aa r X i v : . [ m a t h . R T ] S e p New family of simple gl n ( C )-modules Jonathan Nilsson
Abstract
We construct a new family of simple gl n -modules which depends on n generic param-eters. Each such module is isomorphic to the regular U ( gl n )-module when restrictedthe gl n -subalgebra naturally embedded into the top-left corner. Classification of simple modules is one of the first natural questions which arises whenstudying the representation theory of some (Lie) algebra. Simple modules are, in somesense, “building blocks” for all other modules, and hence understanding simple modulesis important. In some cases, for example for finite dimensional associative algebras,classification of simple modules is an easy problem. However, in most of the cases, theproblem of classification of all simple modules is very difficult. Thus, if we consider simple,finite dimensional, complex Lie algebras, then the only algebra for which some kind ofclassification exists is the Lie algebra sl . This was obtained by R. Block in [Bl], see alsoa detailed explanation in [Maz2, Chapter 6]. However, even in this case the “answer”only reduces the problem to classification of equivalence classes of irreducible elements ina certain non-commutative Euclidean ring.At the moment, the problem of classification of simple modules over simple Lie algebrasseems too hard. However, because of its importance, the problem of construction of newfamilies of modules attracted a lot of attention over the years. The most studied caseseem to be the one of the Virasoro Lie algebras, where many different multi-parameterfamilies of simple modules were constructed by various authors, see, for example, [OW,GLZ, LZ, LLZ, MZ1, MZ2, MW] and references therein.In contrast to the Virasoro case, the “easier” case of simple, complex, finite dimensionalLie algebras does not yet have an equally large variety of families of simple modules.So, let g be a complex, finite dimensional, simple Lie algebra. Some classes of simple g -modules are, of course, well-understood. For example: • simple finite dimensional modules are classified already by Cartan in 1913, see [Ca]; • simple highest weight modules related to a fixed triangular decomposition n − ⊕ h ⊕ n + of g are classified by their highest weights and are extensively studied during last50 years, see, for example, [Di, Hu, BGG]; • simple Whittaker modules in the sense of [Ko], see also [AP, McD1, McD2]; • simple Gelfand-Zeitlin modules, see [DFO1, DFO2, Maz1, FGR]; • simple weight modules with finite dimensional weight spaces were classified in [Mat]extending the previous work in [Fe, Fu];1 simple g -modules which are free of rank one over the universal enveloping algebraof the Cartan subalgebra were constructed and studied in [Ni1, Ni2] (see also [TZ1,TZ2] for similar modules over infinite dimensional Lie algebras).Some further classes of simple modules can be found in [FOS]. We note that the largestknown family of simple gl n -modules is the one of Gelfand-Zeitlin-modules. It depends on n ( n +1)2 generic complex parameters, see [DFO1, DFO2] for details.Based on the above, it seems natural to look for new families of simple g -modules. Thepresent paper contributes with a new large family of simple gl n -modules. This familyis parameterized by invertible n × n complex matrices. Let A , B , C , D be the four Liesubalgebras of gl n of dimension n as indicated in the following figure: (cid:18) A BC D (cid:19) . Then B is nilpotent (and even commutative), and the adjoint action of B on gl n / B isnilpotent, so ( B , gl n ) is a Whittaker pair in the sense of [BM]. The original motivationfor this paper was an attempt to describe generalized Whittaker modules (i.e. moduleson which the action of B is locally finite) for this Whittaker pair. Our main result can besummarized as follows: Theorem 1.
For each non-degenerate complex n × n -matrix Q , there exists a simple gl n module M with the following properties: • M has Gelfand-Kirillov dimension n ; • Res gl n A M is isomorphic to the left regular U ( A ) -module; • Res gl n B M is locally finite. In other words, M is a generalized Whittaker module forthe Whittaker pair ( B , gl n ) ; • With respect to a fixed PBW basis in U ( A ) , the action of each fixed element from A , B , C , D can be written explicitly as maps U ( A ) → U ( A ) of degrees , , , , re-spectively.Moreover, different matrices Q give non-isomorphic modules. The paper is organized as follows. Section 2 introduces notation and lays down some mo-tivation for the construction of our modules. In the same section, for each non-degeneratecomplex n × n -matrix Q , we construct an A + B -module having the first three propertieslisted in Theorem 1. We show that there must exist a simple quotient of the correspondinginduced gl n module that also has the fourth property. In Section 3 we explicitly constructsuch a module for Q being the identity matrix I and show that every other module in ourfamily can be obtained by twisting this module by an explicit automorphism. Finally, wegive explicit formulas for the gl n -action in all cases. Acknowledgements
I am very grateful to Volodymyr Mazorchuk for his ideas and com-ments.
Let g := gl n ( C ). All Lie algebras and vector spaces are over the complex numbers. N denotes the set of nonnegative integers. 2irst we observe that the subalgebras A and D defined above are both isomorphic to gl n while the subalgebras B and C are commutative. Let e i,j be the 2 n × n -matrix witha single 1 in position ( i, j ) and zeros elsewhere. By convention, most indices i, j etc.can be assumed to lie between 1 and n ; in particular our canonical basis for gl n will bewritten [ ≤ i,j ≤ n { e i,j , e n + i,j , e i,n + j , e n + i,n + j } . We denote the identity matrix by I , its size ( n or 2 n ) should be apparent by the context.The transpose of a matrix A is denoted A T and if A is invertible we abbreviate ( A − ) T by A − T .We also recall how to construct twisted modules . For every Lie algebra automorphism ϕ ∈ Aut ( g ) we have a twisting functor F ϕ : g -mod → g -mod which is an auto-equivalence.It maps a module M to ϕ M which is isomorphic to M as a vector space but has modifiedaction: x • v := ϕ ( x ) · v for all x ∈ g and v ∈ ϕ M . gl n Following Kostant’s idea in [Ko] we try to construct some modules on which the actionof B is locally finite.Fix Lie algebra homomorphisms λ A : A → C and λ D : D → C . Let C λ A λ D be the onedimensional ( A + C + D )-module where A acts by λ A , D acts by λ D and C acts trivially.Now define a generalized Verma module M λ A λ D := U ( gl n ) O U ( A + C + D ) C λ A λ A . Denote by M ∗ λ A λ D the full dual of M λ A λ D . This is a gl n module where the action is given( x · f )( m ) = − f ( x · m ) as usual. Proposition 2.
