Irreducible representations of Yangians
aa r X i v : . [ m a t h . R T ] S e p Irreducible representations of Yangians
Sergey Khoroshkin , , Maxim Nazarov and Paolo Papi Institute for Theoretical and Experimental Physics, Moscow 117259, Russia Department of Mathematics, Higher School of Economics, Moscow 117312, Russia Department of Mathematics, University of York, York YO10 5DD, England Dipartimento di Matematica, Sapienza Universit`a di Roma, Roma 00185, Italy
0. Introduction
Take the general linear Lie algebra gl n over the complex field C and considerthe corresponding polynomial current Lie algebra gl n [ x ] . The Yangian Y( gl n )is a deformation of the universal enveloping algebra U( gl n [ x ]) in the class ofHopf algebras. Irreducible finite-dimensional representations of the associativealgebra Y( gl n ) were classified by Drinfeld [D2]. In Subsections 1.1 and 1.2 of thepresent article we recall this classification, together with the original definitionof the Yangian Y( gl n ) coming from the theory of quantum integrable systems.The additive group C acts on Y( gl n ) by Hopf algebra automorphisms, see(1.2) for the definition of automorphism corresponding to the element t ∈ C .The associative algebra Y( gl n ) contains U( gl n ) as a subalgebra, and admits ahomomorphism onto U( gl n ) identical on this subalgebra; see (1.4). By pullingthe representation of U( gl n ) on an exterior power of C n back through thehomomorphism (1.4), and then back through any automorphism (1.2), we getan irreducible representation of Y( gl n ) called fundamental .The work [D2] provided only a parametrization of all irreducible finite-dimensional representations of Y( gl n ) . The results of Akasaka and Kashiwara[AK] showed that up to an automorphism of Y( gl n ) of the form (1.3), each ofthese representations arises as a quotient of a tensor product of fundamentalones; see also the earlier works of Chari and Pressley [CP] and Cherednik [C2].This fact can also be derived from the results of Nazarov and Tarasov [NT1];further results were obtained by Chari [C]. Note that the works [AK] and [C]dealt with representations of quantum affine algebras. For connections to therepresentation theory of Yangians see the work of Molev, Tolstoy and Zhang[MTZ] and the more recent work of Gautam and Toledano-Laredo [GT].Now let g n be one of the two Lie algebras so n , sp n . Regard g n as theLie subalgebra of gl n preserving a non-degenerate bilinear form on the vectorspace C n , symmetric in the case g n = so n , or alternating in the case g n = sp n .For any n × n matrix X let e X be the conjugate of X relative to this form.As an associative algebra, the twisted Yangian Y( g n ) is a deformation of theuniversal enveloping algebra of the twisted polynomial current Lie algebra Khoroshkin, Nazarov and Papi { X ( x ) ∈ gl n [ x ] : X ( − x ) = − e X ( x ) } . This is not a Hopf algebra deformation. However, Y( g n ) is a one-sided coidealsubalgebra in the Hopf algebra Y( gl n ) . Moreover, Y( g n ) contains U( g n ) as asubalgebra, and has a homomorphism onto U( g n ) identical on this subalgebra.The definition of the twisted Yangian Y( g n ) was given by Olshanski [O]with some help from the second author of the present article. This definitionwas motivated by the works of Cherednik [C1] and Sklyanin [S] on the quantumintegrable systems with boundary conditions. Irreducible finite-dimensionalrepresentations of the associative algebra Y( g n ) have been classified by Molev,see the recent book [M]. In Subsections 1.4 and 1.5 we recall this classification,together with the definition of the twisted Yangian Y( g n ) .A new approach to the representation theory of the Yangian Y( gl n ) and ofits twisted analogues Y( so n ) , Y( sp n ) was developed by Khoroshkin, Nazarovand Vinberg in [KN, KNV]. In particular, it was proved in [KN] that up to anautomorphism of Y( sp n ) of the form (1.13), any irreducible finite-dimensionalrepresentation of Y( sp n ) is a quotient of some tensor product of fundamentalrepresentations of Y( gl n ) . The tensor product is regarded as a representationof Y( sp n ) by restriction from Y( gl n ) . Similar result was also proved in [KN]for those representations of Y( so n ) which integrate from so n ⊂ Y( so n ) to thespecial orthogonal group SO n . Moreover, the work [KN] provided new proofsof the above mentioned results of [AK] for Y( gl n ) . We summarize the resultsof [KN] in Subsections 1.3 and 1.6 of the present article.The realizations of irreducible representations of Y( gl n ) and Y( g n ) in [AK]and [KN] were not quite explicit. To make them more explicit is the main aimof the present article. In Subsections 2.6 and 2.7 we give explicit formulas forintertwining operators of tensor products of fundamental representations ofY( gl n ) , such that the quotients by the kernels of these operators realize allirreducible finite-dimensional representations of Y( gl n ) , up to automorphisms(1.3). In Subsections 3.4, 3.5, 3.6 we give analogues of these formulas for Y( g n ) .To obtain the irreducible representations of Y( gl n ) and Y( g n ) as quotientsof tensor products of fundamental representations, in [KN] we imposed certainconditions on the tensor products ; see Theorems 1.1 and 1.2 below. Someof these conditions are not necessary. For Y( gl n ) the condition (1.10) did notappear in [AK] and [NT1], but did appear in [C2]. In Subsection 2.8 we removethis condition from Theorem 1.1 by using the results of Subsection 2.5. InSubsection 3.7 we remove the conditions (1.17),(1.18),(1.19) from Theorem 1.2for Y( g n ) by using the results of Subsection 3.3. Thus we extend the results of[KN] for both Y( gl n ) and Y( g n ) , up to the level of [AK] and [NT1] for Y( gl n ) .
