Radial components, prehomogeneous vector spaces, and rational Cherednik algebras
aa r X i v : . [ m a t h . R T ] J a n RADIAL COMPONENTS, PREHOMOGENEOUS VECTORSPACES, AND RATIONAL CHEREDNIK ALGEBRAS
THIERRY LEVASSEUR
Abstract.
Let ( ˜ G : V ) be a finite dimensional representation of a connectedreductive complex Lie group ( ˜ G : V ) . Denote by G the derived subgroup of ˜ G and assume that the categorical quotient V //G is one dimensional, i.e. C [ V ] G = C [ f ] for a non constant polynomial f . In this situation there exists a homo-morphism rad : D ( V ) G → A ( C ) , the radial component map, where A ( C ) is the first Weyl algebra. We show that the image of rad is isomorphic tothe spherical subalgebra of a rational Cherednik algebra whose multiplicityfunction is defined by the roots of the Bernstein-Sato polynomial of f . In thecase where ( ˜ G : V ) is also multiplicity free we describe the kernel of rad andprove a Howe duality result between representations of G occuring in C [ V ] andlowest weight modules over the Lie algebra generated by f and the “dual” dif-ferential operator ∆ ∈ S ( V ) ; this extends results of H. Rubenthaler obtainedwhen ( ˜ G : V ) is a parabolic prehomogeneous vector space. If ( ˜ G : V ) satisfiesa Capelli type condition, some applications are given to holonomic and equi-variant D -modules on V . These applications are related to results proved byM. Muro or P. Nang in special cases of the representation ( ˜ G : V ) . Contents
1. Introduction 22. Rational Cherednik Algebras of Rank One 42.1. The spherical subalgebra and its restriction 42.2. The one dimensional case 52.3. Algebras similar to U ( sl (2))
73. Representations with a one dimensional quotient 93.1. Prehomogeneous vector spaces 103.2. PHV of rank one 114. Multiplicity free representations 154.1. Generalities 154.2. MF representations with a one dimensional quotient 194.3. A Howe duality 215. D -modules on some PHV 235.1. Representations of Capelli type 235.2. Application to D -modules 245.3. Solutions of invariant differential equations 275.4. Regular holonomic modules 28Appendix A. Irreducible MF representations 32References 33 Mathematics Subject Classification.
Key words and phrases. prehomogeneous vector space, ring of differential operators, Dunkloperator, Cherednik algebra, Capelli operator, radial component, holonomic module. Introduction
The base field is the field C of complex numbers. Let ( G : V ) be a finite dimen-sional representation of a connected reductive Lie group G . The action of G extendsto various algebras: C [ V ] = S ( V ∗ ) the polynomial functions on V , D ( V ) the differ-ential operators on V with coefficients in C [ V ] and S ( V ) identified with differentialoperators on V with constant coefficients. Recall that D ( V ) ∼ = C [ V ] ⊗ S ( V ) as a ( C [ V ] , G ) -module and that g ∈ G acts on D ∈ D ( V ) by ( g.D )( ϕ ) = g.D ( g − .ϕ ) for all ϕ ∈ C [ V ] . We thus obtain algebras of invariants C [ V ] G , S ( V ) G and D ( V ) G .Then C [ V ] G is (by definition) the algebra of regular functions on the categoricalquotient V //G and one can define the algebra D ( V //G ) of differential operators onthis quotient (see [12] or [32]).If D ∈ D ( V ) G and f ∈ C [ V ] G one obviously has D ( f ) ∈ C [ V ] G ; this gives analgebra homomorphism: D ( V ) G −→ D ( V //G ) , D
7→ { f D ( f ) , f ∈ C [ V ] G } . In general
V //G is singular and D ( V //G ) is difficult to describe. We will be in-terested here in the case where V //G is smooth, i.e. isomorphic to C ℓ for some ℓ ∈ N , in which case D ( V //G ) is isomorphic to the Weyl algebra A ℓ ( C ) . Moreprecisely, we want to work with polar representations as defined by J. Dadoc andV. Kac in [5]. In this case there exists a Cartan subspace h ⊂ V , a finite sub-group W ⊂ GL( h ) generated by complex reflections ( W ≃ N G ( h ) /Z G ( h ) ), suchthat the restriction map ψ : C [ V ] G → C [ h ] W , ψ ( f ) = f | h , is an isomorphism. Thus ψ yields the isomorphism V //G ∼−→ h /W ≡ C ℓ and, consequently, an isomorphism D ( V //G ) ∼−→ D ( h /W ) ≡ A ℓ ( C ) . Recall that among the polar representations onefinds two important classes:– the representations with a one dimensional quotient, i.e. dim V //G = 1 ;– the class of “theta groups”.In the latter case there exists a semisimple Lie algebra s and a Z m -grading s = ⊕ m − i =0 s i such that ( G : V ) identifies with the representation of the adjoint groupof s acting on s . This generalizes the case of symmetric pairs ( G : V ) = ( K : p ) where (with obvious notation) s = k ⊕ p is the decomposition associated to acomplexified Cartan involution on s . Here h ⊂ p is a usual Cartan subspace and W is a Weyl group (cf. [15]).Return to a general polar representation ( G : V ) . Combining the morphism D ( V ) G → D ( V //G ) with the isomorphism D ( V //G ) ∼−→ D ( h /W ) we get the radialcomponent map : rad : D ( V ) G −→ D ( h /W ) , rad( D )( f ) = ψ ( D ( ψ − ( f ))) , f ∈ C [ h ] W . The morphism rad has proved to be useful in the representation theory of semisimpleLie algebras, or symmetric pairs ( s : k ) as above, see, e.g. [15, 54, 29, 30]. Twoobvious questions arise: describe the algebra R = Im(rad) ⊂ D ( h /W ) and the ideal J = Ker(rad) ⊂ D ( V ) G . Some answers have been given in particular cases, see forexample [30, 50], and it is expected that the algebra D ( V ) G /J has a representationtheory similar to that of factors of enveloping algebras of semisimple Lie algebras(cf. [51]).It is known that in the case ( G : V ) = ( K : p ) of a symmetric pair, the sub-algebra rad (cid:0) S ( p ) K (cid:1) of R can be described via the introduction Dunkl operators[7, 14, 6, 53]. It is therefore natural to use rational Cherednik algebras [8, 9, 11]to describe R . Recall that to each complex reflection group ( W : h ) is associatedan algebra H ( k ) where k is a “multiplicity function” on the set of reflecting hy-perplanes in h . Denoting by h reg the complement of these hyperplanes, H ( k ) isa subalgebra of the crossed product D ( h reg ) ⋊ C W generated by C [ h ] , C W and a ADIAL COMPONENTS 3 subalgebra C [ T . . . , T ℓ ] ∼ = S ( h ) where each T i is a (generalized) Dunkl operator,see §2.1 for details. If e = | W | P w ∈ W w ∈ C W is the trivial idempotent, e H ( k ) e iscalled the spherical subalgebra. Then one can show that there exists an injectivehomomorphism res : e H ( k ) e −→ D ( h /W ) and we obtain in this way a family U ( k ) = res( e H ( k ) e ) of subalgebras of D ( h /W ) .One would like to obtain information on R by answering the following question: Does there exist a multiplicity function k such that R = U ( k ) ? For instance, suppose that ( G : V ) = ( K : p ) as above. The reflecting hyper-planes are then parametrised by elements of the reduced root system R defined by ( s , h ) and one defines a multiplicity function by: k ( α ) = 12 (cid:0) dim s α + dim s α (cid:1) , α ∈ R , where s β is the root space associated to the root β . For this choice of k one canprove [31]: Theorem (L–Stafford) . One has R = Im(rad) = U ( k ) ∼ = e H ( k ) e . Our aim in this work is to analyse a simpler case, G semisimple and dim V //G =dim h = 1 (hence W ≃ Z /n Z ) and to give some applications of the radial componentmap in this situation. The function k is then given by n − complex parameters k , . . . , k n − , and R , U ( k ) are subalgebras of the fist Weyl algebra C [ z, ∂ z ] . Thepaper is organized as follows.In §2 we recall general facts about Cherednik algebras and their spherical subal-gebras in the one dimensional case. We show (Proposition 2.8) that U ( k ) = e U / (Ω) where e U is an algebra similar to U ( sl (2)) (as defined in [52]) and Ω is a generatorof the centre of e U . This says in particular that the representation of U ( k ) is wellunderstood (and already known).In the third section we assume that V is a representation of the reductive group ˜ G , G is the derived group of ˜ G and C [ V ] G = C [ f ] for a non constant f . Then itis known that: ˜ G acts on V with an open orbit, i.e. ( ˜ G : V ) is a prehomogeneousvector space (PHV), S ( V ) G = C [∆] , ∆( f s +1 ) = b ( s ) f s where b ( s ) = c ( s + 1)( s + α ) · · · ( s + α n − ) is the (Bernstein-)Sato polynomial of f . Choosing k i = α i − in , ≤ i ≤ n − , we prove that R = U ( k ) (Theorem 3.9).In section 4 we assume furthermore that the representation ( ˜ G : V ) is multiplic-ity free (MF). By [18] this is equivalent to the fact that D ( V ) ˜ G = C [ E , . . . , E r ] isa commutative polynomial ring. If Θ is the Euler vector field on V one can findpolynomials b E i ( s ) such that, if Ω i = E i − b E i (Θ) , J = P ri =0 D ( V ) G Ω i (Theo-rem 4.11). We then give a duality (of Howe type) between representations of G andlowest weight modules over the Lie algebra generated by f and ∆ (which is infinitedimensional when deg f ≥ ). This duality recovers, and extends, results obtainedby H. Rubenthaler [49] when ( ˜ G : V ) is of commutative parabolic type.In the last section we specialize further to the case where ( ˜ G : V ) is of “Capellitype”, i.e. ( ˜ G : V ) is an irreducible MF representation such that D ( V ) ˜ G is equalto the image of the centre of U (˜ g ) under the differential τ : ˜ g → D ( V ) of the ˜ G -action. These representations have been studied in [18], they fall into eightcases (see Appendix A). It is not difficult to see that J = [ D ( V ) τ ( g )] G when ( ˜ G : V ) is of Capelli type (Proposition 5.3) . We first apply this result to study D V -modules of the form M ( g, k ) = D ( V ) / (cid:0) D ( V ) τ ( g ) + D ( V ) q (Θ) Q k (cid:1) where q ( s ) is a polynomial and Q k = f k or ∆ k . We show in Theorem 5.9 that M ( g, k ) isholonomic if and only if q ( s ) = 0 . This has the well known consequence that the T. LEVASSEUR space of hyperfunction solutions of M ( g, k ) is finite dimensional. These propertiesgeneralize results obtained by M. Muro [36, 37]. For the second application, recallfirst the classical fact [21] that if ( ˜ G : V ) is MF, there is a finite number of ˜ G -orbits O i , ≤ i ≤ t , in V . Let ˜ C = S ti =1 T ∗ O i V be the union of the conormalbundles to the orbits. P. Nang has shown that, when ( ˜ G : V ) = (SO( n ) × C ∗ : C n ) , (GL( n ) × SL( n ) : M n ( C )) or (GL(2 n ) : V C n ) , the category mod rh˜ C ( D V ) of regularholonomic D V -modules whose characteristic variety is contained in ˜ C is equivalentto the category mod θ ( R ) of finitely generated R -modules on which θ = z∂ z actslocally finitely. These representations are of Capelli type. We conjecture (seeConjecture 5.17) that when ( ˜ G : V ) is of Capelli type the category mod G × C ∗ ( D V ) of ( G × C ∗ ) -equivariant D V -modules is equivalent to mod θ ( R ) . If G is simplyconnected, mod G × C ∗ ( D V ) = mod rh˜ C ( D V ) and the conjecture covers Nang’s results;since mod θ ( R ) can be easily described as a quiver category (i.e. finite diagrams oflinear maps) its validity would give a simple classification of ( G × C ∗ ) -equivariant D V -modules. One can observe (Proposition 5.16) that, as in [39, 41, 43], the proofof the conjecture reduces to show that any M ∈ mod G × C ∗ ( D V ) is generated by its G -fixed points.2. Rational Cherednik Algebras of Rank One
The spherical subalgebra and its restriction.
In this section we summa-rize some of the results we will need about rational Cherednik algebras. We beginwith some general facts, see for example [9, 8, 11, 14].Let h be a complex vector space of dimension ℓ and W ⊂ GL( h ) be an arbitrarycomplex reflection group. Denote by A = { H s } s ∈S the collection of reflectinghyperplanes associated to W (where s ∈ S ⊂ W is a complex reflection). Let α s ∈ h ∗ such that H s = α − s (0) is the reflecting hyperplane associated to s ∈ S .Fix H = H s ∈ A ; recall that the isotropy group W H = { w ∈ W : w | H = id H } is cyclic of order n H (this order only depends on the conjugacy class of s ). Let e H,i ∈ C W H , ≤ i ≤ n H − , be the primitive idempotents of C W H . Fix a family k H s ,i ∈ C , H s ∈ A , ≤ i ≤ n H s − , k H s , = 0 , of complex numbers such that k H s ,i = k H t ,i if s, t ∈ S are conjugate. Such afamily k = ( k H,i ) H,i is called a multiplicity function. Let h reg be the complementof S s ∈S H s and set π = Q s ∈S α s . The group W acts naturally on C [ h ] = S ( h ∗ ) , C [ h reg ] = C [ h ][ π − ] , hence on End C C [ h ] and End C C [ h reg ] . These actions restrictto D ( h ) and D ( h reg ) = D ( h )[ π − ] . Denote by D ( h reg ) ⋊ C W the crossed productof the algebra D ( h reg ) by the group W . Recall that in that algebra we have: wf w − = w.f , w∂ ( y ) w − = ∂ ( w.y ) if f ∈ C [ h ] and ∂ ( y ) is the vector field definedby y ∈ V .Then [8] one can introduce a subalgebra H = H ( W, k ) ⊂ D ( h reg ) ⋊ C W generated by three parts: C [ h ] , W , C [ T ( y ) : y ∈ h ] ∼ = S ( h ) , where T ( y ) is a Dunkloperator defined as follows. Set a H s ( k ) = n H s P n Hs − i =1 k H s ,i e H s ,i ∈ C W H s and T ( y ) = ∂ ( y ) + X H s ∈A h α s , y i α s a H s ( k ) ∈ D ( h reg ) ⋊ C W .
Denote by res : D ( h reg ) ⋊ C W → End C C [ h reg ] the representation given by thenatural action of W and D ( h reg ) on C [ h reg ] . As observed in [8, §2.5] (see also [9,Proposition 4.5]) res( H ) ⊂ End C C [ h ] and this gives a natural structure of faithful ADIAL COMPONENTS 5 H -module on C [ h ] , i.e. we have an injective homomorphism: res : H −→
End C C [ h ] . The group W acts on H , D ( h reg ) ⋊ C W and End C C [ h ] by conjugation, i.e w.u = wuw − . Denote by H W ⊂ ( D ( h reg ) ⋊ C W ) W and (End C C [ h ]) W the algebrasof invariants under this action. Notice that if u ∈ H , w ∈ W and f ∈ C [ h ] ,we have: res( w.u )( f ) = res( wuw − )( f ) = res( wu )( w − .f ) = w. res( u )( w − .f ) =( w. res( u ))( f ) . Thus the homomorphism res is W -equivariant and, in particu-lar, res : H W → (End C C [ h ]) W . Therefore w. res( u )( f ) = w. res( u )( w − .f ) =( w. res( u ))( f ) = res( u )( f ) , for all u ∈ H W , w ∈ W , f ∈ C [ h ] W . We have obtainedthe following representation of H W on C [ h ] W : res : H W −→ End C C [ h ] W . (This morphism is not injective when W = { } .) Let e = 1 | W | X w ∈ W w be the trivial idempotent and define the spherical subalgebra : e H e = e H W ⊂ H W . (2.1)Observe that e H e is an algebra whose unit is equal to e . From the previousdiscussion we obtain e H e ⊂ e ( D ( h reg ) ⋊ C W ) e . It is not difficult to show that e ( D ( h reg ) ⋊ C W ) e = e D ( h reg ) W ∼ = D ( h reg ) W . It follows that u ∈ e H W can be writ-ten u = e d for some d ∈ D ( h reg ) W , hence res( u )( f ) = d ( f ) for all f ∈ C [ h ] W . Thisimplies that res( u ) ∈ End C C [ h ] W acts as the differential operator d on C [ h ] W . Con-sequently, res( u ) ∈ D ( h /W ) = D ( C [ h ] W ) ⊂ D ( h reg /W ) ∼ = D ( h reg ) W . Furthermoreit is easy to see that d = 0 on C [ h ] W implies d = 0 , hence u = 0 . In conclusion:one has the injective restriction morphism (see [14] in the case of a Weyl group): res : e H W −→ D ( h /W ) , ∀ f ∈ C [ h ] W , D ∈ H W , res( e D )( f ) = res( D )( f ) . (2.2)We set: U = U ( W, k ) = res( e H W ) ⊂ D ( h /W ) . (2.3)2.2. The one dimensional case.