For every θ : B → C , there is a unique (up to multiple) eigenvector w in M ∗ λ A λ D with eigenvalue θ for B .Proof. Note that M λ A λ D ≃ U ( B ) as a left and right U ( B )-module. Let C ( θ ) be the 1-dimensional B -module where the action is given by θ . By the tensor-hom adjunction wehave Hom U ( B ) ( C ( θ ) , M ∗ λ A λ D ) = Hom U ( B ) ( C ( θ ) , Hom C (cid:0) M λ A λ D , C ) (cid:1) ≃ Hom U ( B ) ( C ( θ ) , Hom C (cid:0) U ( B ) , C ) (cid:1) ≃ Hom C ( U ( B ) ⊗ U ( B ) C ( θ ) , C ) ≃ Hom C ( C ( θ ) , C ) ≃ C . Thus there is a unique 1-dimensional subspace of M ∗ λ A λ D isomorphic to C ( θ ) in B -mod,which is equivalent to the statement in the proposition.The submodule generated by such an eigenvector must be simple (see [BM]), so we getthe following result. Corollary 3.
There exist simple generalized Whittaker modules for the pair ( B , gl n ) andthey can be realized as simple submodules in the dual of the generalized Verma module M ∗ λ A λ D . M ∗ λ A λ D is very big and inconvenient to workin. A + B -module We now turn to a more explicit construction. Note that B is commutative. Let Q = ( q ij )be a nonsingular n × n matrix and define L Q to be the 1-dimensional U ( B )-module withgenerator v where the action of B is given by Q : e i,n + j · v := q i,j v ≤ i, j ≤ n. Define an induced module M Q := Ind A + BB L Q = U ( A + B ) O U ( B ) L Q . Then M Q is clearly isomorphic to U ( A ) as a left A -module, and for a ∈ U ( A ) we shall writejust av or just a for a ⊗ v . To explicitly see how B acts on M Q , we introduce some morenotation. Consider U ( A ) ⊗ C A as a tensor product in the category of unital associativealgebras. This becomes an infinite dimensional Lie algebra under the commutator bracket.Note that U ( A ) ⊗ A ≃ M at n × n ( U ( A )) in a natural way and we shall even extend thetrace function to U ( A ) ⊗ A by defining tr ( a ⊗ B ) := a tr ( B ). Note also that A embedsinto U ( A ) ⊗ A (as both associative algebra and Lie algebra) by the map A ⊗ A , andwe shall sometimes need to identify elements of A with their images under this map. Toresolve some ambiguity in our notation, for A, B ∈ A we shall write AB for the productin U ( A ) and A.B for the product in the associative algebra A or U ( A ) ⊗ A .Let ψ ′ : A → U ( A ) ⊗ C A be the Lie algebra homomorphism defined by ψ ′ : A A ⊗ I − ⊗ A T . This extends to an algebra homomorphism ψ : U ( A ) → U ( A ) ⊗ C A . Lemma 4.
The action of B on M Q is given by (cid:18) B (cid:19) av = tr ( ψ ( a ) .Q.B T ) v. Proof.
This follows by induction on the degree of a as follows. The lemma clearly holdsfor a = 1 by the definition of the action of B on L Q : we have tr ( Q.B T ) = P ij q ij b ij .Suppose the lemma holds for all monomials a of a fixed degree (with respect to any fixed4BW basis). We then have (cid:18) B (cid:19) ( Aa ) v = A (cid:18) B (cid:19) av + (cid:2) (cid:18) B (cid:19) , (cid:18) A
00 0 (cid:19) (cid:3) av = A (cid:18) B (cid:19) av − (cid:18) A.B (cid:19) av = A tr ( ψ ( a ) .Q.B T ) v − tr ( ψ ( a ) .Q. ( A.B ) T ) v = tr (( A ⊗ I ) .ψ ( a ) .Q.B T ) v − tr ( A T .ψ ( a ) .Q.B T ) v = tr ((( A ⊗ I ) − ⊗ A T ) .ψ ( a ) .Q.B T ) v = tr ( ψ ( A ) .ψ ( a ) .Q.B T ) v = tr ( ψ ( Aa ) .Q.B T ) v. This shows that the lemma holds for all monomials in U ( A ) by induction. Since ψ islinear it holds for all of U ( A ). We proceed to prove that M Q is simple by first proving it for Q = I . Lemma 5.
The following relations hold in U ( A + B ) . [ e j,k + n , e mi,j ] = ( − m e m − i,j e i,k + n for i = j (( e i,j − m − e mi,j ) e i,k + n for i = j. Proof.
This follows easily by induction on m .Fix a PBW basis of U ( A ) of form { e l e l · · · e l n n e l · · · · · · e l n n · · · e l nn nn | l ij ∈ N } , Then U ( A ) ≃ M I has a filtration: M (0) I ⊂ M (1) I ⊂ M (2) I ⊂ · · · where M ( m ) I is the span of all monomials f with deg f := P ij l ij ≤ m . Lemma 6.
