The F-method and a branching problem for generalized Verma modules associated to $({\LieGtwo},{so(7)})$
aa r X i v : . [ m a t h . R T ] M a r The F-method and a branching problem forgeneralized Verma modules associated to(Lie G , so (7)) Todor Milev and Petr Somberg
Abstract
The branching problem for a couple of non-compatible Lie algebrasand their parabolic subalgebras applied to generalized Verma moduleswas recently discussed in [15]. In the present article, we employ the re-cently developed F-method, [10], [11] to the couple of non-compatible Liealgebras Lie G i ֒ → so (7), and generalized conformal so (7)-Verma modulesof scalar type. As a result, we classify the i (Lie G ) ∩ p -singular vectorsfor this class of so (7)-modules. Key words:
Generalized Verma modules, Conformal geometry in dimension5, Exceptional Lie algebra Lie G , F-method, Branching problem. MSC classification:
The subject of our article has its motivation in the Lie theory problem of branch-ing rules for finite dimensional simple Lie algebras and composition structure ofgeneralized Verma modules, and dually in the geometrical problem related to theconstruction of invariant differential operators in parabolic invariant theories.We assume that g , g ′ are complex semisimple Lie algebras and i : g ′ ֒ → g is an injective homomorphism. Then i ( g ′ ) is (complex) reductive in g and wecan choose Borel subalgebras b ′ ⊂ g ′ and b ⊂ g such that i ( b ′ ) ⊂ b . Let p ⊃ b be a parabolic subalgebra of g . Let M gp ( V λ ) be the generalized Verma g -module induced from the irreducible finite dimensional p -module V λ withhighest weight λ . We define the branching problem for M gp ( V λ ) over g ′ to bethe problem of finding all b ′ -singular vectors in M gp ( V λ ), that is, the set of allvectors annihilated by image of the nilradical of b ′ on which the image of theCartan subalgebra of b ′ has diagonal action.In the recent article [15], under certain technical assumptions, we proved that M gp ( V λ ) has (finite or infinite) Jordan-H¨older series over g ′ , and enumerated the b ′ -highest weights µ appearing in the series. We also computed the dimension m ( µ, λ ) of the vector space of b ′ -highest weights of weight µ as a function of µ λ . Further we gave a procedure for producing explicit formulas for some(but not all) b ′ -highest weight vectors.As an example, we discussed Lie G i ֒ → so (7). Restricting our attention tothe parabolic subalgebra p ≃ p (1 , , and the 6 infinite families of highest weights xε , xε + ω , xε + ω , xε + 2 ω , xε + ω + ω , xε + 2 ω we computed in [15]all b ′ -singular vectors with b ′ -dominant weights. From the theory of generalizedVerma modules we know that, depending on the integrality and dominance ofthe b -highest weight, M gp ( V λ ) has b -singular (and therefore b ′ -singular) vectorsother than the highest weight vector. Therefore these vectors give additional b ′ -singular vectors whose weights are not b ′ -dominant (and are not computedin [15]).Fix the pair Lie G i ֒ → so (7) and fix the parabolic subalgebra to be theparabolic subalgebra p (1 , , ⊂ so (7) obtained by crossing out the first (long)root of so (7). Let p ′ (1 , = i − (Lie G ) . In the present article, for the family of so (7)-highest weights of the form xε ,we prove that if x ∈ {− / , − / , / , . . . } , the module M gp (1 , , ( V xε ) has,besides its highest weight vector, exactly one p ′ (1 , -singular vector, and has no p ′ (1 , -singular vectors otherwise. Here we recall that, for an arbitrary parabolicsubalgebra p ′ , a p ′ -singular vector is defined as a vector that is annihilated byall elements of the Levi part of p ′ , and therefore has weight that projects to zeroonto the Levi part of p ′ (“weight of scalar type”). Our result has a somewhatunusually sounding consequence: the p ′ (1 , -singular vector in M so (7) p (1 , , ( V xε )must automatically be p (1 , , -singular. This fact must necessarily fail to gen-eralize for sufficiently large values of a, b and highest weights of the form λ = xω + aω + bω . Indeed, the number of p (1 , , -singular vectors in M so (7) p (1 , , ( V λ )is uniformly bounded, while the number m ( xω , xω + aω + bω ) grows as alinear function of a and b .We would like to note that our example goes beyond the compatible couplesof Lie algebras discussed in [10], [11].A geometric motivation for the branching problem can be described as fol-lows. Let G, G ′ be the connected and simply connected Lie groups with Liealgebras g , g ′ . Let P be the parabolic subgroup of G with Lie algebra p , andlet L ⊂ P be its Levi factor. Then there is a well-known equivalence betweeninvariant differential operators acting on induced representations and homomor-phisms of generalized Verma modules, realized by the natural