We now go the most simplest case of the pre-vious construction: the case when ℓ = dim h = 1 . Notation.
Let h = C v be a one dimensional vector space and W ⊂ GL( h ) be afinite subgroup of order n . We adopt the following notation. • C [ h ] = S ( h ∗ ) = C [ x ] , h x, v i = 1 , D ( h ) = C [ x, ∂ x ] ; • W = h w i ≃ Z /n Z , w.x = ζx where ζ is a primitive n -th root of unity; • C [ h ] W = C [ z ] , z = x n , D ( h /W ) = C [ z, ∂ z ] , θ = z∂ z ; • e = e , e , . . . , e n − ∈ C W are the primitive idempotents (hence e i = n P n − j =0 ζ ij w j ); • k = 0 , k , . . . , k n − ∈ C ; • T = T ( v ) = ∂ x + nx n − X i =1 k i e i ∈ C [ x ± , ∂ x ] ⋊ C W ; • if p ( s ) ∈ C [ s ] is a polynomial, set: τ p ( s ) = p ( s + 1) − p ( s ) , τ j +1 p ( s ) = τ ( τ j p )( s ) , p ∗ ( s ) = p ( s − .The following well known lemma will prove useful (see [26] for a more generalstatement). T. LEVASSEUR
Lemma 2.1.
Let Q ∈ C [ z, ∂ z ] satisfying: ∃ p ∈ Z , ∀ m ∈ N , Q ( z m ) ∈ C z m + p . Then there exists a polynomial ϕ ( s ) ∈ C [ s ] of degree d such that: Q has order d and can be written Q = z p ϕ ( θ ) = d X j =0 q j ( z ) ∂ jz where q j ( z ) = 1 j ! ( τ j ϕ )(0) z j + p and ( τ j ϕ )(0) = 0 if p + j < . Remark 2.2.
One can define the algebra C [ z α : α ∈ Q ] by adjoining roots ofpolynomials of the form t p − z , p ∈ N prime. The derivation ∂ z is naturally definedon this algebra by ∂ z ( z α ) = αz α − . Let Q be as in Lemma 2.1; then Q extends to C [ z α : α ∈ Q ] by Q ( z α ) = P j q j ( z ) ∂ z ( z α ) = ϕ ( α ) z α + p .The next lemma is straightforward by direct computation. Lemma 2.3.
The following formulas hold: (a) [ e i , x ] = x ( e i +1 − e i ) (where e n = e = e ) ; (b) [ T, x ] = 1 + n P n − i =1 k i ( e i +1 − e i ) = 1 + n P n − i =0 ( k i − k i +1 ) e i ; (c) wT w − = ζ − T ; (d) let p ∈ N and define q ∈ { , . . . , n − } by p + q ≡ n ) , then T ( x p ) =( nk q + p ) x p − ; (e) let ≤ j ≤ n and s ∈ N , then T j ( x sn ) = j Y i =1 ( nk i − + sn − i + 1) x sn − j , in particular ( T /n ) n ( z s ) = Q n − i =0 ( s + k i − i/n ) z s − . We now introduce the rational Cherednik algebra, and its spherical subalgebra,in the rank one case.
Definition 2.4.
The rational Cherednik algebra associated to W with parameters k i , ≤ i ≤ n − , is the subalgebra of C [ x ± , ∂ x ] ⋊ C W defined by: H = H ( W, k , . . . , k n − ) = C h x, T, w i Its spherical subalgebra is e H e .Observe that when n = 1 (i.e. W trivial) the algebra H = e H e is nothing but D ( h ) = C [ x, ∂ x ] and all the results we are going to obtain are in this case obvious. We therefore will only be interested in the case n ≥ .It is easily seen that: • e H e = e H W = C h e , e x n , e ( T /n ) n , e xT /n i ; • the image U = res( e H e ) of the injective homomorphism, defined in (2.2), res : e H W −→ D ( h /W ) = C [ z, ∂ z ] , (2.4)is generated by z , res (cid:0) e ( T /n ) n (cid:1) and res( e xT /n ) . • there exists a finite dimensional filtration on e H e such that the associatedgraded algebra gr( e H e ) is isomorphic to S ( h ∗ × h ) W ≡ C [ X, Y, S ] / ( XY − S n ) , cf. [9, p. 262] (one has X ≡ gr( e x n ) , Y ≡ gr( e T n ) , S ≡ gr( e xT ) ). ADIAL COMPONENTS 7
Fix a constant c ∈ C ∗ and set: λ i = k i − in , b ∗ ( s ) = c n − Y i =0 ( s + λ i ) , b ( s ) = b ∗ ( s + 1) = c n − Y i =0 ( s + λ i + 1) (2.5) v ( s ) = − b ( − s ) , ψ ( s ) = 12 ( τ v )( s ) = b ( − s ) − b ( − s − . (2.6) Proposition 2.5.
Set δ = c res (cid:0) e ( T /n ) n (cid:1) . Then U = res( e H e ) = C [ z, θ, δ ] andone has: (1) δ = z − b ∗ ( θ ) = P nj =1 1 j ! ( τ j b ∗ )(0) z j − ∂ jz ; (2) res( e xT /n ) = θ ; (3) [ δ, z ] = ψ ( − θ ) = b ( θ ) − b ( θ −
1) = ( τ b ∗ )( θ ) ; (4) [ θ, z ] = z , [ θ, δ ] = − δ ; (5) zδ + v ( − θ + 1) = 2 (cid:0) zδ − b ∗ ( θ ) (cid:1) = 0 .Proof. The equality U = C [ z, θ, δ ] is clear.(1) From the definition of the map res , cf. (2.2), and Lemma 2.3(e) we deducethat δ ( z s ) = c ( T /n ) n ( z s ) = b ∗ ( s ) z s − . The claim therefore follows from Lemma 2.1applied to Q = δ , ϕ = b ∗ and p = − .(2) By Lemma 2.3(e) again we get that res( xT )( z s ) = xT ( x sn ) = snx sn = nsz s ,i.e. res( e xT /n ) = θ .(3) Using (1) we obtain that [ δ, z ]( z s ) = ( τ b ∗ )( s ) z s = ( τ b ∗ )( θ )( z s ) , hence [ δ, z ] =( τ b ∗ )( θ ) .The formulas in (4) and (5) are obvious. (cid:3) Algebras similar to U ( sl (2)) . We recall here the definition, and some prop-erties, of the algebras similar to U ( sl (2)) introduced in [20] and [52].Let ψ ( s ) ∈ C [ s ] be an arbitrary polynomial of degree ≥ and write ψ = τ v forsome v ∈ C [ s ] of degree n ≥ . Define a C -algebra e U by generators and relationsas follows (cf. [52]): e U = e U ( ψ ) = C h A, B, H i , [ A, B ] − ψ ( H ) = 0 , [ H, A ] − A = 0 , [ H, B ] + B = 0 . Note that when deg ψ = 1 , i.e. n = 2 , one has e U = U ( sl (2)) . The algebra e U hasthe following properties, see [52, 19, 38]. • The centre of e U is Z ( e U ) = C [Ω] , Ω = 2 BA + v ( H + 1) = 2 AB + v ( H ) . • For λ ∈ C one defines the “Verma module” M ( λ ) = e U ⊗ C [ H,A ] C λ , where C λ is the one dimensional module associated to λ over the solvable Lie algebra C A + C H . • Each M ( λ ) has a unique simple quotient L ( λ ) and any finite dimensional e U -module is of the form L ( λ ) for some λ . • The primitive ideals of e U are the annihilators ann L ( λ ) ; the minimal prim-itive ideals are the ann M ( λ ) = (Ω − v ( λ + 1)) , they are completely prime.If I is an ideal strictly containing ann M ( λ ) , then dim C e U /I is finite. • One can define in an obvious way a category O for e U which decomposes as: O = G α O α , O α = { M ∈ O : (Ω − α ) k M = 0 for some k } . Moreover O α ≡ mod A for a finite dimensional C -algebra A .The representation theory of the algebras e U / (Ω − v ( λ + 1)) is therefore quite wellunderstood.We will be interested in the algebra U = e U / (Ω) = C [ a, b, h ] where a, b, h are theclasses of A, B, H . We have in U : [ a, b ] = ψ ( h ) = 12 ( τ v )( h ) , [ h, a ] = a, [ h, b ] = − b, ab = − v ( h ) . T. LEVASSEUR
For simplicity we will assume that v (1) = 0 . Recall then that M (0) ≡ C [ b ] is afaithful U -module where h.b k = − kb k , b.b k = b k +1 , a.b k = (cid:0)P k − i =1 ψ ( − i ) (cid:1) b k − forall k ≥ . We want to study the Lie subalgebra L of ( U, [ , ]) generated by theelements a, b . Recall [52] that when n ≤ this algebra is finite dimensional. Thealgebra L acts on C [ b ] ≡ M (0) and for each i ∈ Z we set: L i = { u ∈ L : u.b k ∈ C b k + i for all k ∈ N } . Clearly: a ∈ L − , h ∈ L , b ∈ L . Lemma 2.6. (1)
The element h is transcendental over C . (2) For g ( h ) ∈ C [ h ] we have [ b, ag ( h )] = − τ ( vg ∗ )( h ) . (3) There exists a sequence ( g m ( h )) m ∈ N ⊂ L ∩ C [ h ] such that deg g m = ( m + 1) n − (2 m + 1) . In particular, deg g m < deg g m +1 when n ≥ .Proof. (1) This follows, for example, from g ( h ) .b k = g ( − k ) b k in M (0) for all g ( h ) ∈ C [ h ] ⊂ U .(2) Recall [52, Appendix] that [ b, g ( h )] = b ( g ( h ) − g ( h − and [ g ( h ) , a ] = aτ ( g )( h ) .Thus: [ b, ag ( h )] = [ b, a ] g ( h ) + a [ b, g ( h )] = −
12 ( τ v )( h ) g ( h ) + ab ( g ( h ) − g ( h − −
12 ( v ( h + 1) − v ( h )) g ( h ) − v ( h )( g ( h ) − g ( h − −
12 ( v ( h + 1) g ( h ) − v ( h ) g ( h − −
12 ( τ ( vg ∗ ))( h ) , as desired.(3) We start with g ( h ) = ψ ( h ) = [ a, b ] ∈ L . Then, deg g = n − and, by (2), [ b, a ( τ g )( h )] = − τ ( v ( τ g ) ∗ )( h ) ∈ L . We thus set g ( h ) = − τ ( v ( τ g ) ∗ )( h ) ; notethat deg τ ( v ( τ g ) ∗ ) = deg v +deg τ g − n − . Suppose that g m ( h ) ∈ L has beenobtained; then [ g m ( h ) , a ] = a ( τ g m )( h ) ∈ L and [ b, [ g m ( h ) , a ]] = − τ ( v ( τ g m ) ∗ )( h ) ∈L . Set g m +1 ( h ) = τ ( v ( τ g m ) ∗ )( h ) . We have deg g m +1 = deg v + deg g m − m + 1) n − (2 m + 1) + n − m + 2) n − (2( m + 1) + 1) , hence the result. (cid:3) Proposition 2.7.
Assume that n ≥ . Then dim C L i = ∞ for all i ∈ Z .Proof. When i = 0 the claim follows from Lemma 2.6(1). Suppose that i > . Let ( g m ( h )) m ⊂ L be as in Lemma 2.6(3). We will now show that a i ( τ i g m )( h ) ∈ L − i for all m . Since deg g m +1 > deg g m (because n ≥ ) the elements a i ( τ i g m )( h ) are linearly independent in the domain U . We argue by induction on i . When i = 1 the claim follows from [ a, g m ( h )] = a ( τ g m )( h ) ∈ L − for all m . Assume that a i ( τ i g m )( h ) ∈ L − i for all m , then: [ a, a i ( τ i g m )( h )] = a i +1 ( τ i +1 g m )( h ) ∈ L − ( i +1) for all m .Observe that there exists an anti-automorphism of e U given by κ ( A ) = B , κ ( B ) = A , κ ( H ) = H . It satisfies κ (Ω) = Ω , therefore κ induces an anti-automorphism of U . Since κ ( a i ( τ i g m )( h )) = ( τ i g m )( h ) b i ∈ L i , we get that dim L i = ∞ . (cid:3) Remark.
The fact that dim C L = ∞ when n ≥ can also be proved by using [48].The next result shows that the spherical subalgebra e H e is isomorphic to aquotient e U / (Ω) for an an obvious choice of ψ ( s ) : Proposition 2.8.
Let b ∗ ( s ) , v ( s ) = − b ( − s ) , ψ ( s ) = ( τ v )( s ) ∈ C [ s ] be as in (2.5) and (2.6) , and denote by U = res( e H e ) = C [ z, θ, δ ] the image of the spherical ADIAL COMPONENTS 9 subalgebra under the restriction map. Let e U = C h A, B, H i be the algebra similar to U ( sl (2)) defined by ψ . Then the morphism π : e U −→ U, π ( A ) = δ, π ( B ) = z, π ( H ) = h = − θ induces an isomorphism π : e U / (Ω) ∼−−→ U . In particular, there exists an isomor-phism e U / (Ω) ∼−→ e H e .Proof. The existence of the surjective homomorphism π clearly follows from (3)and (4) in Proposition 2.5. By Proposition 2.5(5), Ker( π ) contains (Ω) . Since π ( e U ) = U is not finite dimensional, it follows from one of the properties of e U that Ker( π ) = (Ω) . (cid:3) From results of Smith [52], Musson–Van den Bergh [38], et al., one can forinstance deduce the following properties of the the algebra U ∼ = e H e :– The Verma modules over U are the M ( λ ) ’s such that v ( λ + 1) = − b ∗ ( − λ ) = 0 ,i.e. λ = 0 , λ , . . . , λ n − , cf. (2.5).– dim L ( λ ) < ∞ ⇐⇒ b ∗ ( − λ ) = b ∗ ( − λ + j ) = 0 for some j ∈ N ∗ .– The global homological dimension of U is: ∞ if b ∗ ( − s ) has multiple roots; if b ∗ ( − s ) has no multiple root and two roots differing by some j ∈ N ∗ ; otherwise.When a root − λ of the polynomial b ∗ ( s ) is a rational number one can use Re-mark 2.2 to realize the Verma module M ( λ ) under the form C [ z ] z − λ , on which z, θ, δ act as differential operators. This is for example the case for λ = 0 , wherewe have M (0) = C [ z ] . Examples. (1) Take b ( s ) = ( s + 1)( s + 3 / · · · ( s + ( n + 1) / , λ i = i/ , ≤ i ≤ ( n − / . Verma modules: M ( λ i ) = C [ z ] z − i/ with the natural action of z, δ, θ . Irreducible finite dimensional modules: L ( λ i ) = M ( λ i ) /M ( λ i − , i ≥ , ofdimension . The global dimension of U is .(2) Take b ( s ) = ( s + 1)( s + 5)( s + 9) , λ i = 4 i , i = 0 , , . Verma modules: M (4 i ) = C [ z ] z − i , M (0) ⊂ M (4) ⊂ M (8) with quotients L (4 i ) = M (4 i ) /M (4( i − simpleof dimension . The global dimension of U is .(3) Take b ( s ) = ( s + 1)( s + n/ , n ≥ , λ = 0 , λ = n − . Verma modules: M (0) = C [ z ] , M ( n − ) = C [ z ] z − n − . There are two cases: • n = 2 k , then M (0) = C [ z ] ⊂ M ( k −
1) = C [ z ] z − ( k − with quotient L ( k − simple of dimension k − ; the global dimension of U is ; • n = 2 k + 1 , then M (0) = C [ z ] and M ( k − ) = C [ z ] z − ( k − ) are simple; theglobal dimension of U is .(4) Take b ( s ) = ( s + 1) [( s + 2 / s + 4 / s + 3 / s + 5 / s + 5 / s + 7 / × [( s + 7 / s + 9 / s + 11 / s + 13 / ,λ = 0 , λ = − / , λ = 1 / , λ = − / , λ = 1 / , λ = − / , λ = 1 / , λ = − / , λ = − / , λ = 1 / , λ = 3 / . Verma modules: M ( λ i ) = C [ z ] z − λ i which are simple U -modules. The global dimension of U is ∞ .3. Representations with a one dimensional quotient
Prehomogeneous vector spaces.