For each ≤ j, k ≤ n , the element ( e j,k + n − δ j,k ) ∈ U ( B ) has degree − withrespect to the filtration of M I . Moreover, the action on an arbitrary monomial in M ( d ) I isgiven by ( e j,k + n − δ j,k ) · e l · · · e l kj kj · · · e l nn nn = − l kj e l · · · e l kj − kj · · · e l nn nn mod M ( d − I . Proof.
We have( e j,k + n − δ j,k ) · f = f ( e j,k + n − δ j,k ) + [ e j,k + n − δ j,k , f ] = [ e j,k + n , f ] , so the fact that ( e j,k + n − δ j,k ) has degree ≤ − ad e j,k + n is a derivation. 5or the second more precise statement, let f be an arbitrary monomial of degree d . Foreach i let P i , Q i be the monomial factors of f such that f = P i e l ij ij Q i and e ij P i , Q i . Wenow calculate( e j,k + n − δ j,k ) · f = [ e j,k + n , f ] = X i P i [ e j,k + n , e l ij ij ] Q i = P j (( e jj − l jj − e l jj jj ) e j,k + n · Q j + X i = j − l ij P i e l ij − ij e i,k + n · Q i By writing e i,k + n = ( e i,k + n − δ ik ) + δ ik and using the fact that the first term has negativedegree, we see that( e j,k + n − δ j,k ) · f = δ j,k P j (( e jj − l jj − e l jj jj ) Q j + X i = j − δ ik l ij P i e l ij − ij Q i mod M ( d − I = − δ j,k l jj P j e l ij − ij Q j + X i = j − δ ik l ij P i e l ij − ij Q i mod M ( d − I = − X i δ ik l ij P i e l ij − ij Q i mod M ( d − I = − l kj P k e l kj − kj Q k mod M ( d − I . The lemma follows.
Corollary 7.
For each ≤ i, j ≤ n , the action of ( e i,j − δ i,j ) on M I is surjective. Itskernel is spanned by all monomials not divisible by e ij . Proposition 8.
The module M I is simple in U ( A + B ) -mod.Proof. It suffices to show that any f ∈ M I can be reduced to 1 ∈ M I via the B -action.Fix f ∈ M I and let p ∈ M ( d ) I be a nonzero monomial occurring in f with maximaldegree d . If p = Q ij e l ij ij (in the PBW order), it is clear by the previous lemma that B p := Q ij ( e j,n + i − δ ij ) l ij ∈ U ( B ) maps p to a nonzero constant. By the maximality of d , B p annihilates all other monomials occurring in f so in fact B p · f ∈ M (0) I is a nonzeroconstant as desired. Corollary 9.
The module M Q is simple if and only if Q is nonsingular.Proof. For each nonsingular S ∈ A , define ϕ S : A + B → A + B by ϕ S : (cid:18) A B (cid:19) (cid:18) A B.S − (cid:19) . It is easy to verify that ϕ S is a Lie algebra automorphism and that ϕ S ◦ ϕ T = ϕ ST . Itis also clear that the twisted module ϕ Q − T M I is isomorphic to M Q . Since M I is simpleby Proposition 8, and since twisting by automorphisms defines an auto-equivalence on gl n -Mod, M Q is also simple for nonsingular Q .Conversely, assume that Q is singular and let A be a nonzero matrix such that Q T A = 0.We shall show that U ( A ) Av is a proper A + B -submodule of M Q . The subspace U ( A ) Av
6s clearly A -stable. For a ∈ U ( A ) we compute (cid:18) B (cid:19) · aAv = tr ( ψ ( aA ) .Q.B T ) v = tr ( ψ ( a ) .ψ ( A ) .Q.B T ) v = tr ( ψ ( a ) . ( A ⊗ I − ⊗ A T ) .Q.B T ) v = tr ( Q.B T .ψ ( a ) . ( A ⊗ I )) v − tr ( ψ ( a ) .A T .Q.B T ) v = tr ( Q.B T .ψ ( a )) Av − tr ( ψ ( a ) . ( Q T .A ) T .B T ) v = tr ( Q.B T .ψ ( a )) Av.
Thus U ( A ) Av is also B -stable, and is thus a proper submodule of M Q . Our next goal is to prove that for most Q ’s, the module M Q is injective when restrictedto U ( B ). We begin by recalling a result about injective envelopes for the trivial moduleover polynomial rings. For a proof, see for example [L, § Lemma 10.
Let k be a field, let R = k [ x , . . . x n ] and let L be the trivial R -module. Let E be the R -module k [ x − , . . . x − n ] where x i acts by x i · ( x − k · · · x − k n n ) = ( x − k · · · x − k i +1 i · · · x − k n n if k i > otherwise.Then E = E ( L ) is the injective envelope of L . By twisting E by automorphisms we obtain injective envelopes of all 1-dimensional R -modules as follows: Corollary 11.
With notation as in the previous lemma, for scalars q i ∈ k , let L q ,...q n bethe -dimensional R -module with action x i · v = q i v . Then E ( L q ,...q n ) ≃ ϕ E ( L ) where ϕ is the R -automorphism mapping x i x i − q i .Proof. We have L q ,...q n ≃ ϕ L and since twisting by an automorphism is an auto-equivalenceon R -mod, the corollary follows. Proposition 12.
For nonsingular matrices Q , the module Res U ( A + B ) U ( B ) M Q is injective.Proof. Let I ( L Q ) be the injective envelope of L Q . Applying the exact functor Hom B ( − , I ( L Q ))to the exact sequence 0 → L Q → M Q → Coker → → Hom B ( Coker, I ( L Q )) → Hom B ( M Q , I ( L Q )) → Hom B ( L Q , I ( L Q )) → . Hence the morphism L Q → I ( L Q ) mapping L Q into its injective envelope is the imageof some morphism f : M Q → I ( L Q ). Since f is nonzero on span ( v ) = soc ( M Q ), f isinjective. Moreover, for all k ∈ N we havedim soc k ( M Q ) = (cid:18) n + k − k − (cid:19) = dim soc k ( I ( L Q )) , which shows that f is surjective. This shows that f is an isomorphism and in particularthat M Q is the injective envelope of L Q . 7 emark 13. Indecomposable injectives over noetherian rings R correspond to Spec(R)via p injective envelope of ( R/ p ) . Moreover L Q = U ( B ) / m where m is the maximalideal generated by ( e i,n + j − q i,j ) , so if M Q is injective, it must be the injective envelope of U ( B ) / m . Theorem 14.