pairing Ind GP ( V λ ( L ) ∗ ) × M gp ( V λ ) −→ C , (1)where V λ ( L ) denotes the finite-dimensional irreducible L -module, V λ ( L ) ∗ is itsdual, and Ind GP denotes induction from P to G . As a consequence, the singularvectors constructed in the article determine invariant differential operators act-ing between induced representations of i ( G ′ ). It is quite interesting to constructthese invariant differential operators, in particular their curved extensions aslifts to homomorphisms of semiholonomic generalized Verma modules.2ur motivation for the particular example Lie G i ֒ → so (7) comes from anatural problem in conformal geometry of dimension 5 (note that so (7) is thecomplexification of the conformal Lie algebra in dimension 5), see [4] and refer-ences therein. A geometrical characterization of the reduction of the structuregroup with Lie algebra so (7) down to Lie G for a given inducing representa-tion V λ is then given by invariant differential operators acting on sections of theassociated vector bundles, intertwined by actions of so (7) and Lie G .The structure of the article is as follows. In Section 2 we recall basic conven-tions on so (7) , Lie G , i (Lie G ) and the structure of their parabolic subalgebrasrelative to the embedding i . In Section 3, we use (1) to transform the problem offinding differential invariants for ( so (7) , Lie G ) , V λ into an algebraic questionabout homomorphisms between generalized Verma modules, corresponding tosolutions of the branching problem. In Section 4 we fix the conformal parabolicsubalgebra to be p (1 , , ⊂ so (7). Therefore by Lemma 2.1 the subalgebra p ′ is given by i ( p ′ ) = i ( g ′ ) ∩ p and therefore equals the subalgebra p ′ (1 , obtainedby crossing out the first root of Lie G . We note that p ′ (1 , is not compatible( g , p ). We further fix the highest weight to be λε (here we use λ as a scalar).We then apply the distribution Fourier transform (the “F-method”) developedin [10], [11] to obtain our main result Theorem 4.2. Lie G i ֒ → so (7) In the present section we introduce the Lie theoretic conventions for the complexLie algebra so (7), exceptional Lie algebra Lie G , and Levi resp. parabolicsubalgebras p of so (7) relative to parabolic subalgebras i ( p ′ ) of i (Lie G ). Thesewill be used in the subsequent Section 3, where we employ the F-method. Formore detailed review, cf. [15].We start by fixing a Chevalley-Weyl basis of the Lie algebra so (2 n + 1). Letthe defining vector space V of so (2 n + 1) have a basis e , . . . e n , e , e − , . . . e − n ,where the defining symmetric bilinear form B of so (2 n +1) is given by B ( e i , e j ) :=0 , i = − j , B ( e i , e − i ) := 1, B ( e i , e ) := 0, B ( e , e ) := 1, or alternatively definedas an element of S ( V ∗ ), B := n X i = − n e ∗ i ⊗ e ∗− i = ( e ∗ ) + 2 n X i =1 e ∗ i e ∗− i , (2)under the identification v ∗ w ∗ := ( v ∗ ⊗ w ∗ + w ∗ ⊗ v ∗ ).In the basis e , . . . e n , e , e − , . . . e − n , the matrices of the elements of so (2 n +3) are of the form A v ... v n C = − C T w . . . w n − v . . . − v n D = − D T − w ... − w n − A T , i.e., all matrices C such that A t B + BA = 0. We fix e ∗ , . . . e ∗ n , e ∗ , e ∗− , . . . e ∗− n tobe basis of V ∗ dual to e , . . . e n , e , e − , . . . e − n . We identify elements of End ( V )with elements of V ⊗ V ∗ . In turn, we identify elements of End ( V ) with theirmatrices in the basis e , . . . , e n , e , e − , . . . , e − n .Fix the Cartan subalgebra h of so (2 n + 1) to be the subalgebra of diagonalmatrices, i.e., the subalgebra spanned by the vectors e i ⊗ e ∗ i − e − i ⊗ e ∗− i . Thenthe basis vectors e , . . . e n , e , e − , . . . e − n are a basis for the h -weight vectordecomposition of V . Let the h -weight of e i , i >
0, be ε i . Then the h -weight of e − i , i > − ε i , and an h -weight decomposition of so (2 n + 1) is given by theelements g ε i − ε j := e i ⊗ e ∗ j − e − j ⊗ e ∗− i , g ± ( ε i + ε j ) := e ± i ⊗ e ∗∓ j − e ± j ⊗ e ∗∓ i and g ± ε i := √ (cid:0) e ± i ⊗ e ∗ − e ⊗ e ∗∓ i (cid:1) , where i, j > h• , •i g on h ∗ by h ε i , ε j i g = 1 if i = j andzero otherwise.The root system of so (2 n + 1) with respect to h is given by ∆( g ) := ∆ + ( g ) ∪ ∆ − ( g ), where we define∆ + ( g ) := { ε i ± ε j | ≤ i < j ≤ n } ∪ { ε i | ≤ i ≤ n } (3)and ∆ − ( g ) := − ∆ + ( g ). We fix the Borel subalgebra b of so (2 n + 1) to be thesubalgebra spanned by h and the elements g α , α ∈ ∆ + ( g ). The simple positiveroots corresponding to b are then given by η := ε − ε , . . . , η n − := ε n − − ε n , η n := ε n . For the remainder of this Section we fix the odd orthogonal Lie algebra tobe so (7). We order the 18 roots of so (7) in graded lexicographic order withrespect to their simple basis coordinates. We then label the negative roots bythe indices − , . . . , − , . . . ,