1. Representations of Yangians1.1.
First consider the
Yangian Y( gl n ) of the Lie algebra gl n . This is a complexunital associative algebra with a family of generators T (1) ij , T (2) ij , . . . where i , j = 1 , . . . , n . Defining relations for them can be written using the series rreducible representations of Yangians 3 T ij ( x ) = δ ij + T (1) ij x − + T (2) ij x − + . . . where x is a formal parameter. Let y be another formal parameter. Then thedefining relations in the associative algebra Y( gl n ) can be written as( x − y ) [ T ij ( x ) , T kl ( y ) ] = T kj ( x ) T il ( y ) − T kj ( y ) T il ( x ) . (1.1)The algebra Y( gl ) is commutative. By (1.1) for any t ∈ C the assignments T ij ( x ) T ij ( x − t ) (1.2)define an automorphism of the algebra Y( gl n ) . Here each of the formal powerseries T ij ( x − t ) in ( x − t ) − should be re-expanded in x − . Every assignment(1.2) is a correspondence between the respective coefficients of series in x − .Relations (1.1) also show that for any formal power series f ( x ) in x − withcoefficients from C and leading term 1, the assignments T ij ( x ) f ( x ) T ij ( x ) (1.3)define an automorphism of the algebra Y( gl n ) . The subalgebra consisting ofall elements of Y( gl n ) which are invariant under every automorphism of theform (1.3), is called the special Yangian of gl n , and is denoted by SY( gl n ) .Two representations of the algebra Y( gl n ) are called similar if they differ byan automorphism of the form (1.3).Let E ij ∈ gl n be the standard matrix units. By (1.1) the assignments T ij ( x ) δ ij + E ij x − (1.4)define a homomorphism of unital associative algebras Y( gl n ) → U( gl n ) . Thereis also an embedding U( gl n ) → Y( gl n ) , defined by mapping E ij T (1) ij . SoY( gl n ) contains the universal enveloping algebra U( gl n ) as a subalgebra. Thehomomorphism (1.4) is evidently identical on the subalgebra U( gl n ) ⊂ Y( gl n ) .The Yangian Y( gl n ) is a Hopf algebra over the field C . The comultiplication ∆ : Y( gl n ) → Y( gl n ) ⊗ Y( gl n ) is defined by the assignments ∆ : T ij ( x ) n X k =1 T ik ( x ) ⊗ T kj ( x ) . (1.5)When taking tensor products of Y( gl n ) -modules, we will use (1.5). The counithomomorphism Y( gl n ) → C is defined by the assignments T ij ( x ) δ ij . Byusing this homomorphism, one defines the trivial representation of Y( gl n ) .Further, let T ( x ) be the n × n matrix whose i , j entry is the series T ij ( x ) .The antipodal map Y( gl n ) → Y( gl n ) is defined by mapping T ( x ) T ( x ) − . Here each entry of the inverse matrix T ( x ) − is a formal power series in x − with coefficients from the algebra Y( gl n ) , and the assignment T ( x ) T ( x ) − is a correspondence between the respective matrix entries.The special Yangian SY( gl n ) is a Hopf subalgebra of Y( gl n ) . Moreover, itis isomorphic to the Yangian Y( sl n ) of the special linear Lie algebra sl n ⊂ gl n studied in [D1, D2]. For the proofs of these facts see [M, Subsection 1.8]. Khoroshkin, Nazarov and Papi
Up to their equivalence and similarity, the irreducible finite-dimensionalrepresentations of the associative algebra Y( gl n ) are parametrized by sequencesof n − P ( x ) , . . . , P n − ( x ) with complex coefficients. Inparticular, P ( x ) = . . . = P n − ( x ) = 1 for the trivial representation of Y( gl n ) .This parametrization was given by Drinfeld [D2, Theorem 2]. In the presentarticle we will use another parametrization. It can be obtained by combiningthe results of Arakawa and Suzuki [AS] with those of [D1]. Namely, consider theLie algebra gl m . Let t m be a Cartan subalgebra of gl m . Up to