of g v for any v ∈ V . On the other hand, if v is generic, then F ( v ) , . . . , F m ( v ) form a basis for g v . (Cf. Remark 3.2 in [18].)The above inequality ( ♣ ) is crucial for establishing ( F ) in applications. Furthermore,it essentially implies that C to be a free k [ V ] G -module. Theorem 4.9.
Suppose that k [ V ] G is a polynomial algebra freely generated by f , . . . , f l ; V has the codim– property and { ( d f i ) v } are linearly independent for any v ∈ V reg ; Ker ˆ φ is a free k [ V ] -module generated G -equivariant morphisms F , . . . , F m ; a generic fibre of π V is a (closed) G -orbit; N contains finitely many G -orbits; codim N X i ( N ) > i + 1 for any i > .Then gr A = gr C = Sym k [ V ] (Im ˆ φ ) , A = C , and both C and gr C are free (left or right) k [ V ] G -modules. DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 17
Proof.
As the hypotheses imply condition ( F ) for E = Im ˆ φ , only the last assertion re-quires a proof.Recall that k [ Y ] = gr C and Y = Im ( κ ) is a complete intersection of codimension m in V × g ∗ . Consider the map ν : Y → V → V //G ≃ A l , the composition of the projection andthe quotient morphism π V . Its fibre over the origin is Z := { ( v, ξ ) | v ∈ N & h F i ( v ) , ξ i = 0 , i = 1 , . . . , m } . We wish to prove that Z is a variety of pure dimension dim Y − l . Since Z is a fibre ofa dominant map Y → A l , we have dim Z > dim Y − l . On the other hand, consider theprojection p : Z → N . It follows from hypothesis that dim p − ( X i ( N ) \ X i +1 ( N )) dim N − i − g − m + i = dim Y − l − for any i > . Hence Z = p − ( N \ X ( N )) is of pure dimension dim Y − l . [Further-more, it is not hard to prove that p provides a one-to-one correspondence between theirreducible components of Z and N .] Consequently, ν is equidimensional. Since Y isCohen-Macaulay, the ν is also flat. This shows that each (gr C ) n is a flat graded finitelygenerated module over the polynomial ring k [ V ] G , hence a free module. Thus, gr C is afree k [ V ] G -module.The assertion on C can be proved exactly as in Corollary 3.4 in [13]. (cid:3) Remark 4.10.
Conditions ( F d ) can also be considered for Im ˆ ψ or Im ˆ τ whenever Ker ˆ ψ isa free k [ V ] -module or Ker ˆ τ is a free k [ g ] -module. If ( F ) is satisfied, then one obtains asimilar description for the image of κ ψ : V × g → V × V , ( v, x ) ( v, x · v ) or κ τ : g × V → g × V , ( x, v ) ( x, x · v ) , see Theorems 8.8 and 8.11 in [17]. However such descriptionsseem to have no non-commutative counterparts.5. A PPLICATIONS : ISOTROPY REPRESENTATIONS OF SYMMETRIC PAIRS AND BEYOND
In this section, G is a connected semisimple algebraic group. If g = g ⊕ g is a Z -gradingof g , then ( g , g ) is said to be a symmetric pair . Let G be the connected subgroup of G with Lie algebra g . Our goal is to describe a class of Z -gradings that lead to isotropyrepresentations ( G : g ) satisfying the assumptions of Theorems 4.2 and 4.9.Recall necessary results on the representation ( G : g ) . The standard reference for this is[10]. Let N denote the set of nilpotent elements of g . h i Any v ∈ g admits a unique decomposition v = v s + v n , where v s ∈ g is semisimpleand v n ∈ N ∩ g ; v = v s if and only if G · v is closed; v = v n if and only if the closureof G · v contains the origin. For any v ∈ g , there is the induced Z -grading of thecentraliser g v = g ,v ⊕ g ,v , and dim g − dim g = dim g ,v − dim g ,v . h i Let c ⊂ g be a maximal subspace consisting of pairwise commuting semisimpleelements. All such subspaces are G -conjugate and G · c is dense in g ; dim c iscalled the rank of the Z -grading or pair ( g , g ) , denoted rk ( g , g ) . If v ∈ c ∩ ( g ) reg ,then g ,v = c and g ,v is a generic stabiliser for the G -module