9. Finally,we abbreviate the Chevalley-Weyl generator g α ∈ so (7) by g i , where i is thelabel of the corresponding root. For example, g ± = g ± ( ε − ε ) , g ± = g ± ( ε − ε ) , g ± = g ± ( ε ) are the simple positive and negative generators, the element g − = g − ε − ε is the Chevalley-Weyl generator corresponding to the lowest root, andso on. We furthermore set h := [ g , g − ], h := [ g , g − ], h := 1 / g , g − ].Let now g ′ = Lie G . One way of defining the positive root system of Lie G is by setting it to be the set of vectors∆( g ′ ) := {± (1 , , ± (0 , , ± (1 , , ± (1 , , ± (1 , , ± (2 , } . (4)4 α Figure 1: The root system of Lie G We set α := (1 ,
0) and α := (0 , h• , •i g ′ on h ′ ,proportional to the one induced by Killing form by setting (cid:18) h α , α i g ′ h α , α i g ′ h α , α i g ′ h α , α i g ′ (cid:19) := (cid:18) − − (cid:19) . (5)In an h• , •i g ′ -orthogonal basis the root system of Lie G is drawn in Figure 1.Similarly to the so (7) case, we order the 12 roots of Lie G in the gradedlexicographic order with respect to their simple basis coordinates, and label theroots with the indices − , . . . , −
1, 1 , . . . ,
6. We fix a basis for the Lie algebraLie G by giving a set of Chevalley-Weyl generators g ′ i , i ∈ {± , · · · ± } , andby setting h ′ := [ g ′ , g ′− ], h ′ := 3[ g ′ , g ′− ]. Just as in the so (7) case, we ask thatthe generator g ′± i correspond to the root space labeled by ± i .All embeddings Lie G i ֒ → so (7) are conjugate over C . One such embeddingis given via i ( g ′± ) := g ± , i ( g ′± ) := g ± + g ± . As g ′± , g ′± generate Lie G , the preceding data determines the map i and onecan directly check it is a Lie algebra homomorphism. Alternatively, we can use i ( g ′± ) , i ( g ′± ) to generate a Lie subalgebra of so (7), verify that this subalgebrais indeed 14-dimensional and simple, and finally use this 14-dimensional imageto compute the structure constants of Lie G .We denote by ω := ε , ω := ε + ε and ω := ( ε + ε + ε ) thefundamental weights of so (7) and by ψ := 2 α + α , ψ := 3 α + 2 α thefundamental weights of Lie G .Let pr : h ∗ → h ′∗ be the map naturally induced by i . Thenpr( ε − ε | {z } η ) = pr( ε |{z} η ) = α , pr( ε − ε | {z } η ) = α , (6)or equivalently pr( ω ) = pr( ω ) = ψ , pr( ω ) = ψ . Conversely, ι : h ′∗ → h ∗ is the map ι ( α ) = 3 η = 3 ε − ε , ι ( α ) = η + 2 η = ε − ε + 2 ε . (7)According to the usual convention, to an arbitrary subset of the simple positiveroots of so (7) (“crossed-out” roots) we assign a parabolic subalgebra by request-ing that the crossed out root spaces lie outside of the Levi part of p . In turn,5e parametrize the subsets of the simple positive roots of so (7) by triples of 0’sand 1’s with 1 standing for “crossed-out” root. Finally, we index the parabolicsubalgebra by the corresponding triples of 0’s and 1’s. For example, by p (1 , , we denote the parabolic subalgebra of so (7) whose Levi part has roots ± ε .Define the four parabolic subalgebra b ′ ≃ p ′ (1 , , p ′ (1 , , p ′ (0 , , p ′ (0 , ≃ Lie G ofLie G in analogous fashion.We recall from [15] that the pairwise inclusions between the parabolic sub-algebras of so (7) and the embeddings of the parabolic subalgebras of Lie G aregiven as follows. Lemma 2.1
For the pair G i ֒ → so (7) , let h , b , p denote Cartan, Borel andparabolic subalgebras of so (7) and h ′ , b ′ , p ′ denote Cartan, Borel and parabolicsubalgebras of Lie G with the assumptions that i ( h ′ ) ⊂ h ⊂ b , i ( b ′ ) ⊂ b ⊂ p , b ′ ⊂ p ′ . Then we have the following inclusion diagram for the possible values of p , p ′ . p (0 , , ≃ so (7) p (1 , , qqqqqqqqqqqq p (0 , , O O p (0 , , f f ▼▼▼▼▼▼▼▼▼▼▼▼ p ′ (0 , ≃ Lie G k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ p (1 , , O O qqqqqqqqqqqq p (1 , , f f ▼▼▼▼▼▼▼▼▼▼▼▼ qqqqqqqqqqqq p (0 , , f f ▼▼▼▼▼▼▼▼▼▼▼▼ O O p ′ (0 , f f ▼▼▼▼▼▼▼▼▼▼ l l ❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨❨ p (1 , , ≃ b O O qqqqqqqqqqq f f ▼▼▼▼▼▼▼▼▼▼▼ p ′ (1 , k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ O O p ′ (1 , ≃ b ′ O O k k ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ A A ✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄✄ The arrows in the diagram indicate the inclusions between the correspondingparabolic subalgebras. In addition, if an arrow is drawn between the parabolicsubalgebra p ′ of Lie G and a parabolic subalgebra p of so (7) , then p ′ = i − ( i ( g ′ ) ∩ p ) . The structure of so (7) as a module over the Levi part of parabolic subalgebrasof Lie G is described in detail in [15, Lemma 5.2] (the lemma is too large torecall here) and we will implicitly use it throughout Section 4.Note that the conformal parabolic subalgebra p (1 , , ⊂ so (7) and the parabolicsubalgebra p ′ (1 , ⊂ Lie G are not compatible.6 Branching problem and the F-method (alge-braic distribution Fourier transformation)