For each nonsingular matrix n × n -matrix Q there exists a gl n -module M such that • M is generated by a single B -eigenvector with eigenvalues corresponding to the en-tries of Q . • Res U ( gl n ) U ( B ) M ≃ U ( A ) ≃ U ( gl n ) .Proof. As we’ve seen before, we take L Q as the 1-dimensional B -module corresponding to Q and we let M Q = U ( A + B ) N U ( B ) L Q . Then M Q is injective in B -mod. Next we define W := U ( A + B + D ) O U ( A + B ) M Q . Fixing d ∈ D we note that span ( v, d · v ) is a two-dimensional B -submodule of W , andmoreover it is a non-split self-extension of L Q with itself. Now by the injectivity of M Q there exists a morphism ϕ such that the following diagram commutes in B -mod: span ( v, d · v ) ϕ { { M Q L Q ?(cid:31) O O _? o o Thus there exists a d · v ∈ soc ( M Q ) = A · v such that a d · v − d · v spans a 1-dimensional B -submodule S d of W . The module W ′ := W/ P d ∈D U ( A + B + D ) S d is then isomorphicto M Q when restricted to U ( A + B ).Next, let W ′′ := U ( A + B + C + D ) N U ( A + B + D ) W ′ . For a fixed c ∈ C we have a B -submodule B ( c · v ) with simple top and simple socle, both isomorphic to L Q . By similararguments, there exists x ∈ soc ( M Q ) = A · v such that x − c · v spans a B -submodule of W ′′ . Forming the quotient of all these subs we get the module required by the theorem.In the next section we shall give explicit formulas for the elements a d and x of the proofabove in order to write down the action on the simple gl n -modules explicitly. gl n -modules The following formula will be particularly useful for m = 2. Lemma 15.
Let F := ( e j,i ) i,j = P i,j e j,i ⊗ e i,j ∈ U ( A ) ⊗ A . For any A, B ∈ gl n and forall m ∈ N we have [ A, tr ( B.F m )] = tr ([ A, B ] .F m ) in U ( gl n ) 8 roof. We proceed by induction on m . Since tr ( X.F ) = X the equality clearly holds for m = 1. The equation above is linear in both A and B so it suffices to verify it for A = e ij , B = e kl . Note that we explicitly have tr ( e ij .F m +1 ) = X ≤ r ,...,r m ≤ n e ir e r r · · · e r m j . Assume that the equality holds for some fixed m . We now compute[ e ij ,tr ( e kl .F m +1 )] = [ e ij , X r ,...,r m e kr e r r · · · e r m l ]= X r ,...,r m ([ e ij , e kr ] e r r · · · e r m l + e kr [ e ij , e r r · · · e r m l ])= X r ,...,r m ( δ jk e ir − δ r i e kj ) e r r · · · e r m l + X r e kr [ e ij , X r ,...r m e r r · · · e r m l ]= δ jk tr ( e il .F m +1 ) − e kj X r ,...,r m e ir · · · e r m l + X r e kr [ e ij , tr ( e r l .F m )]= δ jk tr ( e il .F m +1 ) − e kj tr ( e il .F m ) + X r e kr tr ([ e ij , e r l ] .F m )= δ jk tr ( e il .F m +1 ) − e kj tr ( e il .F m ) + X r e kr ( δ jr tr ( e il .F m ) − δ il tr ( e r j .F m ))= δ jk tr ( e il .F m +1 ) − e kj tr ( e il .F m ) + e kj tr ( e il .F m ) − δ il X r e kr tr ( e r j .F m ))= δ jk tr ( e il .F m +1 ) − δ il tr ( e kj .F m +1 )= tr ([ e ij , e kl ] .F m +1 ) . By induction the lemma holds.
Remark 16.
Fixing B as the identity matrix above we obtain [ A, tr ( F k )] = 0 for all A in gl n which shows that tr ( F k ) is central in U ( gl n ) . In fact, Z ( gl n ) = C [ tr ( F ) , tr ( F ) , . . . , tr ( F n )] .The elements tr ( F k ) are called Gelfand invariants. We are now ready to state our main result. Define ϕ ′ : A → U ( A ) ⊗ A by ϕ ′ : A A ⊗ I + 1 ⊗ A. This is a Lie algebra homomorphism and it extends to an algebra homomorphism ϕ : U ( A ) → U ( A ) ⊗ A . Also recall that we previously have defined ψ : U ( A ) → U ( A ) ⊗ A which satisfied ψ : A A ⊗ I − ⊗ A T for A ∈ A . Using these two homomorphisms wenow state our main theorem. Theorem 17.
Define an action of gl n on M I ≃ U ( A ) as follows: for any a ∈ U ( A ) , let (cid:18) A BC D (cid:19) · a = Aa − aD + tr ( ψ ( a ) .B T ) − tr ( ϕ ( a ) .F .C ) − tr ( ϕ ( a ) .C ) tr ( F ) . (1) This is a gl n -module structure. roof. First, for all
X, Y ∈ gl n , A ∈ A and a ∈ U ( A ) we have X · Y · Aa − Y · X · Aa == A ( X · Y · a ) + [ XY, A ] a − A ( Y · X · a ) − [ Y X, A ] a = A ( X · Y · a − Y · X · a ) + X · [ Y, A ] a − [ X, A ] · Y a − Y · [ X, A ] a − [ Y, A ] · Xa = A · [ X, Y ] a + [ X, [ Y, A ]] a + [ Y, [ A, X ]] a = A · [ X, Y ] a − [ A, [ X, Y ]] a = [ X, Y ] · Aa.