Let ˜ G be a connected reductive complexalgebraic group. We denote by G = ( ˜ G, ˜ G ) its derived subgroup, which is a con-nected semisimple group. Recall that ˜ G = GC where C = Z ( ˜ G ) , the connectedcomponent of the center of ˜ G , is a a torus.Let ˜ ρ : ˜ G → GL( V ) be a finite dimensional representation of ˜ G . Recall that f ∈ C [ V ] = S ( V ∗ ) is a relative invariant of ( ˜ G : V ) if there exists a rational character χ ∈ X ( ˜ G ) such that g.f = χ ( g ) f for all g ∈ ˜ G .One says, see [25, Chapter 2], that ( ˜ G : V ) = ( ˜ G, ˜ ρ, V ) is a (reductive) prehomo-geneous vector space (PHV) if ˜ G has a dense orbit in V . We denote the complementof the dense orbit by S , it is called the singular set of ( ˜ G : V ) . Then it is known[25, Theorem 2.9] that the one-codimensional irreducible components of V r S areof the form { f i = 0 } , ≤ i ≤ r , for some relative invariants f i . The f i are alge-braically independent and are called the basic relative invariants of ( ˜ G : V ) ; anyrelative invariant f can be (up to a non-zero constant) written as Q ri =1 f m i i . Whenthe singular set is a hypersurface ( ˜ G : V ) is called regular, cf. [25, Theorem 2.28].Let ˜ ρ ∗ : ˜ G → GL( V ∗ ) be the contragredient representation. Then, see [25,Proposition 2.21], ( ˜ G : V ∗ ) is a PHV. Recall that S ( V ) = C [ V ∗ ] can be identifiedwith the algebra of constant coefficients differential operators on V . If ϕ ∈ C [ V ∗ ] we denote by ϕ ( ∂ ) the corresponding differential operator. If f ∈ C [ V ] is a rel-ative invariant of degree n and weight χ ∈ X ( ˜ G ) , there exists a relative invariant f ∗ ∈ C [ V ∗ ] of degree n and weight χ − . The following result summarizes [25,Proposition 2.22] and [23]. Theorem 3.1 (Sato-Bernstein-Kashiwara) . Under the above notation, set ∆ = f ∗ ( ∂ ) ∈ S ( V ) . There exists b ( s ) ∈ R [ s ] of degree n such that: (1) b ( s ) = c Q n − i =0 ( s + λ i + 1) , c > ; (2) ∆( f s +1 ) = b ( s ) f s ; (3) λ i + 1 ∈ Q ∗ + , ≤ i ≤ n − , λ = 0 . The polynomial b ( s ) is called a b -function of f . Since the form of the operator ∆ = f ∗ ( ∂ ) will be important in the proof of Theorem 3.9, we briefly indicate itsexpression in a particular coordinate system (see [25, p. 38]).Denote by a the complex conjugate of a ∈ C and set | a | = aa ∈ R + . Let ˜ K be amaximal compact subgroup of ˜ G , so that ˜ G = ˜ K exp( i ˜ k ) is the complexification of ˜ K (where ˜ k = Lie( ˜ K ) ). Fix a ˜ K -invariant non-degenerate Hermitian form κ on V such that κ ( λv, µw ) = λµκ ( v, w ) , λ, µ ∈ C , v, w ∈ V . We choose a κ -orthonormalbasis { e i } ≤ i ≤ N on V , with dual basis { z i = e ∗ i } i . Define φ : V −→ V ∗ , φ ( v ) = v ∗ = κ ( v, − ) . (3.1)In coordinates we have: φ (cid:0)P i v i e i (cid:1) = P i v i z i . Then φ is a bijective C -antilinearmap such that φ ( h.v ) = h.φ ( v ) for all h ∈ ˜ K . The inverse of φ , also denoted by φ , is given by φ ( v ∗ ) = v , i.e. φ (cid:0)P i a i z i (cid:1) = P i a i e i , and it also satisfies φ ( h.v ∗ ) = h.φ ( v ∗ ) . We can extend φ to S ( V ) = C [ V ∗ ] and S ( V ∗ ) = C [ V ] by multilinearity.Thus we get a bijective ˜ K -equivariant C -antilinear morphism: φ : C [ V ] −→ C [ V ∗ ] , φ ( f ) = f ∗ . (3.2)Now, if f is a relative invariant of ( ˜ G : V ) associated to χ , we obtain: h.f ∗ = h.φ ( f ) = φ ( h.f ) = φ ( χ ( h ) f ) = χ ( h ) φ ( f ) = χ ( h ) − f ∗ for all h ∈ ˜ K , which shows that f ∗ is a relative invariant corresponding to χ − .The expression of φ ( f ) = f ∗ in the chosen basis is given as follows. If i =( i , . . . , i N ) ∈ N N we set | i | = P Nj =1 i j , i ! = Q Nj =1 i j , z i = z i · · · z i N N , e i = ADIAL COMPONENTS 11 e i · · · e i N N and ∂ i = ∂ i z · · · ∂ i N z N . Then, if the polynomial f is written f = X { i ∈ N N : | i | = n } a i z i , a i ∈ C , (3.3)one has f ∗ = X { i ∈ N N : | i | = n } a i e i (3.4)and therefore: ∆ = X { i ∈ N N : | i | = n } a i ∂ i , b (0) = ∆( f ) = c n − Y i =0 ( λ i + 1) = X i ∈ N N i ! | a i | . (3.5) Remarks 3.2.
1) If K is a maximal compact subgroup of the semisimple group G we can embed K in a maximal compact subgroup ˜ K of ˜ G . Note that any relativeinvariant of ( ˜ G : V ) is a G -invariant, and that f is G -invariant if and only if it is K -invariant. The previous construction shows that there exist ˜ K -equivariant bijective C -antilinear morphisms φ : S ( V ) → S ( V ∗ ) and φ : S ( V ∗ ) → S ( V ) such that φ ◦ φ = id . In particular we have φ (cid:0) S ( V ∗ ) K (cid:1) = S ( V ) K , hence φ (cid:0) S ( V ∗ ) G (cid:1) = S ( V ) G .2) Observe that φ : V → V ∗ is K -equivariant, but, in general, there is no G -module isomorphism of between V and V ∗ . For example, in the case ( ˜ G : V ) = (cid:0) GL( n, C (cid:1) , V C n ) one has ( V C n ) ∗ ∼ = V n − C n as a G -module, which is not iso-morphic to V C n when n > .3.2. PHV of rank one.
Let ˜ G = GC be as above and ( ˜ G : V ) be a finite dimen-sional representation of ˜ G . In this subsection we make the following hypothesis: Hypothesis A.
There exists f ∈ S n ( V ∗ ) such that: f / ∈ C [ V ] ˜ G and C [ V ] G = C [ f ] .Remarks. (1) Assume that G is a semisimple group and ( G : V ) is a finite dimen-sional representation of G such that C [ V ] G = C [ f ] , f / ∈ C ∗ . Let the group C ∗ acton V by homotheties. Then ( G × C ∗ : V ) satisfies the hypothesis A. Therefore onecould assume without lost of generality that C = C ∗ .(2) Let f be as in the previous hypothesis. Then, since G is semisimple, thepolynomial f is irreducible. Let g ∈ C , then g.f ∈ C [ V ] G = C [ f ] and deg( g.f ) =deg( f ) , hence g.f = χ ( g ) f for some χ ( g ) ∈ C ∗ . It follows that χ ∈ X ( ˜ G ) , i.e. f isa relative invariant of ( ˜ G : V ) ; note that χ = 1 , since f / ∈ C [ V ] ˜ G . Assume that f is another relative invariant of ( ˜ G : V ) . Then f ∈ C [ V ] G is homogeneous and thisimplies that f = αf m for some α ∈ C , m ∈ N . Proposition 3.3.
Let f be as in hypothesis A. (i) Let C ( V ) be the fraction field of C [ V ] . Then one has: C ( V ) G = C ( f ) , C ( V ) ˜ G = C . In particular, ( ˜ G : V ) is a PHV. (ii) Let f ∗ be the relative invariant obtained in (3.3) , then C [ V ∗ ] G = C [ f ∗ ] . (iii) The representation ( G : V ) is polar.Proof. (i) The equality C ( V ) G = C ( f ) follows from [46, Theorem 3.3]. By theremark (2) above, we know that g.f = χ ( g ) f for all g ∈ C . Observe that C ( V ) ˜ G =( C ( V ) G ) C = C ( f ) C and let ϕ = p ( f ) /q ( f ) ∈ C ( f ) C where p ( f ) , q ( f ) ∈ C [ f ] arerelatively prime polynomials in f . One easily sees that p ( f ) and q ( f ) are relativeinvariants, thus p ( f ) = αf k and q ( f ) = βf ℓ , α, β ∈ C . It follows that χ k − ℓ = 1 ,hence k = ℓ and ϕ ∈ C . From [46, Corollary, p. 156] one gets that ( ˜ G : V ) is aPHV. (ii) Adopt the notation of Remarks 3.2. The map φ : C [ V ] → C [ V ∗ ] defined in (3.2)yields a bijective C -antilinear morphism C [ V ] G = C [ V ] K → C [ V ∗ ] K = C [ V ∗ ] G .Thus C [ V ∗ ] G = C [ f ∗ ] .(iii) Choose v ∈ V regular semisimple, i.e. G.v closed and dim
G.v ≥ dim G.v ′ forall closed orbits G.v ′ . Set h = C v and g = Lie( G ) . Then, see [5], one easily deducesthat h = h v = { x ∈ V : g .x ⊂ g .v } is a Cartan subspace for the G -action on V . (cid:3) From now on, we assume that the hypothesis A is satisfied and we fix a Cartansubspace h = C v for the G -action on V . We set • x = v ∗ , hence C [ h ] = S ( h ∗ ) = C [ x ] ; • W = N G ( h ) /Z G ( h ) .By [5] we know that there exists an isomorphism V //G ∼ = h /W given by the re-striction map ψ : C [ V ] G ∼−−→ C [ h ] W , ψ ( p ( f )) = p ( f ) | h . Since f is homogeneous ofdegree n , ψ ( f ) ∈ C [ x ] is a scalar multiple of z = x n . Therefore, multiplying v by anon zero constant, we may assume that ψ ( f ) = x n , W ≡ h w i ⊂ GL( h ) where w acts on x by w.x = ζx , ζ primitive n -th root of unity. We can thereforeadopt the notation of §2.2. In particular, C [ h ] W = C [ z ] where z = x n = ψ ( f ) .Let b ( s ) = c Q n − i =0 ( s + λ i + 1) ∈ C [ s ] be a b -function of the relative invariant f as in Theorem 3.1. We can then define the rational Cherednik algebra H = H ( W, k , . . . , k n − ) = C h x, T, w i , where the parameters k i are given by k i = λ i + in , cf. (2.5). Recall that the image U = res( e H e ) ⊂ C [ z, ∂ z ] of the spherical subalgebra is generated by z, δ, θ , where θ = z∂ z , δ = z − b ∗ ( θ ) = z − c n − Y j =0 ( θ + λ j ) , see Proposition 2.5.Recall from §1 that we have a radial component map: rad : D ( V ) G −→ D ( h /W ) = C [ z, ∂ z ] , rad( D )( p ( z )) = ψ ( D ( p ( f ))) , for all p ( z ) ∈ C [ h ] W . The algebra of radial components is defined to be R = rad( D ( V ) G ) ⊂ C [ z, ∂ z ] . (3.6)The aim of this section is to prove that R = U , see Theorem 3.9.Before entering the proof, let us give some standard examples. A complete listof pairs ( ˜ G : V ) as above with V irreducible can be found in [25]. Recall that ∆ = f ∗ ( ∂ ) ∈ S ( V ) is the differential operator constructed in (3.5). Examples. (1) ( ˜ G = GL( n ) : V = S C n ) , W = Z n , f = det( x ij ) , ∆ = det( ∂ x ij ) , b ( s ) = Q n − i =0 ( s + i/ .(2) ( ˜ G = E × C ∗ : V = C ) , W = Z , f = cubic form, b ( s ) = ( s + 1)( s + 5)( s + 9) .(3) ( ˜ G = SO( n ) × C ∗ : V = C n ) , W = Z , f = quadratic form, ∆ = Laplacian, b ( s ) = ( s + 1)( s + n/ .(4) ( ˜ G = SL(5) × GL(4) : V = V C ⊗ C ) , W = Z , deg f = 40 , b ( s ) is: ( s + 1) (cid:2) ( s + 2 / s + 4 / s + 3 / s + 5 / s + 5 / s + 7 / (cid:3) (cid:2) ( s + 7 / s + 9 / s + 11 / s + 13 / (cid:3) . (5) ( ˜ G = GL( n ) × SL( n ) : V = M n ( C )) , W = Z n , f = det( x ij ) , ∆ = det( ∂ x ij ) , b ( s ) = Q n − i =0 ( s + i + 1) . ADIAL COMPONENTS 13 (6) ( ˜ G = Sp( n ) × SO(3) × C ∗ : V = M n, ( C )) , W = Z , deg f = 4 , b ( s ) =( s + 1)( s + 3 / s + n )( s + n + 1 / .Remark: The first five examples are regular irreducible PHV, but (6) gives is anexample of an irreducible PHV which is not regular [25]. The description of theVerma modules on U associated to examples (1) to (4) are given in §2.3.Let Θ be the Euler vector field on V ; thus Θ( p ) = np for all p ∈ S n ( V ∗ ) . Inparticular Θ( f ) = nf , which implies that rad(Θ) = nθ . Set ¯Θ = 1 n Θ , (3.7)so that rad( ¯Θ) = θ . Lemma 3.4.
One has: U ⊂ R, U [ z − ] = R [ z − ] = C [ z ± , ∂ z ] . Proof.
Let ∆ = f ∗ ( ∂ ) ∈ S ( V ) be as in in (3.5). By definition and Theorem 3.1we obtain: rad(∆)( z s +1 ) = ψ (∆( f s +1 )) = b ( s ) z s . Hence rad(∆) = δ , see Propo-sition 2.5. From rad( f ) = z and rad( ¯Θ) = θ it follows that U = C [ z, δ, θ ] ⊂ R ⊂ C [ z, ∂ z ] . Observe that U [ z − ] = C [ z ± , δ, z − θ = ∂ z , ] = C [ z ± , ∂ z ] ⊂ R [ z − ] ⊂ C [ z ± , ∂ z ] , thus U [ z − ] = R [ z − ] = C [ z ± , ∂ z ] . (cid:3) Recall that φ : C [ V ] → C [ V ∗ ] , φ ( p ) = p ∗ , is the bijective ˜ K -equivariant C -antilinear morphism defined in (3.2). Lemma 3.5.
The map φ extends to a ˜ K -equivariant C -antilinear anti-automor-phism of D ( V ) , given by φ ( p ) = ∂ ( p ∗ ) . One has: φ (cid:0) D ( V ) G (cid:1) = D ( V ) G , φ = id , φ ( f ) = ∆ = ∂ ( f ∗ ) , φ ( ¯Θ) = ¯Θ . Proof.