In the present section we briefly review the F-method developed in [10], [11].It is based on the analytical tool of algebraic Fourier transformation on thecommutative nilradical n of p , which allows to find singular vectors in gener-alized Verma modules exploiting the algebraic Fourier transform and classicalinvariant theory. The method converts a problem in the universal envelopingalgebra of a Lie algebra into a system of partial or ordinary special differentialequations acting on a polynomial ring. In examples known to us, the conversionto partial differential equations yields a lot more tractable problem than thestarting universal enveloping algebra one.Let ˜ G be a connected real reductive Lie group with the Lie algebra ˜ g , ˜ P ⊂ ˜ G a parabolic subgroup and ˜ p its Lie algebra, ˜ p = ˜ l ⊕ ˜ n the Levi decompositionof ˜ p and ˜ n − its opposite nilradical, ˜ g = ˜ n − ⊕ ˜ p . The corresponding Lie groupsare denoted ˜ N − , ˜ L, ˜ N . Let p denote the fibration p : ˜ G → ˜ G/ ˜ P and let ˜ M := p ( ˜ N − · ˜ P ) denote the big Schubert cell of ˜ G/ ˜ P . Then the exponential map˜ n − → ˜ M , X exp( X ) · o ∈ ˜ G/ ˜ P , o := e · ˜ P ∈ ˜ G/ ˜ P , e ∈ ˜ G. gives the canonical identification of the vector space n − with ˜ M .Given a complex finite dimensional ˜ P -module V (in the present section wedo not indicate explicitly its highest weight), let Ind ˜ G ˜ P ( V ) denote the space ofsmooth sections of the homogeneous vector bundle ˜ G × ˜ P V → ˜ G/ ˜ P , i.e., Ind ˜ G ˜ P ( V ) = C ∞ ( ˜ G, V ) ˜ P := { f ∈ C ∞ ( ˜ G, V ) | f ( g · p ) = p − · f ( g ) , g ∈ ˜ G, p ∈ ˜ P } . Let ˜ π denote the induced representation of ˜ G on Ind ˜ G ˜ P ( V ).Let U (˜ g C ) denote the universal enveloping algebra of the complexified Liealgebra ˜ g C . Let V ∨ be the dual (contragredient) representation to V . Thegeneralized Verma module M ˜ g ˜ p ( V ∨ ) is defined by M ˜ g ˜ p ( V ∨ ) := U (˜ g ) ⊗ U (˜ p ) V ∨ , and there is a (˜ g , ˜ P )-invariant natural pairing between Ind ˜ G ˜ P ( V ) and M ˜ g ˜ p ( V ∨ ),described as follows. Let D ′ ( ˜ G/ ˜ P ) ⊗ V ∨ be the space of all distributions on˜ G/ ˜ P with values in V ∨ . The evaluation defines a canonical equivariant pairingbetween Ind ˜ G ˜ P ( V ) and D ′ ( ˜ G/ ˜ P ) ⊗ V ∨ , and this restricts to the pairingInd ˜ G ˜ P ( V ) × D ′ [ o ] ( ˜ G/ ˜ P ) ⊗ V ∨ → C , (8)where D ′ ( ˜ G/ ˜ P ) [ o ] ⊗ V ∨ denotes the space of distributions supported at the basepoint o ∈ ˜ G/ ˜ P .
As shown in [1], the space D ′ [ o ] ( ˜ G/ ˜ P ) ⊗ V ∨ can be identified, asan U (˜ g )-module, with the generalized Verma module M ˜ g ˜ p ( V ∨ ) . V and V ′ of ˜ P , the space of˜ G -equivariant differential operators from Ind ˜ G ˜ P ( V ) to Ind ˜ G ˜ P ( V ′ ) is isomorphicto the space of (˜ g , ˜ P )-homomorphisms between M ˜ g ˜ p ( V ′ ∨ ) and M ˜ g ˜ p ( V ∨ ). Thehomomorphisms of generalized Verma modules are determined by their singularvectors, and the F-method translates the problem of finding singular vectors tothe study of distributions on ˜ G/ ˜ P supported at the origin, and consequentlyto the problem of finding the solution space for a system of partial differentialequations acting on polynomials Pol(˜ n ) on ˜ n .The representation ˜ π of ˜ G on Ind ˜ G ˜ P ( V ) has the infinitesimal representation d ˜ π of ˜ g C . In the non-compact case, ˜ π acts on functions on the big Schubertcell ˜ n − ≃ ˜ M ⊂ ˜ G/ ˜ P with values in V . The latter representation space canbe identified via the exponential map with C ∞ (˜ n − , V ). The action d ˜ π ( Z ) ofelements Z ∈ ˜ n on C ∞ (˜ n − , V ) is realized by vector fields on ˜ n − with coefficientsin Pol(˜ n − ) ⊗ End V , see [12].By the Poincar´e-Birkhoff-Witt theorem, the generalized Verma module M ˜ g ˜ p ( V ∨ )is isomorphic to U (˜ n − ) ⊗ V ∨ ≃ Diff ˜ N − (˜ n − ) ⊗ V ∨ as an ˜ l -module. In the specialcase when ˜ n − is commutative, Diff ˜ N − (˜ n − ) is the space of holomorphic differ-ential operators on ˜ n − with constant coefficients regarded as a subspace of theWeyl algebra Diff(˜ n − ) of algebraic differential operators on ˜ n − . Moreover, theoperators d ˜ π ∨ ( X ) , X ∈ ˜ g , are realized as differential operators on ˜ n − withcoefficients in End( V ∨ ) . The application of Fourier transform on ˜ n − gives theidentification of the generalized Verma module Diff ˜ N − (˜ n − ) ⊗ V ∨ with the spacePol(˜ n ) ⊗ V ∨ , and the action d ˜ π ∨ of ˜ g on Diff ˜ N − (˜ n − ) ⊗ V ∨ translates to theaction ( d ˜ π ∨ ) F of ˜ g on Pol(˜ n ) ⊗ V ∨ and is realized again by differential operatorswith values in End( V ∨ ). The explicit form of ( d ˜ π ∨ ) F ( X ) is easy to computeby Fourier transform from the explicit form of d ˜ π ∨ . The previous framework can be applied to any pair of couples ˜ P ⊂ ˜ G and˜ P ′ ⊂ ˜ G ′ of Lie groups for which ˜ G ′ ⊂ ˜ G is a reductive subgroup of ˜ G and˜ P ′ = ˜ P ∩ ˜ G ′ is a parabolic subgroup of ˜ G ′ . The Lie algebras of ˜ G ′ , ˜ P ′ aredenoted by ˜ g ′ , ˜ p ′ . In this case, ˜ n ′ := ˜ n ∩ ˜ g ′ is the nilradical of ˜ p ′ , and ˜ L ′ =˜ L ∩ ˜ G ′ is the Levi subgroup of ˜ P ′ . We are interested in the branching problemfor generalized Verma modules M ˜ g ˜ p ( V ∨ ) over, ˜ g , i.e., in the structure of therestriction of M ˜ g ˜ p ( V ∨ ) to ˜ g ′ . Definition 3.1