This shows that it suffices to check that X · Y · − Y · X · X, Y ] · X, Y ∈ gl n in order to prove that the formula in the theorem gives a modulestructure.We first consider the case Y := A ∈ A . We compute (cid:18) A BC D (cid:19) · (cid:18) A
00 0 (cid:19) · − (cid:18) A
00 0 (cid:19) · (cid:18) A BC D (cid:19) · AA − A D + tr (( A ⊗ I − ⊗ A T ) .B T ) − tr (( A ⊗ I + 1 ⊗ A ) .F .C ) − tr (( A ⊗ I + 1 ⊗ A ) .C ) tr ( F ) − (cid:0) A A − A D + A tr ( B T ) − A tr ( F .C ) − A tr ( C ) tr ( F ) (cid:1) = AA − A D + A tr ( B T ) + tr ( A T .B T ) − A tr ( F .C ) − tr ( A .F .C ) − A tr ( C ) tr ( F ) − tr ( A .C ) tr ( F ) − A A + A D − A tr ( B T ) + A tr ( F .C ) + A tr ( C ) tr ( F )= [ A, A ] + tr ( A T .B T ) − tr ( A .F .C ) − tr ( A .C ) tr ( F )= [ A, A ] + tr (( A .B ) T ) − tr ( F .C.A ) − tr ( C.A ) tr ( F )= (cid:18) [ A, A ] A .BC.A (cid:19) · h (cid:18) A BC D (cid:19) , (cid:18) A
00 0 (cid:19) i · . It remains to check that X · Y · − Y · X · X, Y ] · v for X, Y ∈ B , C , D . Moreover, sincethe right side of (1) is linear in A, B, C, and D it suffices to check (1) it for the standardbasis elements of gl n .When X, Y ∈ B the calculation is easy: (cid:18) B (cid:19) · (cid:18) B ′ (cid:19) · − (cid:18) B ′ (cid:19) · (cid:18) B (cid:19) · (cid:18) B (cid:19) · tr ( B ′ ) − (cid:18) B ′ (cid:19) · tr ( B )= tr ( B ) tr ( B ′ ) − tr ( B ′ ) tr ( B ) = 0 = h (cid:18) B (cid:19) , (cid:18) B ′ (cid:19) i · . X, Y ∈ D we have (cid:18) D (cid:19) · (cid:18) D ′ (cid:19) · − (cid:18) D ′ (cid:19) · (cid:18) D (cid:19) · − (cid:18) D (cid:19) · D ′ + (cid:18) D ′ (cid:19) · D = D ′ D − DD ′ = [ D ′ , D ] = (cid:18) D, D ′ ] (cid:19) · h (cid:18) D (cid:19) , (cid:18) D ′ (cid:19) i · . For X ∈ B , Y ∈ D we get (cid:18) B (cid:19) · (cid:18) D (cid:19) · − (cid:18) D (cid:19) · (cid:18) B (cid:19) · − (cid:18) B (cid:19) · D − tr ( B T ) (cid:18) D (cid:19) · − tr (( D ⊗ I − ⊗ D T ) .B T ) + tr ( B T ) D = − Dtr ( B T ) + tr ( D T .B T ) + tr ( B T ) D = tr (( D.B ) T )= (cid:18) D.B (cid:19) · h (cid:18) B (cid:19) , (cid:18) D (cid:19) i · . For X ∈ C , Y ∈ D we apply Lemma 15 for m = 1 , (cid:18) C (cid:19) · (cid:18) D (cid:19) · − (cid:18) D (cid:19) · (cid:18) C (cid:19) v · − (cid:18) C (cid:19) · D + (cid:18) D (cid:19) · ( tr ( C.F ) + tr ( C ) tr ( F ))= tr (( D ⊗ I + 1 ⊗ D ) .F .C ) + tr (( D ⊗ I + 1 ⊗ D ) .C ) tr ( F ) − ( tr ( C.F ) + tr ( C ) tr ( F )) D = D tr ( F .C ) + tr ( D.F .C ) + D tr ( C ) tr ( F ) + tr ( D.C ) tr ( F ) − ( tr ( C.F ) + tr ( C ) tr ( F )) D = [ D, tr ( C.F )] + tr ( C )[ D, tr ( F )] + tr ( D.F .C ) + tr ( D.C ) tr ( F )= tr ([ D, C ] .F )] + tr ( C.D.F ) + tr ( D.C ) tr ( F )= tr ( F .D.C )] + tr ( D.C ) tr ( F )= (cid:18) − D.C (cid:19) · h (cid:18) C (cid:19) , (cid:18) D (cid:19) i · . X ∈ B , Y ∈ C , take X = e i,n + j and Y = e n + k,l . We then have e i,n + j · e n + k,l · − e n + k,l · e i,n + j · − e i,n + j · ( tr ( e kl .F ) + tr ( e kl ) tr ( F )) − e n + k,l · tr ( e Tij )= − e i,n + j · (( n X r =1 e kr e rl ) + δ kl tr ( F )) + δ ij ( tr ( e kl .F ) + tr ( e kl ) tr ( F ))= − n X r =1 (cid:0) tr ( ψ ( e kr e rl ) .e ji ) (cid:1) − δ kl tr ( ψ ( tr ( F )) .e ji ) + δ ij ( tr ( e kl .F ) + tr ( e kl ) tr ( F ))= − n X r =1 tr (cid:0) ( e kr ⊗ I − ⊗ e rk ) . ( e rl ⊗ I − ⊗ e lr ) .e ji (cid:1) − δ kl tr (( tr ( F ) ⊗ I − ⊗ tr ( F )) .e ji )+ δ ij ( tr ( e kl .F ) + tr ( e kl ) tr ( F ))= n X r =1 (cid:16) − tr ( e ji .e rk .e lr ) + e kr tr ( e ji .e lr ) + e rl tr ( e ji .e rk ) − e kr e rl tr ( e ji ) (cid:17) + δ kl (cid:0) tr ( e ji tr ( F )) − tr ( F ) tr ( e ji ) (cid:1) + δ ij ( tr ( e kl F ) + tr ( e kl ) tr ( F ))= (cid:0) − δ kl tr ( e ji tr ( F )) + δ li e kj + δ jk e il − δ ji tr ( e kl F ) (cid:1) + δ kl δ ji − δ kl δ ji tr ( F ) + δ ij tr ( e kl F ) + δ ij δ kl tr ( F )= − δ kl δ ji + δ li e kj + δ jk e il + δ kl δ ji = δ li e kj + δ jk e il = δ jk e il − δ li e n + k,n + j = [ e i,n + j , e n + k,l ] · X, Y ∈ C . Let X = e n + i,j and Y = e n + k,l . In12his case we have e n + i,j · e n + k,l · − e n + k,l · e n + i,j · − e n + i,j · (cid:0) tr ( e kl .F ) + tr ( e kl ) tr ( F ) (cid:1) + e n + k,l · (cid:0) tr ( e ij .F ) + tr ( e ij ) tr ( F ) (cid:1) = − e n + i,j · (cid:0) n X r =1 e kr e rl + δ kl tr ( F ) (cid:1) + e n + k,l · (cid:0) n X r =1 e ir e rj + δ ij tr ( F ) (cid:1) = n X r =1 (cid:16) tr ( e ij .e kr .e rl .F ) + e kr tr ( e ij .e rl .F ) + e rl tr ( e ij .e kr .F ) + e kr e rl tr ( e ij .F )+ (cid:0) tr ( e ij .e kr .e rl ) + e kr tr ( e ij .e rl ) + e rl tr ( e ij .e kr ) + e kr e rl tr ( e ij ) (cid:1) tr ( F ) (cid:17) + δ kl (cid:0) tr ( e ij .tr ( F ) .F ) + tr ( F ) tr ( e ij .F ) + tr ( e ij .tr ( F )) tr ( F ) + tr ( F ) tr ( e ij ) tr ( F ) (cid:1)! − n X r =1 (cid:16) tr ( e kl .e ir .e rj .F ) + e ir tr ( e kl .e rj .F ) + e rj tr ( e kl .e ir .F ) + e ir e rj tr ( e kl .F )+ (cid:0) tr ( e kl .e ir .e rj ) + e ir tr ( e kl .e rj ) + e rj tr ( e kl .e ir ) + e ir e rj tr ( e kl ) (cid:1) tr ( F ) (cid:17) + δ ij (cid:0) tr ( e kl .tr ( F ) .F ) + tr ( F ) tr ( e kl .F ) + tr ( e kl .tr ( F )) tr ( F ) + tr ( F ) tr ( e kl ) tr ( F ) (cid:1)! = n tr ( e ij .e kl .F ) + e kj tr ( e il .F ) + X r e rl tr ( e ij .e kr .F ) + tr ( e kl .F ) tr ( e ij .F )+ (cid:0) n tr ( e ij .e kl ) + e kj tr ( e il ) + X r e rl tr ( e ij .e kr ) + δ ij tr ( e kl .F ) (cid:1) tr ( F )+ δ kl (cid:0) tr ( e ij .F ) + tr ( F ) tr ( e ij .F ) + δ ij tr ( F ) + δ ij tr ( F ) tr ( F ) (cid:1) − n tr ( e kl .e ij .F ) − e il tr ( e kj .F ) − X r e rj tr ( e kl .e ir .F ) − tr ( e ij .F ) tr ( e kl .F )+ (cid:0) − n tr ( e kl .e ij ) − e il tr ( e kj ) − X r e rj tr ( e kl .e ir ) − δ kl tr ( e ij .F ) (cid:1) tr ( F )+ δ ij (cid:0) − tr ( e kl .F ) − tr ( F ) tr ( e kl .F ) − δ kl tr ( F ) − δ kl tr ( F ) tr ( F ) (cid:1) = nδ jk tr ( e il .F ) + e kj tr ( e il .F ) + δ jk X r e rl tr ( e ir .F ) + tr ( e kl .F ) tr ( e ij .F )+ nδ jk δ il tr ( F ) + e kj δ il tr ( F ) + δ jk e il tr ( F ) + δ ij tr ( e kl .F ) tr ( F )+ δ kl tr ( e ij .F ) + δ kl tr ( F ) tr ( e ij .F ) + δ kl δ ij tr ( F ) + δ kl δ ij tr ( F ) − nδ li tr ( e kj .F ) − e il tr ( e kj .F ) − δ li X r e rj tr ( e kr .F ) − tr ( e ij .F ) tr ( e kl .F ) − nδ li δ kj tr ( F ) − e il δ kj tr ( F ) − δ li e kj tr ( F ) − δ kl tr ( e ij .F ) tr ( F ) − δ ij tr ( e kl .F ) − δ ij tr ( F ) tr ( e kl .F ) − δ ij δ kl tr ( F ) − δ ij δ kl tr ( F ) = δ jk X r e rl tr ( e ir .F ) − δ li X r e rj tr ( e kr .F ) + e kj tr ( e il .F )+ nδ jk tr ( e il .F ) + δ kl tr ( e ij .F ) − δ ij tr ( e kl .F ) − nδ li tr ( e kj .F ) − e il tr ( e kj .F )+ [ tr ( e kl .F ) , tr ( e ij .F )]We proceed to compute [ tr ( e kl .F ) , tr ( e ij .F )] separately.13 tr ( e kl .F ) , tr ( e ij .F )] = X r [ e kr e rl , tr ( e ij .F )]= X r (cid:16) e kr [ e rl , tr ( e ij .F )] + [ e kr , tr ( e ij .F )] e rl (cid:17) = X r (cid:16) e kr tr ([ e rl , e ij ] .F ) + tr ([ e kr , e ij ] .F ) e rl (cid:17) = X r (cid:16) δ li e kr tr ( e rj .F ) − δ jr e kr tr ( e il .F ) + δ ri tr ( e kj .F ) e rl − δ kj tr ( e ir .F ) e rl (cid:17) = δ li tr ( e kj .F ) − e kj tr ( e il .F ) + tr ( e kj .F ) e il − δ kj tr ( e il .F )Inserting this into the previous expression gives= δ jk X r e rl tr ( e ir .F ) − δ li X r e rj tr ( e kr .F ) + e kj tr ( e il .F )+ nδ jk tr ( e il .F ) + δ kl tr ( e ij .F ) − δ ij tr ( e kl .F ) − nδ li tr ( e kj .F ) − e il tr ( e kj .F )+ δ li tr ( e kj .F ) − e kj tr ( e il .F ) + tr ( e kj .F ) e il − δ kj tr ( e il .F )= δ jk X r (cid:0) tr ( e ir .F ) e rl + [ e rl , tr ( e ir .F )] (cid:1) − δ li X r (cid:0) tr ( e kr .F ) e rj + [ e rj , tr ( e kr .F )] (cid:1) + nδ jk tr ( e il .F ) + δ kl tr ( e ij .F ) − δ ij tr ( e kl .F ) − nδ li tr ( e kj .F )+ δ li tr ( e kj .F ) + [ tr ( e kj .F ) , e il ] − δ kj tr ( e il .F )= δ jk tr ( e il .F ) + δ jk X r tr ([ e rl , e ir ] .F ) − δ li tr ( e kj .F ) − δ li X r tr ([ e rj , e kr ] .F )+ nδ jk tr ( e il .F ) + δ kl tr ( e ij .F ) − δ ij tr ( e kl .F ) − nδ li tr ( e kj .F )+ δ li tr ( e kj .F ) + tr ([ e kj , e il ] .F ) − δ kj tr ( e il .F )= δ jk (cid:0) δ li tr ( tr ( F ) .F ) − n tr ( e il .F ) (cid:1) − δ li (cid:0) δ jk tr ( tr ( F ) .F ) − n tr ( e kj .F ) (cid:1) + nδ jk tr ( e il .F ) + δ kl tr ( e ij .F ) − δ ij tr ( e kl .F ) − nδ li tr ( e kj .F )+ δ ij tr ( e kl .F ) − δ lk tr ( e ij .F )= δ jk δ li tr ( F ) − δ li δ jk tr ( F ) = 0 = [ e n + i,j , e n + k,l ] · Theorem 18.
Define an action of gl n on M Q ≃ U ( A ) as follows: for any a ∈ U ( A ) , let (cid:18) A BC D (cid:19) · a = Aa − aD + tr ( ψ ( a ) .Q.B T ) − tr ( ϕ ( a ) .F .Q − T .C ) − tr ( ϕ ( a ) .Q − T .C ) tr ( F ) . This is a gl n -module structure.Proof. For each nonsingular S ∈ M at n × n , define ϕ S : gl n → gl n by ϕ S : (cid:18) A BC D (cid:19) (cid:18) A B.S − S.C S.D.S − (cid:19) . It is easy to verify that ϕ S is a Lie algebra automorphism and that ϕ S ◦ ϕ T = ϕ S.T , so themap Ξ :
M at n × n ( C ) ∗ → Aut ( gl n ) with S ϕ S is an injective algebra homomorphism.Let V be the gl n module from in Theorem 17. Now by the action of gl n on the twistedmodule V Q := ϕ Q − T V is precisely as in the statement of this theorem.14he modules V Q now satisfy the conditions of Theorem 1 in the introduction: Proof. (of Theorem 1) The module V Q is simple since Res gl n A + B V Q ≃ M Q is. That theGK-dimension is n and that Res gl n A V Q ≃ U ( A ) follows directly from the definition inTheorem 18. Since the linear maps tr ( ψ ( − ) .B T ) : U ( A ) → U ( A ) never increases thedegree of a monomial, the module Res gl n B V Q is locally finite. The fourth point followsfrom similar arguments: the maps tr ( ψ ( − ) .F .C ) : U ( A ) → U ( A ) have degree 2 and themaps A ( − ) and ( − ) D clearly have degree 1 (compare with Theorem 18). Finally, we notethat any isomorphism ϕ : V Q → V Q ′ must map the generator of V Q to a multiple of thegenerator of V Q ′ . But then q ′ ij ϕ (1) = e i,n + j ϕ (1) = ϕ ( e i,n + j ·
1) = q ij ϕ (1), showing that Q = Q ′ whenever such an isomorphism exists. Since the automorphisms ϕ and ψ themselves are not very explicit, we present another for-mula for how elements of gl n act on monomials of U ( A ). We need some more conventionsin notation for this formula.In the argument of the trace functions, any product is by convention to be taken in M at n × n ( U ( gl n )) (in particular we identify A with M at n × n ( C ) here). Outside the tracefunction all products are in U ( gl n ). When S ⊂ Z , the product Q i ∈ S A i means that theproduct is to be taken in order inherited from Z . For example, Q i ∈{ , , } A i = A A A .For S ⊂ { , . . . , k } , we denote by S ∗ the complement { , . . . , k } \ S and by | S | thecardinality of S . Theorem 19.