In the coordinate system { z i , ∂ z i } ≤ i ≤ N we have: φ ( z i ) = ∂ z i , φ ( ∂ z i ) = z i , φ ( a ) = a for a ∈ C . Since D ( V ) = C [ z i , ∂ z j : 1 ≤ i, j ≤ N ] with relations [ ∂ z j , z i ] = δ ij , it is clear that φ extends to a C -antilinear anti-automorphism of D ( V ) such that φ = id . By construction φ is ˜ K -equivariant, in particular K -equivariantif K ⊂ ˜ K is a maximal compact subgroup of G . From G = K C = K exp( i k ) itfollows that D ( V ) K = D ( V ) G , hence φ (cid:0) D ( V ) G (cid:1) = D ( V ) G . The equality φ ( f ) = ∆ is obvious and φ ( ¯Θ) = ¯Θ is consequence of Θ = P i z i ∂ z i . (cid:3) Recall that rad : D ( V ) G ։ R ; we now want to check that the anti-automorphism φ induces an anti-automorphism on R such that φ ( z ) = δ . Denote by J the kernelof rad , thus: J = (cid:8) D ∈ D ( V ) G : D ( f m ) = 0 for all m ∈ N (cid:9) . Set D = D ( V ) G ⊃ e D = D ( V ) ˜ G . (3.8)Since Θ ∈ D ( V ) G we can decompose D under the adjoint action of Θ : D = M p ∈ Z D [ p ] , D [ p ] = { D ∈ D : [Θ , D ] = pD } . (3.9) Proposition 3.6.
One has φ ( J ) = J .Proof. As φ = id we need to show that φ ( J ) ⊂ J . Since J is an ideal of D ( V ) G it decomposes under the adjoint action of Θ : J = ⊕ p ∈ Z J [ p ] , J [ p ] = J ∩ D [ p ] .Thus we only need to check that φ ( J [ p ]) ⊂ J . In the previous coordinate system { z i , ∂ z i } ≤ i ≤ N we have: D [ p ] = X | i |−| j | = p C z i ∂ j . We can write D ∈ D [ p ] in a unique way under the form D = D + · · · + D t , D k = X | j | = k − p (cid:18) X | i | = k a i , j z i (cid:19) ∂ j , (thus D k = 0 when k < p ). If D t = 0 , we set t = deg z D .Let D = P | i |−| j | = p a i , j z i ∂ j be in J [ p ] r { } . As G acts linearly on V wehave G.D k ⊂ S k ( V ∗ ) S k − p ( V ) . Since D is G -invariant it follows that each D k is G -invariant. We have: φ ( D ) = X k φ ( D k ) , φ ( D k ) = X {| i | = k, | j | = k − p } a i , j z j ∂ i . From the previous expression we get that φ ( D k )(1) = 0 when k > , hence φ ( D )(1) = φ ( D )(1) = φ ( D ) = X | j | = − p a , j z j . Assume that D = P | j | = − p a , j ∂ j ∈ S p ( V ) G is non zero. From Proposition 3.3(ii)we can deduce that p = − ℓn , ℓ ≥ , D = α ∆ ℓ , α = 0 . By hypothesis D ( f ℓ ) = P k D k ( f ℓ ) = 0 . Note that, since | j | = k + nℓ > nℓ implies ∂ j ( f ℓ ) = 0 , we have D k ( f ℓ ) = P {| i | = k, | j | = k + nℓ } a i , j z i ∂ j ( f ℓ ) = 0 if k > . Thus D ( f ℓ ) = D ( f ℓ ) = 0 .If ℓ = 0 we get D = α = D (1) = D ( f ℓ ) = 0 , contradiction. Therefore ℓ ≥ and D ( f ℓ ) = D ( f ℓ ) = α ∆ ℓ ( f ℓ ) = αb ( ℓ − · · · b (0) . It is easily seen that ∆ ℓ ( f ℓ ) = 0 , see [25, Proof of Proposition 2.22] (this is equiv-alent to b ( j ) = 0 for all j ∈ N ), hence a contradiction. Thus: φ ( D )(1) = φ ( D ) = D = 0 .We show that φ ( D ) ∈ J by induction on t = deg z D . (In the case t = 0 one has D = D = 0 .) Since ∆ ∈ S ( V ) G and J is an ideal, one has [ D, ∆] ∈ J . Observethat [ D, ∆] = X k [ D k , ∆] , [ D k , ∆] = X | j | = k − p (cid:0) X | i | = k a i , j [ z i , ∆] (cid:1) ∂ j . But ∆ ∈ S ( V ) G implies that deg z [ z i , ∆] < k = | i | , hence deg z [ D, ∆] < t = deg z D .Then, by induction, φ ([ D, ∆]) = [ φ (∆) , φ ( D )] = [ f, φ ( D )] ∈ J . If m ≥ we thenhave: f, φ ( D )]( f m ) = f φ ( D )( f m ) − φ ( D )( f m +1 ) . This implies, by inductionon m , φ ( D )( f m +1 ) = f φ ( D )( f m ) = f m +1 φ ( D )(1) . It follows from the previousparagraph that φ ( D )( f m +1 ) = φ ( D )(1) = 0 , i.e. φ ( D ) ∈ J . (cid:3) Corollary 3.7. (1)
There exists a C -antilinear anti-automorphism φ : R → R suchthat: φ = id , φ ( z ) = δ, φ ( θ ) = θ, φ ( U ) = U. (2) One has U [ δ − ] = R [ δ − ] .Proof. (1) Let φ : D ( V ) G → D ( V ) G be as in Lemma 3.5. By Proposition 3.6 wecan define φ : R → R by setting φ (rad( D )) = rad( φ ( D )) . Indeed: if rad( D ) = rad( D ′ ) we get D − D ′ ∈ J = Ker(rad) , hence φ ( D ) − φ ( D ′ ) ∈ J and rad( φ ( D )) = rad( φ ( D ′ )) . The equality φ = id is clear; by definition andLemma 3.5: φ ( z ) = rad( φ ( f )) = rad(∆) = δ, φ ( θ ) = rad( φ ( ¯Θ)) = rad( ¯Θ) = θ. From U = C [ z, δ, θ ] we then deduce φ ( U ) = U .(2) Observe that ad( φ ( u )) m ( r ) = ( − m φ (ad( u ) m ( r )) for all u, r ∈ R . Since ad( z ) is a locally nilpotent operator in R , it follows that ad( φ ( z )) = ad( δ ) has the sameproperty. We can therefore construct the C -algebras U [ δ − ] ⊂ R [ δ − ] . ADIAL COMPONENTS 15
Let Q = Frac( U ) be the fraction field of the Noetherian domain U . By Lemma 3.4we know that Q = C ( z, ∂ z ) = Frac( R ) . It is easy to see that φ extends to Q by φ ( s − a ) = φ ( a ) φ ( s ) − for all a ∈ r , = s ∈ R . This gives a C -antilinear anti-auto-morphism of Q . Then φ ( R [ z − ])) = φ ( U [ z − ]) = φ ( U )[ δ − ] = U [ δ − ] , which yields U [ δ − ] = φ ( R [ z − ]) = φ ( R )[ δ − ] = R [ δ − ] , as desired. (cid:3) Let M be a module over a C -algebra A , then the Gelfand-Kirillov of M is denotedby GKdim A M or simply GKdim M , see [32]. Lemma 3.8.
Let r ∈ R . Then: GKdim U ( U + U r ) /U ≤ GKdim U U − . Proof.
From Corollary 3.7 we deduce that there exists ν ∈ N such that z ν r ∈ U and δ ν r ∈ U . Therefore the U -module ( U + U r ) /U is a factor of U/ ( U z ν + U δ ν ) .There exists on U ∼ = e U / (Ω) (cf. Proposition 2.8) a finite dimensional filtration suchthat gr( U ) is isomorphic to the commutative algebra C [ X, Y, S ] / ( XY − S n ) , see§2.2 or [52, 38], where gr( z ) = X , gr( δ ) = Y . It follows that the associated gradedmodule of U/ ( U z ν + U δ ν ) is a factor of gr( U ) / (gr( U ) X ν + gr( U ) Y ν ) , which is finitedimensional. Hence the result. (cid:3) We now can prove the main result of this section.
Theorem 3.9.
One has U = R .Proof. Endow U with a filtration such that gr( U ) ∼ = C [ X, Y, S ] / ( XY − S n ) as in theproof of the previous lemma. Observe that C [ X, Y, S ] / ( XY − S n ) is commutativeGorenstein normal domain. By [3, Theorem 3.9] U is Auslander-Gorenstein andby [55] U is a maximal order. Recall that Q = Frac( U ) and consider the followingfamily of finitely generated U -modules M : F = (cid:8) U ⊂ M ⊂ Q : GKdim U M/U ≤ GKdim U U − (cid:9) . From [3, Theorem 1.14] we know that F contains a unique maximal element ˜ M .By Lemma 3.8 we have U + U r ⊂ ˜ M for all r ∈ R ; hence R ⊂ ˜ M . It follows that R is finitely generated over U with Q = Frac( U ) = Frac( R ) . Thus U = R , since U is a maximal order. (cid:3) Remark 3.10.
Let ( G : V ) be a representation of the connected reductive group G such that dim V //G = 1 . If C [ V ] G = C [ f ] one can define ∆ ∈ S ( V ) G andthe polynomial b ( s ) = c Q n − i =0 ( s + λ i + 1) as in Theorem 3.1. Then, the proof ofTheorem 3.9 can be repeated to show that R = Im(rad) = U ( k ) (where k i = λ i + in , ≤ i ≤ n − ). 4. Multiplicity free representations
Generalities.
Let ( ˜ G : V ) be a connected reductive group. Write ˜ G = GC , C ∼ = ( C ∗ ) c , as in §3.1. We adopt the following notation: • the Lie algebra of an algebraic group is denoted by the corresponding gothiccharacter; • T U is a Borel subgroup of G , T being a maximal torus of G , hence ˜ T U isa Borel subgroup of ˜ G , ˜ T = T C ; • R is the root system of ( g , t ) , B = { α , . . . , α ℓ } is a basis of R and R + is theset of associated positive roots; • Λ is the weight lattice of ( g , t ) , thus Λ = Z ̟ ⊕ · · · ⊕ Z ̟ ℓ where h ̟ i , α j i = δ ij ; Λ + = N ̟ ⊕ · · · ⊕ N ̟ ℓ denotes the dominant weights; • ˜Λ = Λ ⊕ X ( C ) , with X ( C ) ∼ = Z c ; ˜Λ + = Λ + ⊕ X ( C ) ; • if ˜ λ ∈ ˜Λ + , resp. λ ∈ Λ + , we denote by E (˜ λ ) , resp. E ( λ ) , an irreducible ˜ g -module, resp. g -module, with highest weight ˜ λ , resp. λ ; the dual of E (˜ λ ) is isomorphic to E (˜ λ ∗ ) , ˜ λ ∗ = − w (˜ λ ) where w is the longest element ofthe Weyl group of R (similarly for E ( λ ) ∗ ).We fix a finite dimensional representation ( ˜ G : V ) of the reductive group ˜ G .Then the rational ˜ G -module C [ V ] = S ( V ∗ ) decomposes as C [ V ] ∼ = M ˜ λ ∈ ˜Λ + E (˜ λ ) m (˜ λ ) where m (˜ λ ) ∈ N ∪ {∞} . Definition 4.1.
The representation ( ˜ G : V ) is called multiplicity free (MF forshort) if m (˜ λ ) ≤ for all ˜ λ . In this case C [ V ] = M ˜ λ ∈ ˜Λ + V (˜ λ ) m (˜ λ ) , m (˜ λ ) = 0 , where V (˜ λ ) ⊂ S d (˜ λ ) ( V ∗ ) is isomorphic to E (˜ λ ) ; if m (˜ λ ) = 1 , d (˜ λ ) is called thedegree of ˜ λ in C [ V ] . Remark.
The MF representations are classified [21, 1, 27]. We give in Appendix Athe list of ( ˜ G : V ) with V irreducible (see [21]). For instance, the examples (1), (2),(3), (5) given in §3.2 are MF. From now on, let ( ˜ G : V ) be a MF representation. The following results can befound, for example, in [1, 18, 21, 26].– Set ˜Γ = { ˜ λ : m (˜ λ ) = 1 } , then ˜Γ = ⊕ ri =0 N ˜ λ i where the ˜ λ i are linearly indepen-dent over Q .– The algebra of U -invariants C [ V ] U = C [ h , . . . , h r ] is a polynomial ring. If ˜ γ = P i m j ˜ λ j ∈ ˜Γ , one has V (˜ γ ) = U (˜ g ) .h ˜ γ where h ˜ γ = h m · · · h m r r is a highestweight vector of V (˜ γ ) . In particular: h j = h ˜ λ j , d (˜ γ ) = P j m j d (˜ λ j ) .– The representation ( ˜ G : V ) is a prehomogeneous vector space. Let f , . . . , f m be the basic relative invariants of this PHV and let χ j ∈ X ( ˜ G ) = X ( C ) , ≤ j ≤ m ,be their weights. After identification of X ( C ) with a subgroup of ˜Λ as above, onecan number the ˜ λ j so that ˜ λ ≡ χ , . . . , ˜ λ m ≡ χ m , h = f , . . . , h m = f m , thus V (˜ λ j ) = V ( χ j ) is the one dimensional ˜ G -module C f j .Let p : ˜Λ + = Λ + ⊕ X ( C ) ։ Λ + be the natural projection. Set ˜Γ = Γ M Γ , Γ = m M j =0 N ˜ λ j = m M j =0 N χ j , Γ = r M j = m +1 N ˜ λ j . (4.1)Using the results above, the next lemma is easy to prove. Lemma 4.2.
One has: (a) Γ = X ( C ) ∩ ˜Γ = { ˜ γ ∈ ˜Γ : ˜ γ ( t ) = 0 } ; p induces a bijection Γ ∼−→ p (Γ) ; (b) let ˜ γ ∈ ˜Γ , then the G -module V (˜ γ ) is isomorphic to E ( p (˜ γ )) ; (c) let γ, γ ′ ∈ Γ , then the following are equivalent: (i) γ = γ ′ (ii) p ( γ ) = p ( γ ′ ) (iii) V ( γ ) ∼ = V ( γ ′ ) as G -modules; ADIAL COMPONENTS 17 (d) the algebra C [ V ] G of G -invariants is polynomial ring, more precisely: C [ V ] G = C [ f , . . . , f m ] = M γ ∈ Γ C h γ . Set: H ( V ∗ ) = M γ ∈ Γ V ( γ ) . (4.2) Lemma 4.3.
The multiplication map: m : H ( V ∗ ) ⊗ C S ( V ∗ ) G −→ S ( V ∗ ) = C [ V ] is an isomorphism of G -modules.Proof. Let ˜ γ = γ + γ , γ ∈ Γ , γ ∈ Γ . Observe that C acts by scalars on the simple ˜ G -module V (˜ γ ) ; thus, since ˜ g = g ⊕ c , we have: V (˜ γ ) = U (˜ g ) .h ˜ γ = U ( g ) U ( c ) .h ˜ γ = U ( g ) .h ˜ γ = U ( g ) .h γ h γ = ( U (˜ g ) .h γ ) h γ = V ( γ ) h γ . Therefore V (˜ γ ) = m ( V ( γ ) ⊗ C h γ ) with V ( γ ) ⊂ H ( V ∗ ) , h γ ∈ S ( V ∗ ) G . Suppose that m ( P i v i ⊗ h µ i ) = 0 with µ i ∈ Γ , v i ∈ V ( λ i ) , the λ i ∈ Γ being pairwise distinct. Observe that λ i + µ i = λ j + µ j forces λ i − λ j = µ j − µ i ∈ Γ ∩ Γ = (0) . Therefore v i h µ i ∈ V ( λ i + µ i ) and P i v i h µ i = 0 yield v i h µ i = 0 , hence v i = 0 for all i . (cid:3) Recall that we identify S ( V ) with the algebra of differential operators with con-stant coefficients. Consider the non-degenerate pairing S ( V ) ⊗ S ( V ∗ ) −→ C , u ⊗ ϕ
7→ h u | ϕ i = u ( ϕ )(0) , which extends the duality pairing V ⊗ V ∗ → C . It is easily shown that: • h u | S j ( V ∗ ) i = 0 if u ∈ S i ( V ) and i = j ; • h | i is ˜ G -equivariant.Therefore u
7→ h u | i gives a ˜ G -isomorphism from S i ( V ) onto S i ( V ∗ ) ∗ . In partic-ular, the representation ( ˜ G : V ∗ ) is MF and we can write: S i ( V ) = M { ˜ γ ∈ ˜Γ ,d (˜ γ )= i } Y (˜ γ ) , Y (˜ γ ) ∼ = V (˜ γ ) ∗ ∼ = E (˜ γ ∗ ) . Hence, Y (˜ γ ) = U (˜ g ) . ∆ ˜ γ where ∆ ˜ γ is a lowest weight vector (of weight − ˜ γ ). When ˜ γ = ˜ λ j we set ∆ ˜ λ j = ∆ j . Note that ∆ ˜ γ = Q ri =0 ∆ m i i if ˜ γ = P i m i ˜ λ i .If ≤ i ≤ m we have Y (˜ λ i ) = C ∆ i where ∆ i has weight − λ i ≡ χ − i . Clearly,we may take ∆ i = ∂ ( f ∗ i ) where f ∗ i is the relative invariant constructed as in §3.1.We then have S ( V ) G = C [∆ , . . . , ∆ m ] = M γ ∈ Γ C ∆ γ (4.3)(which is a polynomial ring).If µ = P i m i ˜ λ i and ν = P i n i ˜ λ i are elements of ˜Γ , we say that µ ≤ ν if m i ≤ n i for all i . Let k : ˜Γ ։ Γ be the projection associated to the decomposition definedin (4.1); thus each ˜ λ ∈ ˜Λ writes uniquely γ + k (˜ λ ) , γ ∈ Γ , k (˜ λ ) ∈ Γ . Lemma 4.4.