Let V be an irreducible ˜ P -module. Define the ˜ L ′ -module M ˜ g ˜ p ( V ∨ ) ˜ n ′ := { v ∈ M ˜ g ˜ p ( V ∨ ) | dπ ∨ ( Z ) v = 0 for all Z ∈ ˜ n ′ } . (9)The set M ˜ g ˜ p ( V ∨ ) ˜ n ′ is a completely reducible ˜ l ′ -module. Note that for ˜ G =˜ G ′ , M ˜ g ˜ p ( V ∨ ) ˜ n ′ is necessarily finite-dimensional. However for ˜ G = ˜ G ′ , the set M ˜ g ˜ p ( V ∨ ) n ′ will in general (but not necessarily, as illustrated in the next section)be infinite dimensional. An irreducible ˜ L ′ -submodule W ∨ of M ˜ g ˜ p ( V ∨ ) ˜ n ′ givesan injective U (˜ g ′ )-homomorphism from M ˜ g ′ ˜ p ′ ( W ∨ ) to M ˜ g ˜ p ( V ∨ ). Dually, we get8n equivariant differential operator acting from Ind ˜ G ˜ P ( V ) to Ind ˜ G ′ ˜ P ′ ( W ).Using the F-method, the space of ˜ L ′ -singular vectors M ˜ g ˜ p ( V ∨ ) ˜ n ′ is realizedin the ring of polynomials on ˜ n valued in V ∨ and equipped with the action ofthe Lie algebra via ( d ˜ π ∨ ) F . Definition 3.2
We define
Sol (˜ g , ˜ g ′ , V ∨ ) := { f ∈ Pol(˜ n ) ⊗ V ∨ | ( d ˜ π ∨ ) F ( Z ) f = 0 for all Z ∈ ˜ n ′ } . (10)Then the inverse Fourier transform gives an ˜ L ′ -isomorphism Sol (˜ g , ˜ g ′ ; V ∨ ) ∼ → M ˜ g ˜ p ( V ∨ ) ˜ n ′ . (11)An explicit form of the action ( d ˜ π ∨ ) F ( Z ) leads to a system of differential equa-tion for elements in Sol. The transition from M ˜ g ˜ p ( V ∨ ) ˜ n ′ to Sol transforms theproblem of computation of singular vectors in generalized Verma modules intoa system of partial differential equations.In the dual language of differential operators acting on principal series rep-resentation, the set of ˜ G ′ -intertwining differential operators from Ind ˜ G ˜ P ( V ) toInd ˜ G ′ ˜ P ′ ( V ′ ) is in bijective correspondence with the space of all (˜ g ′ , ˜ P ′ )-homomorphismsfrom M ˜ g ′ ˜ p ′ ( V ′ ∨ ) to M ˜ g ˜ p ( V ∨ ) . Lie G ∩ p ′ -singular vectors in the so (7) -generalizedVerma modules of scalar type for the confor-mal parabolic subalgebra In this subsection we determine the i (Lie G ) ∩ p -singular vectors in the familyof ˜ g = so (7) generalized Verma modules M so (7) p (1 , , ( C λ ) induced from character χ λ : ˜ p → C of the weight λε ( ε is the first fundamental weight of so (7)). Inthis way, the results computed in the present section are analytic counterpartrealized by F-method of the algebraic results developed in [15].Denote by v λ the highest weight vector of the generalized Verma so (7)-module M so (7) p (1 , , ( V λ ). Note that as i ( h ′ ) = 3 h = 3 h ε − ε , i ( h ′ ) = h + 2 h = h ε − ε + 2 h ε , h µ, α i = 0 and h µ, α i = λ , we have that the h ′ -weight of v λ is µ = λ ( α + 2 α ).Let n − denote the nilradical opposite to the nilradical of the parabolic sub-algebra p . Then n − is commutative, U ( n − ) ⊗ V ∨ ≃ Pol (cid:18) ∂∂x , . . . , ∂∂x (cid:19) ⊗ C λ and the variables ∂∂x , . . . , ∂∂x denote the following so (7)-root space generators: ∂∂x := g − ε + ε = g − , ∂∂x := g − ε − ε = g − , ∂∂x := g − ε = g − , ∂∂x := g − ε + ε = g − , ∂∂x := g − ε − ε = g − . x i , ∂∂x j ] = − [ ∂∂x j , x i ] = (cid:26) i = j − i = j is the adjointaction of the differential operator x i on the differential operator ∂∂x j .By Lemma 2.1, the simple part of the Levi factor of i ( p ′ ) is isomorphic to sl (2) and its action on n − can be extended to action on U ( n − ) ≃ S ⋆ ( n − ). Theelements h := h , e := g , f := g − give the standard h, e, f -basis of sl (2), i.e.,[ e, f ] = h, [ h, e ] = 2 e, [ h, f ] = − f . Then the action of h on n − is the adjointaction of x ∂∂x + x ∂∂x − x ∂∂x − x ∂∂x , the action of e is the adjoint actionof x ∂∂x − x ∂∂x and the action of f is the adjoint action of − x ∂∂x + x ∂∂x .We now proceed to generate all l ′ -invariant singular vectors in M so (7) p (1 , , ( C λ ),i.e., the singular vectors that induce i (Lie G )-generalized Verma modules in-duced from character (scalar generalized Verma modules). To do that we needthe following lemma from classical invariant theory of reductive Lie algebras. Lemma 4.1
Then the sl (2) -invariants of S ⋆ ( n − ) are an associative algebragenerated by the elements u := ∂∂x ∂∂x + ∂∂x ∂∂x = g − g − + g − g − u := ∂∂x = g − . (12) Proof.