g . h i The algebra k [ g ] G is polynomial and dim g //G = rk ( g , g ) . The quotient map π g : g → g //G is equidimensional. The generic fibre is the orbit of a G -regularsemisimple element. Each fibre of π g contains finitely many G -orbits and eachclosed G -orbit in g meets c . The null-cone in g equals N ∩ g . h i If v ∈ ( g ) reg and f , . . . , f dim c ∈ k [ g ] G are basis invariants, then the { ( d f i ) v } i arelinearly independent.A Z -grading (or a symmetric pair ( g , g ) ) is said to be N - regular if g contains a regularnilpotent element of g . By a result of Antonyan [1], a Z -grading is N -regular if and onlyif g contains a regular semisimple element. Then dim g − dim g = rk g − c .Now we are interested in properties of(5.1) → Ker ˆ φ → Mor ( g , g ) ˆ φ → Mor ( g , g ) for N -regular Z -gradings. Item h i above shows that a generic stabiliser for ( G : g ) iscommutative if and only if c contains a regular semisimple element. Therefore a Z -grading is N -regular if and only if a generic stabiliser for ( G : g ) is commutative. Inview of Remark 4.8, these are the only Z -gradings, where one could expect that Ker ˆ φ isgenerated by G -equivariant morphisms. The following is [18, Theorem 5.8]. Theorem 5.1.
Suppose the Z -grading g = g ⊕ g is N -regular. Then Ker ˆ φ is a free k [ g ] -module generated by G -equivariant morphisms. In this case rk (Ker ˆ φ ) = rk g − dim c . Recall the construction of such a basis for
Ker ˆ φ . By [16, Theorem 4.7], Z = G · g is anormal complete intersection in g , codim Z = rk g − dim c , and the ideal of Z in k [ g ] isgenerated by certain homogeneous basis invariants in k [ g ] G . Let f , . . . , f m be such basisinvariants ( m = rk g − dim c ). As in Section 4, any F i ∈ Ker ˆ φ determines the polynomial b F i ∈ k [ g × g ∗ ] = k [ g × g ] ≃ k [ g ] and vice versa. In this case, b F i is defined to be thebi-homogeneous component of f i of degree (1 , deg f i − . (Here ”1” is the degree withrespect to g .) Since b F i is clearly G -invariant, F i is G -equivariant. From this description,it follows that ( d f i ) v = F i ( v ) if v ∈ g .For g simple, the list of N -regular symmetric pairs consists of symmetric pairs of max-imal rank (when dim c = rk g and hence Ker ˆ φ = 0 ) and the following 4 cases:1 ( sl n , sl n ∔ sl n ∔ t ) m = rk (Ker ˆ φ ) = n − ;2 ( sl n +1 , sl n ∔ sl n +1 ∔ t ) m = rk (Ker ˆ φ ) = n ;3 ( so n , so n − ∔ so n +1 ) , n > m = rk (Ker ˆ φ ) = 1 ;4 ( E , A × A ) m = rk (Ker ˆ φ ) = 2 . DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 19
Theorem 5.2.
Suppose the Z -grading g = g ⊕ g is N -regular, and ( g , g ) = ( sl n +1 , sl n ∔ sl n +1 ∔ t ) . Then inequality ( ♣ ) holds for N g = N ∩ g and therefore ( F ) is satisfied for Im ˆ φ inEq. (5.1) .Proof. In the maximal rank case,
Ker ˆ φ = 0 and therefore condition ( F ) is vacuous. Inthe remaining four cases, we have to resort to explicit calculations. By results of [10],Proposition 4.7 applies in this situation. Hence our goal is to verify whether Eq. (4.3) and(4.4) are satisfied for the elements of N ∩ g .Let F , . . . , F m be a basis for Ker ˆ φ . The above (classification-free) construction of the F i ’s implies that dim( span { F ( v ) , . . . , F m ( v ) } ) = m for any v ∈ ( g ) reg (see the beginning ofSection 5 in [18]). Therefore Eq. (4.3) is true in all four cases. It remains to handle Eq. (4.4).In the N -regular case, each nilpotent G -orbit meets g [1]. Therefore we can argue interms of nilpotent G -orbits in g . Consider all the cases in turn.For the first two cases, the explicit form of the F i ’s is pointed out in [18, Example 5.6].Namely, regarding elements v ∈ g as matrices (of order n or n + 1 ), we set F i ( v ) = v i ,the usual matrix power with i m . Remark.