Let ˜ λ ∈ Γ and ˜ γ ∈ ˜Γ . Then: (a) ∆ ˜ γ ( h ˜ γ ) = 0 ; (b) ∆ ˜ λ ( h ˜ γ ) = 0 ⇐⇒ ˜ λ ≤ k (˜ γ ) , and in this case ∆ ˜ λ gives an isomorphism of G -modules, ϕ ∆ ˜ λ ( ϕ ) , from V (˜ γ ) onto V (˜ γ − ˜ λ ) .Proof. Set ˜ λ = P mi =0 p i ˜ λ i , ˜ γ = P ri =0 q i ˜ λ i .(a) Recall that we have an isomorphism of ˜ G -modules, β : Y (˜ γ ) ∼−→ V (˜ γ ) ∗ , β ( u ) = h u | i . Thus β (∆ ˜ γ ) is a lowest vector in V (˜ γ ) ∗ , which implies β (∆ ˜ γ )( h ˜ γ ) = ∆ ˜ γ ( h ˜ γ )(0) = 0 . But ∆ ˜ γ ∈ S d (˜ γ ) ( V ) where d (˜ γ ) is the degree of ˜ γ , therefore ∆ ˜ γ ( h ˜ γ ) ∈ C . Thus ∆ ˜ γ ( h ˜ γ ) = ∆ ˜ γ ( h ˜ γ )(0) = 0 .(b) Since ∆ ˜ λ ∈ S ( V ) G we have ∆ ˜ λ ( V (˜ γ )) = ∆ ˜ λ ( U ( g ) h ˜ γ ) = U ( g )∆ ˜ λ ( h ˜ γ ) . ByLemma 4.2 we know that V (˜ γ ) is a simple G -module, it follows that the map ∆ ˜ λ : V (˜ γ ) → ∆ ˜ λ ( V (˜ γ )) is either or an isomorphism of G -modules.Notice that ∆ ˜ λ ( h ˜ γ ) ∈ C [ V ] U has weight ˜ γ − ˜ λ = k (˜ γ ) − ˜ λ + P ri = m +1 q i ˜ λ i where k (˜ γ ) − ˜ λ = P mi =0 ( q i − p i )˜ λ i . Therefore if q i < p i for some i = 0 , . . . , m we must have ∆ ˜ λ ( h ˜ γ ) = 0 , i.e. ∆ ˜ λ ( h ˜ γ ) = 0 implies ˜ λ ≤ k (˜ γ ) . Conversely, suppose that ˜ λ ≤ k (˜ γ ) ;then, by (a), = ∆ ˜ γ ( h ˜ γ ) = ∆ ˜ γ − ˜ λ ∆ ˜ λ ( h ˜ γ ) which forces ∆ ˜ λ ( h ˜ γ ) = 0 .Now assume ˜ λ ≤ k (˜ γ ) . Then = ∆ ˜ λ ( h ˜ γ ) ∈ C [ V ] U implies that ∆ ˜ λ ( h ˜ γ ) is ahighest weight vector in V (˜ γ − ˜ λ ) , hence ∆ ˜ λ : V (˜ γ ) ∼−−→ ∆ ˜ λ ( V (˜ γ )) = V (˜ γ − ˜ λ ) . (cid:3) Recall the definition of H ( V ∗ ) given in (4.2) and set S + ( V ) = ⊕ i> S i ( V ) . Thenext proposition identifies H ( V ∗ ) with harmonic elements. Proposition 4.5.
We have: H ( V ∗ ) = { ϕ ∈ C [ V ] : ∆ ( ϕ ) = · · · = ∆ m ( ϕ ) = 0 } = (cid:8) ϕ ∈ C [ V ] : D ( ϕ ) = 0 for all D ∈ S + ( V ) G (cid:9) . Proof.
From (4.3) we know that S + ( V ) G = L =˜ λ ∈ Γ C ∆ ˜ λ . Let ϕ ∈ V (˜ γ ) for some ˜ γ ∈ Γ and let = ˜ λ ∈ Γ . We have k (˜ γ ) = 0 , thus ∆ ˜ λ ( ϕ ) = 0 by Lemma 4.4(b).This shows that H ( V ∗ ) ⊂ (cid:8) ϕ ∈ C [ V ] : D ( ϕ ) = 0 for all D ∈ S + ( V ) G (cid:9) .Conversely assume that ϕ = P ˜ γ ∈ ˜Γ ϕ ˜ γ , ϕ ˜ γ ∈ V (˜ γ ) , satisfies ∆ i ( ϕ ) = 0 for all i = 0 , . . . , m . Fix i ∈ { , . . . , m } . By Lemma 4.4(b) we get that ∆ i ( V (˜ γ )) =0 if ˜ λ i k (˜ γ ) and ∆ i : V (˜ γ ) ∼−→ V (˜ γ − ˜ λ i ) if ˜ λ i ≤ k (˜ γ ) . Therefore ∆ i ( ϕ ) = P ˜ γ ∈ ˜Γ ∆ i ( ϕ ˜ γ ) belongs to L { ˜ γ ∈ ˜Γ , ˜ λ i ≤ k (˜ γ ) } V (˜ γ − ˜ λ i ) . Since ∆ i ( ϕ ) = 0 we can deducethat ∆ i ( ϕ ˜ γ ) = 0 for all ˜ γ such that ˜ λ i ≤ k (˜ γ ) . By the previous remark this implies ϕ ˜ γ = 0 when ˜ λ i ≤ k (˜ γ ) , thus ϕ = P { ˜ γ ∈ ˜Γ , ˜ λ i k (˜ γ ) } ϕ ˜ γ . Observe that ˜ λ i k (˜ γ ) means that the weight ˜ λ i does not appear in ˜ γ . Since this holds for all i = 0 , . . . , m we deduce that ϕ = P ˜ γ ∈ Γ ϕ ˜ γ . Hence the result. (cid:3) Remark 4.6.
If we set H ( V ) = L γ ∈ Γ Y ( γ ) we obtain that S ( V ) ∼ = H ( V ) ⊗ S ( V ) G as G -modules, with an analogous characterization of H ( V ) .We now recall some facts about invariant differential operators on MF representa-tions, cf. [1, 18, 26]. Recall that the C [ V ] -module D ( V ) identifies with S ( V ∗ ) ⊗ S ( V ) through the multiplication map m : S ( V ∗ ) ⊗ S ( V ) ∼−→ D ( V ) , ϕ ⊗ f ϕf ( ∂ ) . The isomorphism m is also ˜ G -equivariant, hence D ( V ) ˜ G ∼ = L ˜ γ ∈ ˜Γ [ V (˜ γ ) ⊗ Y (˜ γ )] ˜ G .But, since Y (˜ γ ) ∼ = V (˜ γ ) ∗ , [ V (˜ γ ) ⊗ Y (˜ γ )] ˜ G = C E ˜ γ is one dimensional. Let E ˜ γ ( x, ∂ x ) = 1dim V (˜ γ ) m ( E ˜ γ ) ∈ D ( V ) ˜ G be the operator corresponding to E ˜ γ . The E ˜ γ ( x, ∂ x ) are called the normalizedCapelli operators . Set E j = E ˜ λ j ( x, ∂ x ) , ≤ j ≤ r. (4.4) ADIAL COMPONENTS 19
It is known [18, Proposition 7.1] that ( ˜ G : V ) multiplicity free is equivalent to D ( V ) ˜ G commutative. The operators E j give a set of generators for this algebra,cf. [18, Theorem 9.1] or [1, Corollary 7.4.4]: Theorem 4.7 (Howe-Umeda) . e D = D ( V ) ˜ G = C [ E , . . . , E r ] = L ˜ γ ∈ ˜Γ C E ˜ γ ( x, ∂ x ) is a commutative polynomial ring. Notice for further use the following property of the Capelli operators, see [26,Corollary 4.4] or [1, Proposition 8.3.2]:
Proposition 4.8.
Set a ∗ = C ⊗ Z Z ˜Γ = ⊕ ri =0 C ˜ λ i , a = ⊕ ri =0 C a i where { a i } i isthe dual basis of { ˜ λ i } i . For each ˜ γ ∈ ˜Γ there exists a polynomial function b ˜ γ = b ˜ γ ( a , . . . , a r ) ∈ C [ a ∗ ] = S ( a ) = C [ a , . . . , a r ] such that E ˜ γ ( x, ∂ x )( h ˜ λ ) = b ˜ γ (˜ λ ) h ˜ λ for all ˜ λ ∈ a ∗ . Remarks 4.9. (1) Suppose that ( ˜ G : V ) is irreducible. Then we can assume that V = V (˜ γ r ) . If dim V = N we have E r = E r ( x, ∂ x ) = ¯Θ = N Θ where Θ is theEuler vector field.(2) If j ∈ { , . . . , m } we may take E j = f j ∆ j . Recall that f j = h ˜ λ j and ∆ j = ∂ ( f ∗ j ) . By Theorem 3.1 there exists b j ( s ) ∈ C [ s ] such that ∆ j ( f mj ) = b ∗ j ( m ) f m − j ;thus E j ( f mj ) = b ∗ j ( m ) f mj . This shows that b ˜ λ j ( s, , . . . ,
0) = b ∗ j ( s ) .(3) Let D ∈ e D ; then D ( V (˜ λ )) = U (˜ g ) .D ( h ˜ λ ) is either (0) or equal to V (˜ λ ) .Indeed, the ˜ G -invariance of D implies that g.D ( h ˜ λ ) = D ( g.h ˜ λ ) = ˜ λ ( g ) D ( h ˜ λ ) for all g ∈ ˜ T U , where we have considered here ˜ λ as a character of the Borel subgroup ˜ T U of ˜ G ; thus D ( h ˜ λ ) ∈ C h ˜ λ is either or a highest weight vector of V (˜ λ ) .4.2. MF representations with a one dimensional quotient.
In this subsec-tion we will work under the following hypothesis:
Hypothesis B. ( ˜ G : V ) is a multiplicity free representation which satisfies Hy-pothesis A. In the notation of §4.1, this condition means that m = 0 , i.e. Γ = N ˜ λ . Set f = f , n = d (˜ λ ) , ∆ = ∆ = ∂ ( f ∗ ) , then we have f ∈ S n ( V ∗ ) , V (˜ λ ) = C f , Y (˜ λ ) = C ∆ and: C [ V ] G = C [ f ] , S ( V ) G = C [∆] (see Lemma 4.2 and (4.3)). By Remark 4.9(2) we have E = f ∆ , b ˜ λ j ( s, , . . . ,
0) = b ∗ ( s ) = b ( s − where b ( s ) is the b -function of f . Recall from (3.8) the followingnotation: D = D ( V ) G ⊃ D ( V ) ˜ G = e D . Recall also that ( G : V ) is polar and that we have studied in §3.2 the image ofradial component map rad : D ( V ) G → D ( h /W ) = C [ z, ∂ z ] . We now want todescribe J = Ker(rad) . Lemma 4.10.
Let P ∈ e D . Then there exists a polynomial b P ( s ) ∈ C [ s ] such that P ( f m ) = b P ( m ) f m , rad( P ) = b P ( θ ) , P − b P ( ¯Θ) ∈ J = Ker(rad) . Proof.
Write P = P γ ∈ ˜Γ p γ E ˜ γ ( x, ∂ x ) , cf. Theorem 4.7, and define a polynomialfunction by b P ( s ) = P ˜ γ ∈ ˜Γ p ˜ γ b ˜ γ ( s, , . . . , , where b ˜ γ ∈ S ( a ) is as in Proposition 4.8.Since f m = h m ˜ λ we obtain that P ( f m ) = b P ( m ) f m . It follows that rad( P )( z m ) = b P ( m ) z m for all m ∈ N and Lemma 2.1 yields rad( P ) = b P ( θ ) . Since rad( ¯Θ) = θ we have rad( P − b P ( ¯Θ)) = 0 . (cid:3) Notice that ¯Θ ∈ e D ; for j ∈ { , . . . , r } we set Ω j = E j − b E j ( ¯Θ) ∈ J ∩ e D . (4.5)Thus we have: e D = C [ E , . . . , E r ] = C [Ω , Ω , . . . , Ω r , ¯Θ] . Recall that for j = 0 one has E = f ∆ , hence b E ( s ) = b ∗ ( s ) where b ( s ) is the b -function of f . Thus Ω = f ∆ − b ∗ ( ¯Θ) ; observe that we have already shown in§3.2 that rad( f ∆ − b ∗ ( ¯Θ)) = zδ − b ∗ ( θ ) = 0 .When V is irreducible we adopt the notation of Remark 4.9(1) and we obtain E r = ¯Θ , b E r ( s ) = s , thus Ω r = 0 . Therefore in this case one has e D = C [ ¯Θ , Ω , . . . , Ω r − ] . (4.6)The next result gives a description of Ker(rad) and another proof of Theorem 3.9in the case of MF representations. When ( ˜ G : V ) = (GL( n ) : S C n ) , part (i) ofTheorem 4.11 is proved in [36, Proposition 2.1]. Theorem 4.11.