equivalence andsimilarity, we will parametrize the non-trivial irreducible finite-dimensionalrepresentations of Y( gl n ) by m = 1 , , . . . and by certain orbits in t ∗ m × t ∗ m under the diagonal shifted action of the Weyl group of gl m .Pick the Cartan subalgebra t m of gl m with the basis E , . . . , E mm of thediagonal matrix units. For any weight λ ∈ t ∗ m define the sequence λ , . . . , λ m of its labels by setting λ a = λ ( E aa ) for a = 1 , . . . , m . In particular, for thehalf-sum ρ ∈ t ∗ m of the positive roots we have ρ a = m/ − a + 1 / gl m is the symmetric group S m . It acts on the Cartan subalgebra t m by permuting the basis vectors E , . . . , E mm . Hence it acts on any weight λ ∈ t ∗ m by permuting its labels. The shifted action of w ∈ S m on t ∗ m is given by w ◦ λ = w ( λ + ρ ) − ρ . (1.6)For our parametrization we will be using only the orbits of the pairs ( λ , µ ) ∈ t ∗ m × t ∗ m such that all labels of the weight ν = λ − µ belong to { , . . . , n − } .Given any such a pair ( λ , µ ) define a sequence P ( x ) , . . . , P n − ( x ) of monicpolynomials with complex coefficients, as follows. For each i = 1 , . . . , n − P i ( x ) = Y ν a = i ( x − µ a − ρ a ) (1.7)where µ , . . . , µ m and ν , . . . , ν m are the labels of µ and ν respectively. Notethat the simultaneous shifted action of the group S m on the weights λ and µ gives the usual permutational action of S m on the labels of µ + ρ and of ν . Therefore each polynomial (1.7) depends only on the S m -orbit of the pair( λ , µ ) ∈ t ∗ m × t ∗ m . Moreover, the orbit is determined by the polynomials (1.7)uniquely. Furthermore, any sequence of monic polynomials P ( x ) , . . . , P n − ( x )of the total degree m with complex coefficients arises in this way.In the next subsection, to each of these orbits we will attach an irreducibleY( gl n ) -module. Its Drinfeld polynomials P ( x ) , . . . , P n − ( x ) will be given by(1.7). For the definition of the Drinfeld polynomials of an arbitrary irreduciblefinite-dimensional Y( gl n ) -module used here see [KN, Subsection 5.1]. For k = 0 , , . . . , n consider the exterior power Λ k ( C n ) of the defining gl n -module C n . Using the homomorphism (1.4), regard it as a module overthe Yangian Y( gl n ) . For t ∈ C denote by Φ kt the Y( gl n ) -module obtained bypulling the Y( gl n ) -module Λ k ( C n ) back through the automorphism (1.2).Now take any ( λ , µ ) ∈ t ∗ m × t ∗ m such that all labels of the weight ν = λ − µ belong to the set { , . . . , n − } . Let the weights λ and µ vary so that ν is fixed. rreducible representations of Yangians 5 For the Cartan subalgebra t m of gl m the weight µ is generic if µ a − µ b / ∈ Z whenever a = b . Consider the tensor product of Y( gl n ) -modules Φ ν µ + ρ + ⊗ . . . ⊗ Φ ν m µ m + ρ m + . (1.8)It is known that the Y( gl n ) -module (1.8) is irreducible if (but not only if) theweight µ is generic, see [NT2, Theorem 4.8] for a more general result. Moreover,if (but not only if) µ is generic then all the Y( gl n ) -modules obtained from (1.8)by permuting the tensor factors, are equivalent to each other. In particular,for every generic µ there is a unique non-zero Y( gl n ) -intertwining operator Φ ν µ + ρ + ⊗ . . . ⊗ Φ ν m µ m + ρ m + → Φ ν m µ m + ρ m + ⊗ . . . ⊗ Φ ν µ + ρ + (1.9)corresponding to the permutation of tensor factors of the maximal length. Wewill denote this operator by I ( µ ) ; it is unique up to a multiplier from C \ { } .For all generic weights µ , the source vector spaces of the operators I ( µ ) arethe same. The target vector spaces of all I ( µ ) also coincide with each other.Hence I ( µ ) is a