Strictly speaking, this formula for F i is only valid if the big Lie algebra is gl N .For sl N , one have to add a correcting term in order to ensure zero trace: F i ( v ) = v i − (tr ( v i ) /N ) I . However, the correcting term vanishes if v is nilpotent, which is the casebelow. No. 1
Let ( η , η , . . . ) be the partition of n corresponding to v ∈ N ∩ g . Then v i = 0 if and only if i η − , and the nonzero terms are easily seen to be linearly indepen-dent. Therefore dim( span { F ( v ) , . . . , F m ( v ) } ) = (cid:4) η − (cid:5) . Write (ˆ η , ˆ η , . . . , ˆ η s ) for the dualpartition. Then s = η . The term g v occurring in Eq. (4.4) now becomes g ,v . The generalequality dim G · v = 2 dim G · v means in this case that dim g v = 2 dim g ,v + 1 . Because dim g v = dim( sl n ) v = P si =1 ˆ η i − , the required inequality looks as follows: s X i =1 ˆ η i + (cid:22) η − (cid:23) − n > , if v ( N ∩ g ) reg , i.e., if ˆ η > . Since P si =1 ˆ η i = 2 n , it is readily transformed into s X i =1 (ˆ η i − + (cid:0)(cid:22) s − (cid:23) − s (cid:1) > , which is true if ˆ η > . Indeed, if v is subregular, then ˆ η = 2 , ˆ η j = 1 for j > , and s =2 n − . Hence the LHS is zero. For all other non-regular SL n -orbits the LHS is positive. No. 2
In this case, the numerical data are slightly different: P si =1 ˆ η i = 2 n + 1 , m = n ,and dim g v = 2 dim g ,v . The required inequality is s X i =1 (ˆ η i − + (cid:0)(cid:22) s − (cid:23) − s + 12 (cid:1) > , if v ( N ∩ g ) reg . It fails to hold only if v is subregular, i.e., ˆ η = 2 , ˆ η j = 1 for j > ,and s = 2 n . This means that codim N X ( N ) = 1 and ( ♣ ) in Proposition 4.7 is not satisfied.Using this, one can prove that condition ( F ) is not satisfied for Im ˆ φ here. No. 3
Since F is the only basis morphism, the validity of Eq. (4.4) reduces to theassertion that F ( v ) = 0 whenever G · v is an orbit of codimension 1 in N ∩ g . The map F arises from the pfaffian, Pf , a skew-symmetric matrix of order n , and, as explainedabove, F ( u ) = d ( Pf ) u . If u ∈ so n is nilpotent, then d ( Pf ) u = 0 if and only if u has at leastthree Jordan blocks [22, Lemma 4.4.1]. However G · v ⊂ N is the subregular nilpotent orbitand the corresponding partition is (2 n − , . No. 4
Here m = 2 and there are two basis morphisms in Ker ˆ φ . These two are associ-ated with basis invariants of G = E of degree 5 and 9. Therefore their degrees are equalto 4 and 8. Call them F (4) and F (8) , respectively. Here the validity of Eq. (4.4) reduces tothe assertions that if codim N ∩ g G · v = 1 , then F (4) ( v ) = 0 and F (8) ( v ) = 0 ,if codim N ∩ g G · v = 2 , then F (4) ( v ) = 0 or F (8) ( v ) = 0 .In the first case, G · v is the subregular nilpotent orbit, usually denoted E ( a ) . In thesecond case, G · v is the unique orbit of codimension 4 in N , denoted D . The computationswe need have been performed by Richardson, see [22, Appendix]. He computed the”exponents” for all but one nilpotent orbits in the exceptional Lie algebras. In particular,for the G -orbit of type E ( a ) (resp. D ), the exponents include and (resp. include ).This is exactly what we need. (cid:3) It follows from the previous exposition that if g = g ⊕ g is an N -regular grading,then, modulo one exception, all the hypotheses of Theorem 4.9 are satisfied for the G -module g . Indeed, by above-mentioned results of Kostant and Rallis [10], the hypotheses1, 2, 4, and 5 hold for all Z -gradings. The third (resp. sixth) assumption is verified inTheorem 5.1 (resp. Theorem 5.2).Thus, applying results of Section 4 to our situation, we obtain Theorem 5.3.