The following properties hold. (i) D = e D [ f, ∆] = C [ E , . . . , E r ][ f, ∆] = C [Ω , . . . , Ω r ][ f, ∆ , ¯Θ] . (ii) D = (cid:0)L p ∈ N e D f p (cid:1) ⊕ (cid:0)L p ∈ N ∗ e D ∆ p (cid:1) . (iii) For k ∈ Z , set D [ k ] = (e D f k if k ≥ , e D ∆ − k if k < ;then D [ k ] = f k e D , ik k ≥ , or ∆ − k e D , if k < . (iv) R = rad( D ) = U = C [ z, δ, θ ] . (v) J = Ker(rad) = P ri =0 D Ω i = P ri =0 Ω i D .Proof. Endow D ( V ) , D and e D with the “Bernstein filtration”, i.e.: F p D ( V ) = X i + j ≤ p S i ( V ∗ ) S j ( V ) , F p D = F p D ( V ) ∩ D ⊃ F p e D = F p D ( V ) ∩ e D . Then, since ˜ G and G are reductive, e S = (cid:2) gr F D ( V ) (cid:3) ˜ G = (cid:2) S ( V ∗ ) ⊗ S ( V ) (cid:3) ˜ G , S = (cid:2) gr F D ( V ) (cid:3) G = (cid:2) S ( V ∗ ) ⊗ S ( V ) (cid:3) G . Denote by σ j = gr F ( E j ) ∈ (cid:2) V (˜ λ j ) ⊗ Y (˜ λ j ) (cid:3) ˜ G the principal symbol of E j for F . Then e S = C [ σ , . . . , σ r ] , see for example [1]. Recall that E = f ∆ , hence σ = f f ∗ . ByLemma 4.3 and Remark 4.6 we know that S ( V ∗ ) = H ( V ∗ ) ⊗ C [ f ] , S ( V ) = H ( V ) ⊗ C [ f ∗ ] , hence S = [ H ( V ∗ ) ⊗ H ( V )] G ⊗ C [ f, f ∗ ] (as vector spaces). Let γ, λ ∈ Γ ; recallthat the G -module V ( γ ) is isomorphic to E ( p ( γ )) and that p ( γ ) = p ( λ ) if and onlyif γ = λ , cf. Lemma 4.2. It follows that [ V ( γ ) ⊗ Y ( γ )] G = [ V ( γ ) ⊗ Y ( γ )] ˜ G = C E γ and [ H ( V ∗ ) ⊗ H ( V )] G = M γ ∈ Γ C E γ ⊂ e S = C [ σ , . . . , σ r ] . Thus: e S [ f, f ∗ ] ⊂ S = [ H ( V ∗ ) ⊗ H ( V )] G ⊗ C [ f, f ∗ ] ⊂ e S [ f, f ∗ ] . Since the centre C of ˜ G acts trivially on e S and via χ j , resp. χ − j , on f j , resp. ( f ∗ ) j ,we obtain: S = e S [ f, f ∗ ] = (cid:16)M j ≥ e S f i (cid:17) ⊕ (cid:16)M i> e S ( f ∗ ) i (cid:17) = C [ σ , . . . , σ r ] ⊗ C [ f, f ∗ ] . ( ⋆ ) ADIAL COMPONENTS 21
Then, by a filtration argument, one deduces that D = P p e D f p + P p e D ∆ p = e D [ f, ∆] = C [ E , . . . , E r ][ f, ∆] = C [Ω , . . . , Ω r ][ f, ∆ , ¯Θ] (recall that Ω j = E j − b E j ( ¯Θ) ). This proves (i).Observe that e D f p and f p e D , resp. e D ∆ p and ∆ p e D , are contained in the χ p -weightspace, resp. χ − p -weight space, for the action of C on D . This implies easily, asin ( ⋆ ), that these one dimensional subspaces are equal to the corresponding weightspaces. This proves (ii) and (iii).Since Ω i ∈ J = Ker(rad) , we obtain rad( D ) = C [rad( f ) , rad(∆) , rad( ¯Θ)] = C [ z, δ, θ ] , hence (iv).(v) Let P ∈ D and write P = P k P k , P k ∈ D [ k ] with P k = Q k f k or Q ′ k ∆ − k and Q k , Q ′ k ∈ e D = C [Ω , . . . , Ω r , ¯Θ] . Set: Q p = X i ≥ Q p,i ¯Θ i , Q p = X i ≥ Q ′ p,i ¯Θ i where Q p,i , Q ′ p,i ∈ C [Ω , . . . , Ω r ] . Since Q p,i ∈ Q p,i (0) + P j Ω j e D , Q ′ p,i ∈ Q ′ p,i (0) + P j Ω j e D , Q p,i (0) , Q ′ p,i (0) ∈ C , we obtain by applying rad : rad( P ) = X k ≥ (cid:16)X i ≥ Q k,i (0) θ i (cid:17) z k + X p> (cid:16)X i ≥ Q ′ p,i (0) θ i (cid:17) δ p . Recall from §2.2 that there exists a filtration on R such that gr( R ) is isomorphic to C [ X, Y, S ] / ( XY − S n ) where gr( z ) ≡ X, gr( δ ) ≡ Y, gr( θ ) ≡ S (up to some scalar).This implies easily that R = (cid:0)L k ≥ C [ θ ] z k (cid:1) ⊕ (cid:0)L p> C [ θ ] δ p (cid:1) . Now suppose that P ∈ J , then rad( P ) = 0 = P k ≥ (cid:0)P i ≥ Q k,i (0) θ i (cid:1) z k + P p> (cid:0)P i ≥ Q ′ p,i (0) θ i (cid:1) δ p forces P i ≥ Q k,i (0) θ i = P i ≥ Q ′ p,i (0) θ i = 0 , hence Q k,i (0) = Q ′ p,i (0) = 0 forall k, p, i . This shows that Q k , Q ′ p ∈ P ri =0 Ω e D + · · · + Ω r e D and therefore P ∈ P ri =0 Ω i D . Writing P k ∈ f k e D or ∆ − k e D yields P ∈ P ri =0 e D Ω + · · · + e D Ω r . (cid:3) Remark 4.12. (1) In the case where ( ˜ G : V ) is irreducible we have noticed that Ω r = 0 , see (4.6), thus J = r − X i =0 D Ω i = r − X i =0 Ω i D . (2) From [ ¯Θ , f k ] = kf k and [ ¯Θ , ∆ k ] = − k ∆ k we can deduce that D [ k ] = { D ∈ D : [ ¯Θ , D ] = kD } . A Howe duality.
The Howe duality for the Weil representation gives a bijec-tion between irreducible finite dimensional representations of
SO( n ) and irreduciblelowest weight U ( sl (2)) -modules. Algebraically, this result corresponds to the case ( ˜ G = SO( n ) × C ∗ : V = C n ) : here the Lie subalgebra of D ( V ) generated by f (quadratic form) and ∆ (Laplacian) is isomorphic to sl (2) . More precisely we havethe following result. Let A ∼ = sl (2) be this Lie algebra, denote by H d ⊂ H ( V ∗ ) thespace of harmonic polynomials of degree d , i.e.: H d = { q ∈ S d ( V ∗ ) : ∆( q ) = 0 } . Each H d is an irreducible SO( n ) -module and the A ×
SO( n ) -module C [ V ] decom-poses as C [ V ] = M d X ( d + n/ ⊗ H d where X ( d + n/ is the irreducible lowest sl (2) -module of lowest weight d + n/ .This kind of duality has been extended by H. Rubenthaler [49] to a class of PHV,the so-called commutative parabolic PHV. They are associated to short gradings s = s − ⊕ s ⊕ s on simple Lie algebras. The commutative parabolic PHV are irreducible, MF and satisfy dim V //G = 1 , thus Hypothesis B holds. We want togeneralize the Howe duality to the more general class of representations ( ˜ G : V ) satisfying only Hypothesis B. We therefore fix a representation ( ˜ G : V ) satisfyingthis hypothesis, see §4.2. We have indicated in the last column of the table ofAppendix A the irreducible MF representations ( ˜ G : V ) which are of commutativeparabolic type.Let A = Lie h f, ∆ i ⊂ (cid:0) D ( V ) , [ , ] (cid:1) (4.7)be the Lie subalgebra generated by f, ∆ . Notice that A ⊂ D . Let ˜ γ = P rj =0 a j ˜ λ j .Recall that V (˜ γ ) = U ( g ) .h ˜ γ ; we then set: a = ( a , . . . , a r ) , h ˜ γ = h a = f a h a · · · h a r r , V (˜ γ ) = V ( a ) = V ( a , . . . , a r ) . Let b ∈ N , by Lemma 4.4(b), the operator ∆ b acts as follows: ∆ b ( V ( a )) = 0 if a < b , ∆ b : V ( a , . . . , a r ) ∼−→ V ( a − b, a , . . . , a r ) if b ≤ a , and in the latter case ∆ b ( h a ) ∈ C ∗ f a − b h a · · · h a r r is a highest weight vector in V ( a − b, . . . , a r ) . Obvi-ously, the multiplication by f b gives an isomorphism f b : V ( a , . . . , a r ) ∼−→ V ( a + b, a , . . . , a r ) of ˜ G -modules. Define: A j = { D ∈ A : D ( V ( a , . . . , a r )) ⊂ V ( a + j, a , . . . , a r ) for all a ∈ N r +1 } . (4.8)It is clear that L j ∈ Z A j ⊂ A and, by the previous remarks, f b ∈ A b , ∆ b ∈ A − b . Remarks.
1) It is difficult to compute in the Lie algebra A because [∆ , f ] = ψ ( − ¯Θ)+ Q for some Q ∈ Ker(rad) which is not easy to calculate (recall that here ψ ( s ) = b ( − s ) − b ( − s − , cf. (2.6)). For example when ( ˜ G : V ) = (GL( n ) × SL( n ) : M n ( C )) P. Nang [41] has shown that (up to some scalar): Q = trace( x ∂ ) where x ,resp. ∂ , is the adjoint matrix of x = [ x ij ] ij , resp. ∂ = [ ∂ x ij ] ij . (See also [43,Proposition 7] for the case (GL(2 m ) : V C m ) .)2) When n = deg f = 2 one has A ∼ = sl (2) , thus dim C A = 3 .The assertion (2) of the next proposition should be compared with [49, Théorè-me 3.1]. Proposition 4.13. (1)
One has A k = A∩ D [ k ] , A = L k ∈ Z A k . The Lie subalgebra A is abelian. (2) Suppose n ≥ , then dim C A k = ∞ for all k ∈ Z .Proof. (1) Since f, ∆ ∈ A the relations [ ¯Θ , f ] = f and [ ¯Θ , ∆] = − ∆ show that ad( ¯Θ) : D → D induces an endomorphism of A . From A ⊂ D we deduce that A = L k ∈ Z A ∩ D [ k ] . let P ∈ D [ k ] , P = f k D or ∆ − k D be an element of A ∩ D [ k ] .Then, using Remark 4.9(3), we see that P ∈ A k . The desired equalities followeasily. Since A ⊂ e D and e D is a commutative algebra, cf. Theorem 4.7, A isabelian.(2) We claim that rad( A j ) = L j , where L j is defined as in §2.3, i.e.: L =Lie h f, ∆ i , L i = { u ∈ L : u ( z m ) ∈ C z m + j for all m ∈ N } (with the convention that C z m + j = (0) when m + j < ). Note first that rad( A ) = Lie h rad( f ) , rad(∆) i = L .Let D ∈ A j , then rad( D )( z m ) = ψ ( D ( f m )) . Observe that f m ∈ V ( m, , . . . , ,hence D ( f m ) ∈ V ( m + j, , . . . , , which is (0) is m + j < and C f m + j if m + j ≥ .Thus rad( D ) ∈ C z m + j and rad( A j ) ⊂ L j . It follows that rad( A ) = P j rad( A j ) ⊂ ⊕ j L j ⊂ L = rad( A ) . Therefore rad( A j ) = L j for all j (and L = ⊕ j L j ). Now,Proposition 2.5 yields the desired assertion. (cid:3) Set A + = L k> A k , A − = L k< A k . We then have a triangular decomposition A = A − ⊕ A ⊕ A + which enables us to introduce the notion of a lowest weight A -module, see [49]. As usual in this situation the weights will be elements of thelinear dual space A ∗ of the abelian Lie algebra A . ADIAL COMPONENTS 23
Definition 4.14.
The A -module X is a lowest weight module if there exist x ∈ X and ϕ ∈ A ∗ such that: X = U ( A ) .x , A − .x = 0 , a.x = ϕ ( a ) x for all a ∈ A .The next theorem generalizes [49, Proposition 4.2]; it gives a Howe duality forMF representations with a one dimensional quotient (see also [10, Corollary 4.5.17]).Recall that H ( V ∗ ) = L γ ∈ Γ V ( γ ) , where V ( γ ) ∼ = E ( p ( γ )) , is equal to the space ofharmonic elements, i.e. { ϕ ∈ C [ V ] : ∆ ( ϕ ) = · · · = ∆ m ( ϕ ) = 0 } , cf. (4.2) andProposition 4.5. Theorem 4.15.
Let ( ˜ G : V ) satisfying Hypothesis B. The A × g -module C [ V ] de-composes as C [ V ] ∼ = L γ ∈ Γ X ( γ ) ⊗ E ( p ( γ )) , where X ( γ ) = U ( A ) .h γ is an irreduciblelowest weight A -module. Moreover: X ( γ ) ∼ = X ( γ ′ ) ⇐⇒ γ = γ ′ .Proof. Recall that ˜Γ = N ˜ λ ⊕ Γ , Γ = ⊕ ri =1 N ˜ λ i . Let γ = P ri =1 a i ˜ λ i ∈ Γ ; set P γ = γ + N ˜ λ . Then h γ = h a · · · h a r r and V ( γ ) = U ( g ) .h γ ∼ = E ( p ( γ )) , see Lemma 4.2.From f b h γ = h ( b,a ,...,a r ) and ∆ b ( f a h a · · · h a r r ) ∈ C ∗ f a − b h a · · · h a r r when a ≥ b ,and when a < b , we get that X ( γ ) = U ( A ) .h γ = L µ ∈ P γ C h µ is an irreducible U ( A ) -module. Let D ∈ A − j = A ∩ D [ − j ] , j > , and write D = D ′ ∆ j , D ′ ∈ e D .Then D ( h γ ) = D ′ ∆ j ( h γ ) and we have noticed that ∆ j ( h γ ) = 0 , thus A − .h γ = 0 .When D ∈ A = A ∩ e D , Remark 4.9(3) gives that D ( h γ ) = ϕ ( D ) h γ ∈ C h γ . Sinceit is obvious that ϕ ∈ A ∗ , X ( γ ) is an irreducible lowest weight module.By [10, Theorem 4.5.16] we know that the D × g -module C [ V ] has the followingdecomposition: C [ V ] = M λ ∈ Λ + Hom g ( E ( λ ) , C [ V ]) ⊗ C E ( λ ) where Hom g ( E ( λ ) , C [ V ]) is either (0) or a simple D -module (the action being givenby D ( φ )( x ) = D ( φ ( x )) for all φ ∈ Hom g ( E ( λ ) , C [ V ]) , x ∈ E ( λ ) ). Since the G -module V (˜ γ ) , ˜ γ ∈ ˜Γ , is isomorphic to E ( p (˜ γ )) , we obtain that Hom g ( E ( λ ) , C [ V ]) =(0) if λ / ∈ p (Γ) = p (˜Γ) . If λ ∈ p (Γ) the non zero elements of this D -module identifywith the g -highest weight vectors of weight λ in C [ V ] through the map φ φ ( v λ ) ,where v λ is a highest weight vector in E ( λ ) . It is easily seen that the g -highestweight vectors of weight λ in C [ V ] are the h ˜ γ with ˜ γ = k ˜ λ + γ ′ , p ( γ ′ ) = λ . Recallthat for γ, γ ′ ∈ Γ , p ( γ ) = p ( γ ′ ) ⇐⇒ γ = γ ′ ; therefore these g -highest weightvectors are the h ˜ γ with ˜ γ ∈ P γ = γ + N ˜ λ , where γ ∈ Γ is such that p ( γ ) = λ .From the previous paragraph we then obtained that Hom g ( E ( λ ) , C [ V ]) ∼ = X ( γ ) as A -module. The last assertion follows from [10, Theorem 4.5.12]. (cid:3) D -modules on some PHV In this section we continue with a representation ( ˜ G : V ) of the connected re-ductive group ˜ G as in §3.1.5.1. Representations of Capelli type.
Let τ : ˜ g = Lie( ˜ G ) −→ D ( V ) be the differential of the ˜ G -action. The elements τ ( ξ ) are linear derivations on C [ V ] given by: τ ( ξ )( ϕ )( v ) = ddt | t =0 (e tξ .ϕ )( v ) = ddt | t =0 ϕ (e − tξ .v ) , for all ϕ ∈ C [ V ] , v ∈ V . They are homogeneous of degree zero in the sense that [Θ , τ ( ξ )] = 0 . The map τ yields a homomorphism τ : U (˜ g ) → D ( V ) .Recall that the group ˜ G acts naturally on D ( V ) ; the differential of this actionis given by D [ τ ( ξ ) , D ] , ξ ∈ ˜ g , D ∈ D ( V ) . Therefore, a subspace I ⊂ D ( V ) isstable under ˜ G , resp. G , if and only if [ τ (˜ g , I ] ⊂ I , resp. [ τ ( g , I ] ⊂ I . It is then clear that e D = D ( V ) ˜ G = { D ∈ D ( V ) : [ τ (˜ g ) , D ] = 0 } ⊂ D = D ( V ) G = { D ∈D ( V ) : [ τ ( g ) , D ] = 0 } . In particular, if Z (˜ g ) = U (˜ g ) ˜ G is the centre of U (˜ g ) , then τ ( Z (˜ g )) ⊂ e D .Following [18, (10.3)] we make the following definition: Definition 5.1.