Direct computation shows that u , u are invariants. Alternatively, asthe direct sum of two two-dimensional sl (2)-modules gives a natural embedding sl (2) ֒ → sl (2) × sl (2), we can view u as the invariant element induced by thedefining symmetric bilinear form of so (4) ≃ sl (2) × sl (2). Let the positive rootof sl (2) be η , and the multiplicity of the sl (2)-module with highest weight t η in S l ( n − ) be b ( l, t ). Denoting by x, z a couple of formal variables, we have that P l ∈ Z ≥ ,t ∈ Z ≥ b ( l, t )( z l x t + z l x − − t ) is the power series expansion of the rationalfunction (1 − x − ) 1(1 − zx ) − zx − ) − z ) . Direct computation shows that b ( l, t ) equals − / t + 1 + 1 / tl + 1 / l + 1 / t whenever l + t is even and − / t + 1 / / tl + 1 / l whenever l + t is odd, and l , t satisfy the inequalities l ≥ t ≥
0. Finally, substituting with t = 0, we get b ( l,
0) = 1+ l/ l and b ( l,
0) = 1 / l/
2. For a fixed l , this is exactly thedimension of the vector space generated by the linearly independent invariants u q u r ∈ S l ( n − ) with r + 2 q = l , which completes the proof of our Lemma. (cid:3) η is of course the projection of long Lie G -root α = pr( ε − ε ) from the dual ofthe two-dimensional Cartan subalgebra of Lie G to the dual of the one-dimensional Cartansubalgebra of a long-root sl (2)-subalgebra of Lie G i it follows thatad( i ( g ′ )) = − x ∂∂x + x ∂∂x , ad( i ( g ′− )) = − x ∂∂x + x ∂∂x ,
13 ad( i ( h ′ )) = ad( h ) = [ad( i ( g ′ )) , ad( i ( g ′− ))]= x ∂∂x + x ∂∂x − x ∂∂x − x ∂∂x , ad( i ( h ′ )) = − x ∂∂x + x ∂∂x + 3 x ∂∂x + 2 x ∂∂x , and thereforead( i (2 h ′ + h ′ )) = x ∂∂x + 3 x ∂∂x + 2 x ∂∂x + 3 x ∂∂x + x ∂∂x (13)represents the central element of the Levi factor i ( l ′ ). Its action therefore nat-urally induces a grading gr on the Weyl algebra of n − in the variables (cid:26) x , x , x , x , x , ∂∂x , ∂∂x , ∂∂x , ∂∂x , ∂∂x (cid:27) , via − gr ( x ) = gr (cid:16) ∂∂x (cid:17) = − , − gr ( x ) = gr (cid:16) ∂∂x (cid:17) = − , − gr( x ) = gr (cid:16) ∂∂x (cid:17) = − , − gr( x ) = gr (cid:16) ∂∂x (cid:17) = − , − gr ( x ) = gr (cid:16) ∂∂x (cid:17) = − . (14)In particular, we get that the invariants u = ∂∂x ∂∂x + ∂∂x ∂∂x and u = (cid:16) ∂∂x (cid:17) are homogeneous with respect to the gr-grading.Let ξ , . . . ξ be formal variables, Fourier-dual with respect to x , . . . , x n . Let ∂ := ∂∂ξ , . . . , ∂ := ∂∂ξ , denote the derivatives in the ξ i -variables. We recall that the distributive Fouriertransform F maps the Weyl algebra generated by x , . . . , x n , ∂∂x , . . . , ∂∂x to theWeyl algebra generated by ∂ , . . . , ∂ , ξ , . . . , ξ via F ( x i ) := ∂ i F (cid:18) ∂∂x i (cid:19) := ξ i . As the Fourier transform is a Lie algebra homomorphism, by Lemma 4.1 thesubalgebra of l ′ s = sl (2)-invariants with respect to the Fourier dual representa-tion is the polynomial ring P ol [ ξ ξ + ξ ξ , ξ ]. Theorem 4.2
Let v λ be the highest weight vector of the so (7) -generalized Vermamodule M so (7) p (1 , , ( C λ ) induced from character χ λ , λ ∈ C . Let N ∈ N be a positive nteger and A i ∈ C , i ∈ N a collection of complex numbers such that at leastone of them is non-zero. Let u · v λ := N X k =0 A k u k u N − k · v λ , (15) where u , u are given by (12) .1. A vector u · v λ is i (Lie G ) ∩ p -singular (“singular vector of scalar type”)of homogeneity N if and only if λ = N − / and u = (2 u + u ) N =(2 u + u ) λ +5 / .2. M so (7) p (1 , , ( C λ ) has no i (Lie G ) ∩ p -singular vector of homogeneity N + 1 .3. A vector v ∈ M so (7) p (1 , , ( C λ ) , not proportional to v λ , is so (7) ∩ p -singular ifand only if λ = N − / and v = u · v λ is the vector given in 1. Proof.