Suppose that g is simple, g = g ⊕ g is an N -regular Z -grading, and ( g , g ) =( sl n +1 , sl n ∔ sl n +1 ∔ t ) . Set m = rk g − dim c . Then (i) Sym k [ g ] (Im ˆ φ ) is a factorial domain of Krull dimension dim g + dim g − m . DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 21 (ii)
Spec (
Sym k [ g ] (Im ˆ φ )) ≃ Im ( κ ) , where κ : g × g → g × g is defined by ( v, v ′ ) ( v, [ v, v ′ ]) . (iii) Im ( κ ) is an irreducible factorial complete intersection and its ideal is generated by b F i , i = 1 , . . . , m . (iv) A = C , i.e., C = Cent D ( g ) ( k [ g ] G ) is the algebra generated by k [ g ] and ς ( g ) . (v) Both C and gr C = Sym k [ g ] (Im ˆ φ ) are free (left or right) k [ g ] G -modules. Remark 5.4.
For the symmetric pairs of maximal rank, the equality A = C is proved in anunpublished manuscript of Levasseur [11, Theorem 4.4]. In our exposition, this assertionalso appears as a special case of Proposition 4.6. Remark 5.5. As ( F ) is not satisfied for Im ˆ φ in case of ( sl n +1 , sl n ∔ sl n +1 ∔ t ) , one mightexpect that some assertions of Theorem 5.3 are wrong for that symmetric pair. However,condition ( F ) still holds, and this is sufficient for proving that Y = Im ( κ ) and it is acomplete intersection whose ideal is generated by b F , . . . , b F m . It seems to be hard to checkdirectly what is happening with assertion (iv). We are only able to prove that Im ( κ ) isnot factorial for ( sl , gl ) . For, here Im ( κ ) is a hypersurface in the 8-dimensional space g × g , and the defining relation can be written up.As Theorems 4.2 and 4.9 have rather general formulations (the group G even is notsupposed to be reductive!), it is instructive to have natural illustrations to it, which lieoutside the realm of the isotropy representations of symmetric pairs. In view of Proposi-tion 4.6, many representations with trivial generic stabiliser will work. So, we concentrateon examples with non-trivial stabiliser, i.e., with Ker ˆ φ = 0 . Example . Take G = SL × SL and V = R ( ̟ ) ⊗ R ( ̟ ) , where ̟ i is the i -th funda-mental weight. This representation is associated with an automorphism of order 3 of E ,and Vinberg’s theory of θ -groups [25], which is an extension of the Kostant-Rallis the-ory, provides a lot of information about it. In particular, k [ V ] G is polynomial (with threegenerators) and N V contains finitely many G -orbits. Here m = 1 and Proposition 4.7 isapplicable. The situation here resembles very much that for N -regular Z -gradings. Thebasis covariant F : V → g ∗ in Ker ˆ φ is associated with the basis E -invariant f of degree . Therefore deg F = 9 and F ( v ) = ( d f ) v for v ∈ V . Since m = 1 , it suffices to ver-ify that dim N V − dim X ( N V ) > . In other words, if O is a G -orbit of codimension 1 in N V , then we need F | O = 0 . An explicit classification of G -orbits in N V [4, § E -orbits denoted by E ( a ) and E ( a ) . Finally, using again Richardson’s calculations [22, Appendix], we obtain the re-quired non-vanishing assertion.We omit most details for this example, since we are going to consider applications ofour theory to θ -groups in a forthcoming article. Example . We describe non-reductive Lie algebras whose coadjoint representation sat-isfies the hypotheses of Theorem 4.2. This yields an incarnation of the “ A = C phe-nomenon” in the non-reductive case. We detect such examples among -wonderful alge-bras. By a previous discussion (Remarks 2.1 and 2.5), hypotheses 1–3 are always satisfiedfor them. Thus, it remains only to have condition ( F ) for Im ˆ φ . Our examples exploitthe semi-direct product construction (see Example 2.6). We start with s = sl and set q = sl ⋉ sl . Then q is a quadratic -wonderful algebra and m = ind q = 2 . There are twobasis Q -equivariant morphisms F i : q ∗ → q in Ker ˆ φ . Identifying q ∗ and q , we may regard F i as elements of Mor Q ( q , q ) . Representing elements of q as pairs ( x, y ) , where x, y ∈ sl ,we obtain the following explicit formulae: F ( x, y ) = ( x, y ) and F ( x, y ) = (0 , x ) . Then X = { (0 , y ) | y ∈ sl } and X = { (0 , } . Hence condition ( F ) is satisfied. We have alsochecked ( F ) for sl ⋉ sl , sp ⋉ sp and G ⋉ G .Hopefully, this could be true if s is any simple Lie algebra, but we unable to prove it.6. S OME SPECULATIONS
There are two different generalisations of Dixmier’s result on adjoint vector fields in thecontext of the adjoint representation of semisimple Lie algebras.One possibility is provided by the Levasseur-Stafford description of
Cent D ( g ) ( k [ g ] G ) [13,Theorem 1.1], see discussion in Section 4. On the other hand, the formulation given inRemark 2.4 suggests the following question, which was raised by Dixmier himself [6,1.2]. Question.
Suppose D ∈ D ( g ) and D ( f ) = 0 for all f ∈ k [ g ] G . Is it true that D ∈ D ( g ) · ς ( g ) ?The affirmative answer is obtained by Levasseur and Stafford [12]. They proved that K := { D ∈ D ( g ) | D ( f ) = 0 ∀ f ∈ k [ g ] G } is the left ideal of D ( g ) generated by ς ( g ) . Then a similar result was obtained for theisotropy representation of some symmetric pairs [14]. To state that result, we need somepreparations. Let Σ be the restricted root system of ( g , g ) . The following condition on Σ was considered by Sekiguchi [23]: ( ♥ ) dim g α + dim g α for any α ∈ Σ .Sekiguchi also obtained the list of corresponding symmetric pairs. The following is [14,Theorem C]: Suppose that Σ( g , g ) satisfies ( ♥ ) . Then K ( g ) := { D ∈ D ( g ) | D ( f ) = 0 ∀ f ∈ k [ g ] G } isthe left ideal generated by ς ( g ) . DJOINT VECTOR FIELDS AND DIFFERENTIAL OPERATORS 23
Furthermore, it is proved in [14, Section 6] that K ( g ) = D ( g ) ς ( g ) for ( so n +1 , so n ) ,while the inclusion D ( g ) ς ( g ) ⊂ K ( g ) is strict for ( sl , gl ) . It is curious that accord-ing to Sekiguchi’s classification, ( ♥ ) is satisfied precisely if ( g , g ) is N -regular except for ( sl n +1 , sl n ∔ sl n +1 ∔ t ) .This raises the following questions for representations of connected (reductive?)groups. There are two properties of representations ( G : V ) :1) The algebra Cent D ( V ) ( k [ V ] G ) is generated by k [ V ] and ς V ( g ) ;2) The ideal { D ∈ D ( V ) | D ( f ) = 0 ∀ f ∈ k [ V ] G } is generated by ς V ( g ) .Is it true that one of them implies another (under appropriate constraints)? At least, isthere a relationship in case of isotropy representations of symmetric pairs? Remark.
The only ”bad” N -regular symmetric pair ( sl n +1 , sl n ∔ sl n +1 ∔ t ) is also distin-guished by a bad behaviour of the commuting variety. Recall that the commuting varietyis E ( g ) = { ( x, y ) ∈ g × g | [ x, y ] = 0 } , and it is irreducible for all N -regular pairs but thatone. In the maximal rank case, the irreducibility is proved in [15]. The four remainingcases (see the list in Section 5) are considered in [19]. This is of certain interest becausethere is a relationship between the irreducibility of E ( g ) and properties of the ideal K ( g ) ,see [14, Prop. 4.6]. R EFERENCES [1]
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