We say that ( ˜ G : V ) is of Capelli type if: • ( ˜ G : V ) is irreducible and multiplicity free; • τ ( Z (˜ g )) = e D . Remarks 5.2. (1) By Howe and Umeda [18], in the list of ( ˜ G : V ) which areirreducible and MF we have: three among the thirteen cases are not of Capellitype; two among the ten cases such that dim V //G = 1 are not of Capelli type.(Thus we are interested in eight cases of the table in Appendix A.)(2) This definition originates in the case ( ˜ G = GL( n ) × SL( n ) : V = M n ( C )) where the writing of E = f ∆ = det( x ij ) det( ∂ ij ) as an element of τ ( Z (˜ g )) is the“classical” Capelli identity.Recall the morphism rad : D ( V ) G → D ( V //G ) . By definition τ ( g )( C [ V ] G ) = 0 ,thus one always has: [ D ( V ) τ ( g )] G ⊂ J = Ker(rad) . When ( ˜ G : V ) satisfies Hypothesis B, we have computed in Theorem 4.11 the ideal J ⊂ D : J = r X i =0 D Ω i = r X i =0 Ω i D where the Ω i ’s are defined in (4.5). When ( ˜ G : V ) is irreducible we observed inRemark 4.12(1) that we can number these operators so that Ω r = 0 , hence J = P r − i =0 D Ω i ; the next proposition gives a more useful description if, moreover, ( ˜ G : V ) is of Capelli type, i.e. one of the eight cases mentioned in Remark 5.2(1). Proposition 5.3.
Let ( ˜ G : V ) be of Capelli type and such that dim V //G = 1 .Then:
Ker(rad) = [ D ( V ) τ ( g )] G . Proof.
In the irreducible case the centre C of ˜ G acts by scalars on V and we mayassume that: ˜ G = GC with C = C ∗ . Write ˜ g = g ⊕ c , c = Lie( C ) = C ζ . Since C [ V ] G = C [ f ] and f is not ˜ G -invariant one can also suppose that τ ( ζ ) = ¯Θ = n Θ .Write Z (˜ g ) = Z ( g )[ ζ ] and Z ( g ) = C ⊕ Z + ( g ) where Z + ( g ) = [ U ( g ) g ] G . Theprevious paragraph implies that τ ( Z (˜ g )) = C [ ¯Θ] + τ ( Z + ( g ))[ ¯Θ] with τ ( Z + ( g )) =[ U ( τ ( g )) τ ( g )] G ⊂ [ D ( V ) τ ( g )] G . As recalled above we already know that J = P r − i =0 D Ω i , Ω i ∈ e D , ≤ i ≤ r − . By hypothesis τ ( Z (˜ g )) = e D , thus we canwrite each Ω j as follows: Ω j = X k ≥ ¯Θ k P j,k , P j,k = p j,k + P ′ j,k , p j,k ∈ C , P ′ j,k ∈ τ ( Z + ( g )) . Recall that [ D ( V ) τ ( g )] G ⊂ J ; thus we have rad( P ′ j,k ) = 0 and we obtain: rad(Ω j ) = P k ≥ θ k p j,k = 0 in R = C [ z, δ, θ ] . Therefore p j,k = 0 for all k ≥ , which gives Ω j = P k ≥ ¯Θ k P ′ j,k ∈ [ D ( V ) τ ( g )] G and J ⊂ [ D ( V ) τ ( g )] G . (cid:3) Application to D -modules. In this subsection we assume that ( ˜ G : V ) sat-isfies Hypothesis B, hence ( ˜ G : V ) is MF, C [ V ] G = C [ f ] , f / ∈ C [ V ] ˜ G . Recall from Theorem 4.11 that D = L k ∈ Z D [ k ] . We have seen (Lemma 4.10)that if D ∈ e D there exists a polynomial b D ( s ) such that D = b D ( ¯Θ) + D , D ∈ J = Ker(rad) . ADIAL COMPONENTS 25
Fix P ∈ D [ k ] and write P = DQ k , Q k = f k if k ≥ , Q k = ∆ − k if k < , D ∈ e D .Then: P = b D ( ¯Θ) Q k + P , P = D Q k ∈ J. (5.1)Observe that, since ∆( f m ) = b ∗ ( m ) f m − , P ( f m ) = ( b D ( m + k ) f m + k if k ≥ ; b ∗ ( m ) b ∗ ( m − · · · b ∗ ( m + k + 1) b D ( m + k ) f m + k if k < .Therefore if we set a P ( s ) = b D ( s + k ) if k ≥ , or a P ( s ) = b D ( s + k ) Q k +1 j =0 b ∗ ( s + j ) if k < , then deg a P = deg b P or deg b P + nk , and P ( f m ) = a P ( m ) f m + k . Notice that a Q k ( s ) = 1 if k ≥ , a Q k ( s ) = Q k +1 j =0 b ∗ ( s + j ) if k < , thus a P ( s ) = b D ( s + k ) a Q k ( s ) .When ( ˜ G : V ) = (GL( n ) : S C n ) , similar results were obtained by MasakazuMuro in [36, Proposition 3.8], where our polynomial a P ( s ) is denoted by b P ( s ) . Definition 5.4.
Let P ∈ D [ k ] be as above and define the D -module associated to P by: M P = D ( V ) (cid:14) ( D ( V ) τ ( g ) + D ( V ) P ) = D ( V ) /I P , where I P = D ( V ) τ ( g ) + D ( V ) P . Remark.
Let I ⊂ D ( V ) be a left ideal containing D ( V ) τ ( g ) . Since the condi-tion [ τ ( g ) , I ] ⊂ I is satisfied, the group G acts naturally on I and therefore on M = D ( V ) /I . Furthermore, the differential of this action is given by the left mul-tiplication on M by τ ( ξ ) , ξ ∈ g . It follows, see [17, §II.2, Theorem], that M is a G -equivariant D -module on V (cf. [17] for the definition). This is in particular truefor M P .Following [33, 34, 35, 36] we want to study the solutions of the differential systemassociated to M P . We first need to study the characteristic variety on M P .Recall that D ( V ) is filtered by the order of differential operators, see [4, 16],and that the associated graded ring of D ( V ) identifies with C [ T ∗ V ] , where T ∗ V = V × V ∗ is the cotangent bundle of V . If u ∈ D ( V ) we denote its order by ord u and itsprincipal symbol by σ ( u ) ∈ C [ T ∗ V ] = S ( V ∗ ) ⊗ S ( V ) . Let M be a finitely generated D ( V ) -module, then one can endow M with a good filtration and one defines thecharacteristic variety of M , denoted by Ch M , as being the support in T ∗ M of theassociated graded module (see, e.g., [16, §I.3]). When M = D ( V ) /I , Ch M ⊂ T ∗ M is the variety of zeroes of the symbols σ ( u ) , u ∈ I . Recall that if dim Ch M ≤ dim V the D -module M is called holonomic (one always have dim Ch M ≥ dim V if M = (0) ). Remark.
Let v ∈ V, v ∗ ∈ V ∗ , ξ ∈ ˜ g . One has: σ (Θ)( v, v ∗ ) = h v ∗ , v i , σ ( τ ( ξ ))( v, v ∗ ) = −h ξ.v ∗ , v i = h v ∗ , ξ.v i . Lemma 5.5.
Let k ∈ Z and P = DQ k be as above. (a) There exists Q ′ k ∈ J and q k ( s ) ∈ C [ s ] such that Q k Q − k = q k ( ¯Θ) + Q ′ k , ord( Q k Q − k ) = ord( Q − k Q k ) = deg q k = | k | n . (b) Write P = b D ( ¯Θ) Q k + P as in (5.1) and set P = b D ( ¯Θ) Q k . Then P Q − k = b D ( ¯Θ) q k ( ¯Θ) + P with P ∈ J and ord P ≤ ord( P Q − k ) = deg( b D q k ) .Proof. (a) Let m ∈ N . Then: Q k Q − k ( f m ) = a Q − k ( m ) Q k ( f m − k ) = a Q − k ( m ) a Q k ( m − k ) f m . Set q k ( s ) = a Q − k ( s ) a Q k ( s − k ) ∈ C [ s ] . The previous computation yields Q k Q − k = q k ( ¯Θ) + Q ′ k with Q ′ k ∈ J . Since deg a Q k = 1 or − nk (if k ≥ or < ) we get that deg q k = | k | n . On the other hand, Q k Q − k = f k ∆ − k or ∆ k f − k has order ord ∆ | k | ,i.e. | k | n = deg q k . This implies in particular that ord Q ′ k ≤ ord( Q k Q − k ) . (b) From (a) we obtain P Q − k = b D ( ¯Θ) q k ( ¯Θ) + P , P = b D ( ¯Θ) Q ′ k ∈ J . Onehas: ord( P Q − k ) = ord P + ord Q − k = deg b D + ord Q k + ord Q − k = deg b D +ord( Q k Q − k ) . Therefore, ord P = deg b D + ord Q ′ k ≤ deg b D + ord( Q k Q − k ) =ord( P Q − k ) , as desired. (cid:3) As in [45, Section 3] define the commuting varieties of ( ˜ G : V ) and ( G : V ) by: ˜ C ( V ) = (cid:8) ( v, v ∗ ) ∈ T ∗ V : h v ∗ , ˜ g .v i = 0 (cid:9) , C ( V ) = (cid:8) ( v, v ∗ ) ∈ T ∗ V : h v ∗ , g .x i = 0 (cid:9) . Recall that ( ˜ G : V ) is MF; this implies [21] that ˜ G has finitely many orbits in V , denoted by O , . . . , O t . Set T ∗ O i V = { ( v, v ∗ ) ∈ T ∗ V : v ∈ O i , h v ∗ , ˜ g .v i = 0 } . By[47], see also [45, Theorem 3.2], we have the following result: Theorem 5.6.
The irreducible components of ˜ C ( V ) are the closures of the conormalbundles of the orbits, i.e. the C i = T ∗ O i V . In particular, ˜ C ( V ) is equidimensional ofdimension dim V . Remark 5.7.
Set C ( V ) ′ = (cid:8) ( v, v ∗ ) ∈ T ∗ V : σ ( u )( v, v ∗ ) = 0 for all u ∈ D ( V ) τ ( g ) } .Thus C ( V ) ′ is the characteristic variety of the D ( V ) -module N = D ( V ) (cid:14) D ( V ) τ ( g ) (5.2)Let P ∈ D [ k ] . Then we clearly have: ˜ C ( V ) ⊂ C ( V ) , Ch M P ⊂ C ( V ) ′ ⊂ C ( V ) . We will now assume that ( ˜ G : V ) is irreducible. This means that ( ˜ G : V ) is oneof the cases (1) to (10) in the table of Appendix A. We may assume here that ˜ G = GC , C ∼ = C ∗ . We then write ˜ g = g ⊕ C ζ , c = Lie( C ) = C ζ where ζ is chosensuch that τ ( ζ ) = ¯Θ (i.e. ζ acts as n id V on V ). Corollary 5.8.
Under the previous hypothesis:– dim C ( V ) = dim V + 1 ;– the module N is not holonomic, i.e. dim C ( V ) ′ = dim V + 1 .Proof. By [21, Theorem 1] the representation ( G : V ) is visible, i.e. { v ∈ V : f ( v ) =0 } contains a finite number of G -orbits. Then, since C ( V ) G = C ( f ) and f is nonconstant, [44, Theorems 2.3 & 3.1, Corollary 2.5] yield dim C ( V ) = dim V + 1 .Recall that if M is any D ( V ) -module one can identify Hom D ( V ) ( N , M ) withthe space { x ∈ M : τ ( g ) .x = 0 } . In particular, Hom D ( V ) ( N , C [ V ]) ≡ C [ V ] G = C [ f ] . If N were holonomic we would have dim C Hom D ( V ) ( N , C [ V ]) < ∞ , cf. [32,Theorem 9.5.5], which is absurd. (cid:3) Assume that ( ˜ G : V ) is of Capelli type and let P = b D ( ¯Θ) Q k + P , P ∈ J , asin (5.1). By Proposition 5.3, I P = D ( V ) τ ( g ) + D ( V ) b D ( ¯Θ) Q k . Therefore: M ( b D , k ) = M P = D ( V ) (cid:14) ( D ( V ) τ ( g ) + D ( V ) b D ( ¯Θ) Q k ) (5.3)depends only on the polynomial b D ( s ) and the integer k . We need to know in whichcase M ( b D , k ) is holonomic. Theorem 5.9.
Assume that ( ˜ G : V ) is of Capelli type and let P = b D ( ¯Θ) Q k + P with P ∈ Ker(rad) . The following are equivalent: (i) b D ( s ) = 0 ; (ii) M ( b D , k ) = M P is holonomic.In this case Ch M P ⊂ ˜ C ( V ) is a union of C i ’s.Proof. Suppose that b D = 0 , then M ( b D , k ) = N is not holonomic. Conversely,suppose b D = 0 . We are going to show that M P ⊂ ˜ C ( V ) , then Theorem 5.6 willgive the result. ADIAL COMPONENTS 27
Set α = σ ( ¯Θ) = σ ( τ ( ζ )) ∈ C [ T ∗ V ] , hence α ( v, v ∗ ) = n h v ∗ , v i . Since ˜ g = g ⊕ C ζ ,we have ˜ C ( V ) = C ( V ) ∩ α − (0) . Using the notation of Lemma 5.5 we set P = b D ( ¯Θ) Q k , P Q − k = b D ( ¯Θ) q k ( ¯Θ) + P with P ∈ J . Notice that M P = M P . Since b D = 0 , we can write h ( s ) = b D ( s ) q k ( s ) = h d s d + h d − s d − + · · · , d = deg h ( s ) =deg b D ( s ) + deg q k ( s ) = deg b D ( s ) + | k | n ≥ . Recall that ord P ≤ ord P Q k = d ,therefore σ ( P Q k ) = σ ( h ( ¯Θ)) or σ ( h ( ¯Θ)) + σ ( P ) (depending on ord P < d or ord P = d ).If d = 0 , we get k = 0 , b D ∈ C ∗ , thus M P = (0) and the claim is obvious.Suppose d ≥ and let ( v, v ∗ ) ∈ Ch M P ⊂ Ch N = C ( V ) ′ . From P ∈ J =[ D ( V ) τ ( g )] G ⊂ D ( V ) τ ( g ) we know that σ ( P )( v, v ∗ ) = 0 . Therefore, σ ( P )( v, v ∗ ) =0 implies σ ( P )( v, v ∗ ) σ ( Q − k )( v, v ∗ ) = σ ( P Q − k )( v, v ∗ ) = σ ( h ( ¯Θ))( v, v ∗ )= h d σ ( ¯Θ) d ( v, v ∗ ) = h d α ( v, v ∗ ) d . Hence α ( v, v ∗ ) = 0 and this proves ( v, v ∗ ) ∈ C ( V ) ′ ∩ α − (0) ⊂ ˜ C ( V ) , as desired. (cid:3) Remark 5.10.
The previous result generalizes the main step in the proof of [36,Theorem 4.1] (see also [36, Theorem 6.1 and Remark 6.1]). Indeed the homogeneouselements of degree kn in [36] are the elements of D [ k ] and [36, Lemma 4.1] shows,when ( ˜ G : V ) = (GL( n ) : S C n ) , that M P is holonomic when P = 0 and deg a P =ord P . Since a P ( s ) = b D ( s + k ) a Q k ( s ) = 0 ⇐⇒ b D ( s ) = 0 , Theorem 5.9 ensuresthat a more general result holds for representations of Capelli type; notice that theexample given [36, Remark 4.1] is P = Ω = f ∆ − b ∗ ( ¯Θ) ∈ J , hence M P = N isnot holonomic. Observe also that Theorem 5.9 is proved in [33, Proposition 2.1] inthe case ( b D = 1 , k ≥ .5.3. Solutions of invariant differential equations.