1. By Lemma 4.1 and Section 2 a p ′ -singular vector must be polynomialin u and u and therefore a homogeneous p ′ -singular vector of homogeneity 2 N must be of the form (15).First we determine the action of the second simple positive root g in theFourier dual representation d ˜ π (ad( i ( g ′ ))), acting on P ol [ ξ , . . . , ξ ].Let n i be non-negative integers. Then i ( g ′ ) · ( ξ n ξ n ξ n ξ n ξ n · v λ ) = (cid:0) ( − n + n ) ξ n − ξ n ξ n ξ n ξ n − n ξ n ξ n − ξ n +13 ξ n ξ n + n λξ n − ξ n ξ n ξ n ξ n + ( n − n ) ξ n ξ n ξ n − ξ n +14 ξ n +2 n ξ n ξ n ξ n − ξ n ξ n +15 − n n ξ n − ξ n ξ n ξ n ξ n + n n ξ n ξ n − ξ n ξ n +14 ξ n − − n n ξ n − ξ n ξ n ξ n ξ n − n n ξ n − ξ n ξ n ξ n ξ n (cid:1) · v λ = ( − ξ ∂ − ξ ∂ + λ∂ + ξ ∂ + 2 ξ ∂ − ξ ∂ ∂ + ξ ∂ ∂ − ξ ∂ ∂ − ξ ∂ ∂ ) · ( ξ n ξ n ξ n ξ n ξ n ) · v λ , (16)Let P ( λ ) denote the differential operator on C [ ξ , ξ , ξ , ξ , ξ ] obtained in thefollowing computation:( − ξ ∂ − ξ ∂ + λ∂ + ξ ∂ + 2 ξ ∂ − ξ ∂ ∂ + ξ ∂ ∂ − ξ ∂ ∂ − ξ ∂ ∂ )= ( − ξ ∂ + ξ ∂ + 2 ξ ∂ + ( − ξ ∂ + ξ ∂ ) ∂ − ( ξ ∂ + ξ ∂ + ξ ∂ − λ ) ∂ )= ( − ξ ∂ + ξ ∂ + 2 ξ ∂ + ∂ ( − ξ ∂ + ξ ∂ ) − ( ξ ∂ + ξ ∂ + ξ ∂ − λ − ∂ ) .
12e compute ∂ · ( u b u b ) = b ξ u b − u b ,∂ · ( u b u b ) = b ξ u b − u b , ( ξ ∂ + ξ ∂ ) · ( u b u b ) = b u b u b ,∂ · ( u b u b ) = 2 b ξ u b u b − ,∂ · ( u b u b ) = 2 b (2 b − u b u b − , and so ( − ξ ∂ + ξ ∂ + 2 ξ ∂ + ∂ ( − ξ ∂ + ξ ∂ ) − ( ξ ∂ + ξ ∂ + ξ ∂ − λ − ∂ ) · ( u b u b )= ( − ξ ∂ + ξ ∂ + 2 ξ ∂ − ( ξ ∂ + ξ ∂ + ξ ∂ − λ − ∂ ) · ( u b u b )= − b ξ ξ u b − u b + 2 b (2 b − ξ u b u b − + 4 b ξ ξ u b u b − − ( ξ ∂ + ξ ∂ + ξ ∂ − λ − · ( b ξ u b − u b )= − b ξ ξ u b − u b + 2 b (2 b − ξ u b u b − + 4 b ξ ξ u b u b − +( − b + 1 + λ + 1 − b ) b ξ u b − u b = 2 b ((2 b − ξ + 2 ξ ξ ) u b u b − + b (( − b − b + λ + 2) ξ − ξ ξ ) u b − u b . (17)The operator P ( λ ) is homogeneous with respect to the grading in (14), andits application to a homogeneous polynomial in u = u ( ξ , . . . , ξ ), u = u ( ξ , . . . , ξ ) yields P ( λ )( P Nk =0 A k u k u N − k )= P Nk =0 A k (2( N − k )((2( N − k ) − ξ + 2 ξ ξ ) u k u N − k − + k (( − k − N − k ) + λ + 2) ξ − ξ ξ ) u k − u N − k )= P N +1 s =1 A s − ( N − ( s − N − ( s − − ξ +2 ξ ξ ) u ( s − u N − ( s − − + P Nk =0 kA k (( − k − N − k ) + λ + 2) ξ − ξ ξ ) u k − u N − k = P Ns =1 (2 A s − ( N − s + 1)((2 N − s + 1) ξ + 2 ξ ξ )+ sA s (( s − N + λ + 2) ξ − ξ ξ )) u s − u N − s ) . The 2 N summands of the form ξ u s − u N − s and ξ ξ u s − u N − s are linearlyindependent and therefore the above sum is zero if and only if2 A s − ( N − s + 1)((2 N − s + 1) ξ + 2 ξ ξ )+ sA s (( s − N + λ + 2) ξ − ξ ξ )) (18)equals zero for all values of s . When s = N , the above sum becomes2 A N − ( ξ + 2 ξ ξ ) + N A N (( − N + λ + 2) ξ − ξ ξ ) . It is a straightforward check that if A N vanishes, then A N − , A N − , . . . mustalso vanish; therefore we may assume A N = 0. The vanishing of the coefficient13n front of ξ implies A N − = − N A N ( − N + λ + 2) and in turn, the vanishingof the coefficient in front of ξ ξ implies − N − λ = 0. Therefore λ = N − / . Substituting λ back into (18), we get2 A s − ( N − s + 1)((2 N − s + 1) ξ + 2 ξ ξ )+ sA s (( − N + s − / ξ − ξ ξ ) = 0 . This implies A s = N − s +1) s A s − = · · · = 4 s (cid:0) Ns (cid:1) A , which completes the proofof 1).2. A homogeneous i (Lie G ) ∩ p -singular vector is, in particular, sl (2) ≃ i ([ l ′ , l ′ ])-singular and by Lemma 4.1 must be of the form u = ξ P Nk =0 A k u k u N − k .The application of 2 ξ ∂ converts A N ( ξ ξ + ξ ξ ) N ξ into 2 A N ( ξ ξ + ξ ξ ) N ξ .Furthermore 2 A N ( ξ ξ + ξ ξ ) N ξ contains in its binomial expansion 2 A N ( ξ ξ ) N ξ .Direct check shows that the action of P ( λ ) on ( ξ ξ + ξ ξ ) N − i ξ i for i > ξ ξ ) N ξ . This implies that A N = 0 and byinduction, the polynomial is trivial. Consequently, there is no nontrivial oddhomogeneity polynomial solving the differential equation P ( λ ).As an illustration, for N = 0 we have P ( λ )( A ξ ) = 2 A ξ . This vanishesprovided A = 0, which implies the polynomial is trivial.3. An so (7) ∩ p -singular vector must be i (Lie G ) ∩ p -singular. As the gradingelement from (13) maps i (Lie G ) ∩ p -singular to i (Lie G ) ∩ p -singular vectors,it quickly follows that an i (Lie G ) ∩ p -singular vector is a linear combinationof gr-homogeneous elements (see (14)). From the explicit form of u and u itimmediately follows that a homogeneous i (Lie G ) ∩ p -singular vector is of theform (15).From 1) we know that, other than v λ , there is at most one more homogeneous i (Lie G ) ∩ p -singular vector, and thus the vector (15) is the only candidate for a so (7) ∩ p -singular vector. The simple part of l is isomorphic to so (5) and inducesthe quadratic form with matrix in the coordinates ξ , . . . , ξ Q = , i.e., the metric of the form g ( ξ , ξ , ξ , ξ , ξ ) = ( dξ ) + 2( dξ ⊗ dξ + dξ ⊗ dξ ) + 2( dξ ⊗ dξ + dξ ⊗ dξ ) . The Fourier transform of the so (5)-invariant Laplace operator associated to Q is F ( △ ξ ) = Q ( ξ , ξ , ξ , ξ , ξ ) = 4( ξ ξ + ξ ξ ) + ξ . △ ξ and the binomial formula for (4( ξ ξ + ξ ξ ) + ξ ) s , we see thatthe Lie G ∩ p -singular vector constructed 1) is indeed so (7) ∩ p -singular. Theproof is complete. (cid:3) Remark.