Let a = Q ki =1 A i be a monomial in V Q (see Theorem 18). The action of gl n on the monomial a can be written explicitly as follows. (cid:18) A BC D (cid:19) · k Y i =1 A i := A k Y i =1 A i − k Y i =1 A i ( Q − T .D.Q T )+ X S ⊂{ ,...,k } (cid:16) Y i ∈ S ∗ A i (cid:17)(cid:16) ( − | S | tr ( B T . Y i ∈ S A Ti .Q ) − tr ( Q − T .C. Y i ∈ S A i .F ) − tr ( Q − T .C. Y i ∈ S A i ) tr ( F ) (cid:17) Proof.
This follows by induction on k by comparing with the formula in Theorem 18. Theverification is omitted here. References [AP] D. Arnal, G. Pinczon.
On algebraically irreducible representations of the Lie algebra sl (2) . J. Math. Phys. (1974) 350–359.[BGG] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand. A certain category of g -modules .Funkcional. Anal. i Prilozen. , 1–8.[Bl] R. Block. The irreducible representations of the Lie algebra sl (2) and of the Weylalgebra . Adv. in Math. (1981), no. 1, 69–110.[BM] P. Batra, V. Mazorchuk. Blocks and modules for Whittaker pairs . J. Pure Appl.Algebra (2011), no. 7, 1552–1568.[Ca] E. Cartan.
Les groupes projectifs qui ne laissent invariante aucune multiplicit´eplanet.
Bull. Soc. Math. France vol. (1913) pp. 53–96.15Di] J. Dixmier. Enveloping Algebras.
American Mathematical Society, 1977.[DFO1] Yu. Drozd, S. Ovsienko, V. Futorny.
On Gelfand-Zetlin modules . Proceedingsof the Winter School on Geometry and Physics (Srni, 1990). Rend. Circ. Mat.Palermo (2) Suppl. No. (1991), 143–147.[DFO2] Yu. Drozd, S. Ovsienko, V. Futorny. Harish-Chandra subalgebras andGel’fandZetlin modules.
In: Finite-Dimensional Algebras and Related Topics (Ot-tawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. , KluwerAcad. Publ., Dordrecht, 1994, 79–93.[Fe] S. Fernando.
Lie algebra modules with finite dimensional weight spaces, I.
Trans.Amer. Mas. Soc., (1990), 757–781.[FGR] V. Futorny, D. Grantcharov, L.E. Ramirez.
Irreducible Generic Gelfand???TsetlinModules of gl ( n ). SIGMA Symmetry Integrability Geom. Methods Appl. (2015),Paper 018, 13 pp.[FOS] V. Futorny, S. Ovsienko, M. Saor´ın. Torsion theories induced from commutativesubalgebras . J. Pure Appl. Algebra (2011), no. 12, 2937–2948.[Fu] V. Futorny.
Weight representations of semisimple finite dimensional Lie algebras .Ph. D. Thesis, Kiev University, 1987.[GLZ] X.Guo, R. Lu, K. Zhao.
Irreducible modules over the Virasoro algebra.
Doc. Math. (2011), 709–721.[Hu] J. E. Humphreys. Representations of Semisimple Lie Algebras in the BGG Category O . American Mathematical Society, 2008.[Ko] B. Kostant.
On Whittaker vectors and representation theory . Invent. Math. Lectures on Modules and Rings.
Graduate Texts in Math., Vol. 189,Springer-Verlag, 1999.[LLZ] G. Liu, R. Lu, K. Zhao.
A class of simple weight Virasoro modules.
J. Algebra (2015), 506–521.[LZ] R. Lu, K. Zhao.
Irreducible Virasoro modules from irreducible Weyl modules.
J.Algebra (2014), 271–287.[Mat] O. Mathieu.
Classification of irreducible weight modules , Ann. Inst. Fourier (Greno-ble) (2000), no. 2, 537–592.[Maz1] V. Mazorchuk. On Gelfand-Zetlin modules over orthogonal Lie algebras.
AlgebraColloq. (2001), no. 3, 345–360.[Maz2] V. Mazorchuk. Lectures on sl ( C ) -modules . Imperial College Press, London, 2010.[MW] V. Mazorchuk, E. Wiesner. Simple Virasoro modules induced from codimensionone subalgebras of the positive part.
Proc. Amer. Math. Soc. (2014), no. 11,3695–3703.[MZ1] V. Mazorchuk, K. Zhao.
Classification of simple weight Virasoro modules with afinite-dimensional weight space.
J. Algebra (2007), no. 1, 209–214.[MZ2] V. Mazorchuk, K. Zhao.
Simple Virasoro modules which are locally finite over apositive part.
Selecta Math. (N.S.) (2014), no. 3, 839–854.16McD1] E. McDowell. On modules induced from Whittaker modules.
J. Algebra (1985),161–177.[McD2] E. McDowell. A module induced from a Whittaker module.
Proc. Amer. Math.Soc. (1993), 349–354.[Ni1] J. Nilsson.
Simple sl n +1 -module structures on U ( h ) . J. Algebra (2015), 294–329.[Ni2] J. Nilsson. U ( h ) -free modules and coherent families. Preprint arXiv:1501.03091.[OW] M. Ondrus, E. Wiesner.
Whittaker modules for the Virasoro algebra.
J. AlgebraAppl. (2009), no. 3, 363–377.[TZ1] H. Tan, K. Zhao. Irreducible modules over Witt algebras W n and over sl n +1 ( C ).Preprint arXiv:1312.5539.[TZ2] H. Tan, K. Zhao. W + n and W n -module structures on U ( h ) . J. Algebra,424