We continue with a rep-resentation ( ˜ G : V ) of Capelli type. Let ( ˜ G R : V R ) be a real form of ( ˜ G : V ) inthe sense of [25, §4.1, Proposition 4.1], cf. also [33, §1.2] and [13, §4.2 & §4.3].Such a form always exists. Let M be a finitely generated D ( V ) -module of the form D ( V ) /I . Denote by B V R the space of hyperfunctions on V R (see, e.g., [13, §4.1]).Then B V R is a D ( V ) -module and the “hyperfunction solutions space” of M is: Sol( M, B V R ) = { T ∈ B V R : D.T = 0 for all D ∈ I } ≡ Hom D V ( M, B V R ) . Notice that since any distribution on V R can be viewed as a hyperfunction, the“distribution solutions space” of M is contained in Sol( M, B V R ) . An element T of B V R such that τ ( g ) .T = 0 is said to be g -invariant; it is called quasi-homogeneousif there exist t ∈ N , µ ∈ C such that (Θ − µ ) t .T = 0 . Assume that I = D ( V ) τ ( g ) + D ( V ) P for some P ∈ D ( V ) , then Sol( M, B V R ) identifies with the space of g -invarianthyperfunctions T which are solutions of the equation P.T = 0 .The next corollary has been proved by M. Muro [36, Theorem 4.1] for the realform ( ˜ G R = GL( n, R ) , V R = Sym n ( R )) of ( ˜ G : V ) = (GL( n ) : S C n ) when P = 0 and deg a P = ord P . (See Remark 5.10 for more details.) Corollary 5.11.
Let ( ˜ G : V ) be of Capelli type and let ( ˜ G R : V R ) be a real formof ( ˜ G : V ) . Let P ∈ D [ k ] and write P = b D ( ¯Θ) Q k + P , P ∈ Ker(rad) , asin (5.1) . Assume that b D = 0 . Then, Sol( P, B V R ) = Sol( M P , B V R ) = { T ∈ B V R : T g -invariant, P.T = 0 } is finite dimensional and has a basis of quasi-homogeneouselements; it depends only on the polynomial b D ( s ) and the integer k .Proof. We merely repeat the proof of M. Muro (loc. cit.). A well-known result ofM. Kashiwara (see [23, Théorème 5.1.6]) says that if M holonomic Sol( M, B V R ) is afinite dimensional C -vector space. Therefore by combining the remarks above andTheorem 5.9 we obtain that S = Sol( P, B V R ) is finite dimensional. Now observe that [Θ , P ] = kP and [Θ , τ ( g )] = 0 imply that S is stable under the action of Θ .Therefore S decomposes as a direct sum of spaces of the form Ker(Θ − µ id S ) t . Wehave noticed after (5.3) that M P depends only on b D ( s ) and k , therefore it is alsothe case for S = Hom D V ( M P , B V R ) . (cid:3) Regular holonomic modules.
Assume that ( ˜ G : V ) is of Capelli type and dim V //G = 1 .We filter D ( V ) by order of differential operators and we set D ( V ) j = { D ∈D ( V ) : ord D ≤ j } . For sake of completeness we now recall some known (or easy)results. Definition 5.12.
Let M be a finitely generated D ( V ) -module.– M is monodromic if dim C C [Θ] .x < ∞ for all x ∈ M .– M is homogeneous if there exists a Θ -stable good filtration on M , i.e. a goodfiltration F M = ( F p M ) p ∈ N such that Θ .F p M ⊂ F p M for all p .– x ∈ M is quasi-homogeneous (of weight λ ) if there exists j ∈ N and λ ∈ C suchthat (Θ − λ ) j .x = 0 ; we set M λ = S j ∈ N Ker M (Θ − λ ) j .Recall the following result proved in [39, Theorem 1.3] (which holds in the ana-lytic case). Theorem 5.13.
Let M be a homogeneous D ( V ) -module. Then: (1) M = D ( V ) .E , E finite dimensional and generated by quasi-homogeneouselements; (2) if F M = ( F p M ) p ∈ N is a Θ -stable good filtration on M , the space F p M ∩ M λ is finite dimensional for all p ∈ N , λ ∈ C . Using this result it is not difficult to obtain the next corollary.
Corollary 5.14.
Let M be a finitely generated D ( V ) -module. The following asser-tions are equivalent: (i) M is monodromic; (ii) M is homogeneous; (iii) M = D ( V ) .E , E finite dimensional such that Θ .E ⊂ E . Recall that a holonomic D ( V ) -module is regular if there exists a good filtration F M on M such that the ideal ann C [ T ∗ V ] gr F ( M ) is radical, see [24, Corollary 5.1.11].Denote by:– mod rh˜ C ( D V ) the category of regular holonomic whose characteristic variety iscontained in ˜ C ( V ) ;– mod Θ˜ C ( D V ) the category of monodromic modules with characteristic varietycontained in ˜ C ( V ) .Let G be the simply connected cover of G and set ˜ G = G × C (recall that C ∼ = C ∗ is the connected component of the centre of ˜ G ). One has: Lie( ˜ G ) =˜ g = g ⊕ c , c = C ζ , where τ ( ζ ) = ¯Θ as above. The group ˜ G maps onto G × C ,and therefore onto ˜ G = GC . It follows that ˜ G and G × C act on V ; the orbitsin V then coincide with the ˜ G -orbits O , . . . , O t . The category of ˜ G -equivariant D ( V ) -modules is denoted by mod ˜ G ( D V ) . Observe that if mod G × C ( D V ) is the category of ( G × C ) -equivariant D ( V ) -modules, any object in mod G × C ( D V ) can be naturally considered as an object of mod ˜ G ( D V ) . When G is simply con-nected, e.g. G = SL( n ) , SL( n ) × SL( n ) , SL( n ) × Sp( m ) or G , we have ˜ G = G × C and these two categories are the same. ADIAL COMPONENTS 29
Lemma 5.15.
Let M be a finitely generated D ( V ) -module. Then: (i) M ∈ mod ˜ G ( D V ) ⇐⇒ (ii) M ∈ mod rh˜ C ( D V ) = ⇒ (iii) M ∈ mod Θ˜ C ( D V ) . In particular, mod G × C ( D V ) = mod rh˜ C ( D V ) when G is simply connected.Proof. (i) ⇒ (ii): By [4, Theorem 12.11], or [17, §5], M is regular holonomic. Itscharacteristic variety Ch M is therefore a ˜ G -stable subvariety of T ∗ V . Let X bean irreducible component of Ch M ; then X is a Lagrangian conical closed ˜ G -stablesubvariety of T ∗ V and, if π : T ∗ V ։ V is the natural projection, [22, § 5, Lemme 1],implies that X = T ∗ π ( X ) reg V . But it is easy to see that π ( X ) reg (the smooth locusof π ( X ) ) is equal to O j for some ≤ j ≤ t . Hence X = C j and Ch M ⊂ ˜ C ( V ) .(ii) ⇒ (iii) and (i): (We mimic the proof of [39, Proposition 1.6].) Choose a goodfiltration such that I ( M ) = ann C [ T ∗ V ] gr F ( M ) is radical. Since Ch M ⊂ ˜ C ( V ) ,the principal symbols α = σ ( ¯Θ) and σ ( τ ( ξ )) , ξ ∈ g , belong to I ( M ) , that is tosay σ ( τ ( ξ )) gr Fj ( M ) = α gr Fj ( M ) = (0) ⊂ gr Fj +1 ( M ) ; in other words: ¯Θ .F j M =Θ .F j M ⊂ F j M and τ ( ξ ) .F j M ⊂ F j M . In particular, M is homogeneous, i.e. mon-odromic. Let x ∈ M . Since dim C [Θ] .x < ∞ there exist j ∈ M and λ , . . . , λ l suchthat x ∈ P li =1 F j M ∩ M λ i . From [ τ ( g ) , Θ] = 0 and τ ( g ) .F j M ⊂ F j M it follows that τ ( g ) .F j M ∩ M λ i ⊂ F j M ∩ M λ i ; hence U ( g ) .x is contained in the finite dimensionalspace P li =1 F j M ∩ M λ i , cf. Theorem 5.13. This shows that the action of ˜ g on M given by the τ ( ξ ) , ξ ∈ ˜ g , is locally finite. The formula e tξ .x = exp( tτ ( ξ )) .x = X k ≥ t k k ! τ ( ξ ) k .x then yields a rational action of ˜ G on M whose differential is given by multiplicationby the elements τ ( ξ ) . It remains to check that this action is compatible with theaction of ˜ G on D ( V ) , which is an easy exercise. (cid:3) Following [39, 40, 41, 42, 43] we want to describe the category mod rh˜ C ( D V ) interms of a category of modules over U = R = C [ z, δ, θ ] . A finitely generated U -module N is called monodromic if, for all v ∈ N , dim C C [ θ ] .v < ∞ . Denote by mod θ ( U ) the category of monodromic modules. Observe that N ∈ mod θ ( U ) decomposes as N = M λ ∈ C N λ , N λ = [ j ≥ Ker N ( θ − λ ) j . Recall that [ θ, z ] = z , [ θ, δ ] = − δ , zδ = b ∗ ( θ ) , δz = b ( θ ) and that the roots of b ( − s ) are λ + 1 = 1 , λ + 1 , . . . , λ n − + 1 , cf. Theorem 3.1. We then obtain: • z.N λ ⊂ N λ +1 , δ.N λ ⊂ N λ − ; • dim N λ < ∞ ( N is finitely generated); • zδ , resp. δz , is bijective on N λ if and only if λ = − λ j , resp. λ = − ( λ j + 1) ,therefore z : N λ ∼−→ N λ +1 , δ : N λ +1 ∼−→ N λ if λ = − ( λ j + 1) .From these properties it is easy to give a description of the category mod θ ( U ) interms of “finite diagrams of linear maps” as in [39, 40, 41, 43].Let M ∈ mod G × C ( D V ) . Since the differential of the G -action on M is given by x ξ.x , ξ ∈ g , x ∈ M , one has: Φ M = M G = { x ∈ M : τ ( g ) .x = 0 } . (5.4)Therefore [ D ( V ) τ ( g )] G .M = 0 and, via rad : D ( V ) G / [ D ( V ) τ ( g )] G ∼−→ U (Theo-rem 3.9), Φ M can be considered as a U -module by: u.x = D.x if u = rad( D ) , x ∈ M G . Observe also that the isomorphism rad yields a natural structure of right U -module on the module N = D ( V ) / D ( V ) τ ( g ) by: ¯ a.u = aD if ¯ a ∈ N and u = rad( D ) ∈ U . For N ∈ mod θ ( U ) we set: Ψ N = N ⊗ U N. (5.5)With these notation we have: Proposition 5.16. (1)
Let M ∈ mod G × C ( D V ) and N ∈ mod θ ( U ) , then: Φ M ∈ mod θ ( U ) , Ψ N ∈ mod G × C ( D V ) , ΦΨ N = N. (2) Suppose that any M ∈ mod G × C ( D V ) is generated by M G as a D ( V ) -module.Then the categories mod G × C ( D V ) and mod θ ( U ) are equivalent via the functors Φ and Ψ . If furthermore G is simply connected, we obtain: mod rh˜ C ( D V ) ≡ mod θ ( U ) .Proof. (1) From G reductive and M finitely generated, one deduces that the D ( V ) G -module M G is finitely generated. Recall that M is monodromic (Lemma 5.15); since θ.x = rad( ¯Θ) .x = Θ .x it follows that Φ M is monodromic. Thus Φ N ∈ mod θ ( U ) .It is clear that Ψ N is finitely generated over D ( V ) . The group G acts naturallyon D ( V ) and this action passes to N (note that D ( V ) τ ( g ) is G -stable). One easilychecks that one can endow N ⊗ U N with a rational G -module structure by setting: g. (¯ a ⊗ U x ) = g.a ⊗ U x for ¯ a ∈ N , g ∈ G, x ∈ N . Notice that since N is monodromicthe group C = exp( C ζ ) acts on N by e tζ .x = exp( tθ ) .x . One can then verify that C acts on N ⊗ U N by: e tζ . (¯ a ⊗ U x ) = e tζ . ¯ a ⊗ U exp( tθ ) .x , t ∈ C . One showswithout difficulty that this G × C -action is compatible with the ˜ G -action on D ( V ) .Moreover, with the previous notation we get that: ddt | t =0 e tξ . (¯ a ⊗ U x ) = [ τ ( ξ ) , a ] ⊗ U x = τ ( ξ ) a ⊗ U x = τ ( ξ ) . (¯ a ⊗ U x ) ,ddt | t =0 e tζ . (¯ a ⊗ U x ) = [ ¯Θ , a ] ⊗ U x + ¯ a ⊗ U θ.x = ¯Θ a ⊗ U x − a ¯Θ ⊗ u x + ¯ a ⊗ U θ.x = ¯Θ . (¯ a ⊗ U x ) − a ⊗ U θ.x + ¯ a ⊗ U θ.x = ¯Θ . (¯ a ⊗ U x ) = τ ( ζ ) . (¯ a ⊗ U x ) . This shows that Ψ N ∈ mod G × C ( D V ) . The equality (Ψ N ) G = ¯1 ⊗ U N follows easilyfrom the definition of the G -action on Ψ N , hence ΦΨ N = N .(2) Note that there is a surjective ( G × C ) -equivariant morphism of D ( V ) -modules m : ΨΦ M = N ⊗ U M G ։ M given by m (¯ a ⊗ U x ) = a.x . Set L = Ker m , hence m : (ΨΦ M ) /L ∼−→ M . Then L ∈ mod G × C ( D V ) is generated by L G , by hypothesis,and we obtain ( G is reductive): M G ∼ = (cid:0) ΨΦ M/L (cid:1) G = (ΨΦ M ) G /L G = (ΦΨΦ M ) /L G = M G /L G . This implies L G = (0) , thus L = 0 and ΨΦ M ≡ M . The last statement followsfrom Lemma 5.15 (cid:3) The previous proposition and work of P. Nang lead to the following:
Conjecture 5.17.
Let ( ˜ G : V ) be of Capelli type with dim V //G = 1 . Then thecategories mod G × C ( D V ) and mod θ ( U ) are equivalent via the functors Φ and Ψ .By Proposition 5.16, this conjecture is equivalent to showing that M = D ( V ) M G for all M ∈ mod G × C ( D V ) .P. Nang [39, 41, 43] has proved Conjecture 5.17 in the cases: (SO( n ) × C ∗ : C n ) , (GL( n ) × SL( n ) : M n ( C )) , (GL(2 m ) : V C m ) . It would be interesting toobtain a uniform proof in the eight cases where ( ˜ G : V ) is of Capelli type and dim V //G = 1 (see Appendix A). As observed above the category mod θ ( U ) has anice combinatorial description, which would give, when G is simply connected, a ADIAL COMPONENTS 31 classification of the regular holonomic modules on V whose characteristic variety iscontained in ˜ C ( V ) . T . L E VA SS E U R A ppe n d i x A . I rr e d u c i b l e M F r ep r e s e n t a t i o n s ( ˜ G : V ) deg f b ( s ) Capelli com. parabolic(1) (SO( n ) × C ∗ : C n ) 2 ( s + 1)( s + n/ yes yes(2) (GL( n ) : S C n ) n Q ni =1 ( s + ( i + 1) / yes yes(3) (GL( n ) : V C n ) , n even n/ Q n/ i =1 ( s + 2 i − yes yes(4) (GL( n ) × SL( n ) : M n ( C )) n Q ni =1 ( s + i ) yes yes(5) (Sp( n ) × GL(2) : M n, ( C )) 2 ( s + 1)( s + 2 n ) yes no(6) (SO(7) × C ∗ : spin = C ) 2 ( s + 1)( s + 4) yes no(7) (SO(9) × C ∗ : spin = C ) 2 ( s + 1)( s + 8) no no(8) (G × C ∗ : C ) 2 ( s + 1)( s + 7 / yes no(9) (E × C ∗ : C ) 3 ( s + 1)( s + 5)( s + 9) no yes(10) (GL(4) × Sp(2) : M ( C )) 4 ( s + 1)( s + 2)( s + 3)( s + 4) yes yes(3’) (GL( n ) : V C n ) , n odd — — yes no(4’) (GL( n ) × SL( m ) : M n,m ( C )) , n = m — — yes no(11) (Sp( n ) × GL(1) : C n ) — — yes no(12) (Sp( n ) × GL(3) : M n, ( C )) — — no no(10’) (GL( n ) × Sp(2) : M n, ( C )) , n = 4 — — yes no(13) (SO(10) × C ∗ : spin = C ) — — yes no ADIAL COMPONENTS 33
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