As noted in the proof of 3) every i (Lie G ) ∩ p -singular is a linearcombination of homogeneous i (Lie G ) ∩ p -singular vectors, and therefore The-orem 4.2, 1) and 2) give all i (Lie G ) ∩ p -singular vectors (namely, the linearcombinations of v λ and the vector given by (15)).We note that an alternative proof of Theorem 4.2, 3) can be given as fol-lows. From a well known example (see e.g., [3], [10], [11]) of singular vectors inconformal geometry of dimension 5 describing conformally invariant powers ofthe Laplace operator, we know that for λ ∈ {− / , − / , / , . . . } there existsone so (7) ∩ p -singular vector in M so (7) p (1 , , ( C λ ). On the other hand points 1) and2) of Theorem 4.2 present us with only one such candidate, so that candidatemust be the so (7) ∩ p -singular vector in question.For λ ∈ {− / − / , / , . . . } , the h -weight of the so (7) ∩ p -singular vectorin M so (7) p (1 , , ( C λ ) given by Theorem 4.2 equals ( λ − N ) ε = ( λ − λ + 5 / ε =( − λ − ε . Therefore the vector from Theorem 4.2 corresponds to the homo-morphism of generalized Verma modules M so (7) p (1 , , ( C − λ − ) ֒ → M so (7) p (1 , , ( C λ ) . (19)In an analogous fashion we conclude that Theorem 4.2 gives a homomorphismof generalized Verma modules M Lie G p ′ (1 , ( C ( − λ − ψ ) ֒ → M Lie G p ′ (1 , ( C λψ ) . (20)We note that the existence of the above homomorphisms was proved in [14].We conclude this article by the following. Proposition 4.3
The homomorphisms (19) , (20) are standard. Proof. [2, Chapter 7] implies that a (non-generalized) Verma module M gb ( C µ )lies in a (non-generalized) Verma module M gb ( C ν ) if and only if there ex-ists a sequence of roots α , . . . , α k such that s α k . . . s α ( µ + ρ ) − ρ = ν and s α j +1 . . . s α ( µ + ρ ) − s α j . . . s α ( µ + ρ ) is a positive integer multiple of α j +1 forall j . Here, s α i denotes reflection in the root α i and ρ is the half-sum of thepositive roots.Computation using the above criterion shows that, for λ = − / , − / , / , . . . ,we have that M so (7) b ( C − λ − ) ⊂ M so (7) b ( C λ ) (21)and M Lie G b ′ ( C ( − λ − ψ ) ⊂ M Lie G b ′ ( C λψ ) . (22)Computation furthermore shows that M so (7) b ( C − λ − ) * M so (7) b ( C µ ) (23)15or any µ = λ of the form w ( λ + ρ ) − ρ , where w is in the Weyl group of so (7)and ρ is the half-sum of the positive roots of so (7). Similarly, M Lie G b ′ ( C − λ − ) * M Lie G b ′ ( C µ ) (24)for any µ = λψ of the form w ( λψ + ρ ′ ) − ρ ′ , where w is in the Weyl group ofLie G and ρ ′ is the half-sum of the positive roots of Lie G .(21), (22), (23), (24), together with [13, Proposition 3.3] now imply thatthe standard homomorphism maps from M so (7) p (1 , , ( C − λ − ) to M so (7) p (1 , , ( C λ ) andfrom M Lie G p ′ (1 , ( C ( − λ − ψ ) to M Lie G p ′ (1 , ( C λψ ) are non-zero. On the other hand,our main Theorem 4.2 shows that there is a unique b ′ -singular vector of weight( − − λ ) ψ in M Lie G p ′ (1 , ( C λψ ) and therefore also a unique b -singular vector ofweight ( − − λ ) ε in M so (7) p (1 , , ( C λ ). Therefore the homomorphisms (20), (19)are standard. (cid:3) Acknowledgment.
The authors gratefully acknowledge the support by theCzech Grant Agency through the grant GA CR P 201/12/G028.We would also like to thank Toshihisa Kubo for discovering an error andsuggesting the correction to an earlier version of Proposition 4.3.
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