function of µ taking values in the space of linear operatorsΛ ν ( C n ) ⊗ . . . ⊗ Λ ν m ( C n ) → Λ ν m ( C n ) ⊗ . . . ⊗ Λ ν ( C n )between tensor products of exterior powers of C n . The multipliers from C \{ } can be chosen so that I ( µ ) becomes a rational function of µ , see Section 2 for aparticular choice. Any such a choice allows to determine intertwining operators(1.9) for those non-generic µ where I ( µ ) is regular. Thus the weight µ need notbe generic anymore, so that the Y( gl n ) -module (1.8) may be reducible. Thisway of determining intertwining operators between Y( gl n ) -modules goes backto the work of Cherednik [C2] and is commonly called the fusion procedure .For the Cartan subalgebra t m of gl m the weight λ will be called dominant if λ a − λ b = − , − , . . . whenever a < b . The pair ( λ , µ ) ∈ t ∗ m × t ∗ m will becalled good if the weight λ + ρ is dominant and moreover ν a > ν b whenever λ a + ρ a = λ b + ρ b and a < b . (1.10)The orbit of any ( λ , µ ) ∈ t ∗ m × t ∗ m under the shifted action of S m on t ∗ m doescontain a good pair. Here we assume only that ν , . . . , ν m ∈ { , . . . , n − } . Inits present form, the next theorem was proved in [KN, Subsection 5.1]. It willbe generalized in Subsection 2.8 of the present article. Theorem 1.1.
For the fixed weight ν = λ − µ the multipliers from C \{ } ofthe operators (1.9) for all generic weights µ can be chosen so that the rationalfunction I ( µ ) is regular and non-zero whenever the pair ( λ , µ ) is good. Thenfor any good pair ( λ , µ ) the quotient of (1.8) by the kernel of operator I ( µ ) isan irreducible Y( gl n ) -module with the Drinfeld polynomials given by (1.7) . Let g n be one of the two Lie algebras so n , sp n . We will regard g n as theLie subalgebra of gl n preserving a non-degenerate bilinear form h , i on C n ,symmetric in the case g n = so n , or alternating in the case g n = sp n . In the Khoroshkin, Nazarov and Papi latter case, the positive integer n has to be even. When considering so n , sp n simultaneously we will use the following convention. Whenever the double sign ± or ∓ appears, the upper sign will correspond to the case of a symmetricform on C n so that g n = so n . The lower sign will correspond to the case of analternating form on C n so that g n = sp n .Let e T ( x ) be the conjugate of the matrix T ( x ) relative to the form h , i on C n .An involutive automorphism of the algebra Y( gl n ) is defined by the assignment T ( x ) e T ( − x ) . (1.11)This assignment is a correspondence between respective matrix entries. Nowconsider the matrix product S ( x ) = e T ( − x ) T ( x ) . Its ij entry is the series S ij ( x ) = n X k =1 e T ik ( − x ) T kj ( x ) = δ ij + S (1) ij x − + S (2) ij x − + . . . (1.12)with coefficients from the algebra Y( gl n ) . The twisted Yangian correspondingto the Lie algebra g n is the subalgebra of Y( gl n ) generated by the coefficients S (1) ij , S (2) ij , . . . where i , j = 1 , . . . , n . We denote this subalgebra by Y( g n ) .The algebras Y( so n ) corresponding to different choices of the symmetricform h , i on C n are isomorphic to each other, and so are the algebras Y( sp n )corresponding to different choices of the alternating form h , i on C n . Theseisomorphisms can be described explicitly, see for instance [M, Corollary 2.3.2].The subalgebra Y( g n ) ∩ SY( gl n ) of Y( gl n ) is denoted by SY( g n ) , and iscalled the special twisted Yangian corresponding to g n . The automorphism(1.3) of Y( gl n ) determines an automorphism of Y( g n ) which maps S ( x ) f ( x ) f ( − x ) S ( x ) . (1.13)The subalgebra SY( g n ) of Y( g n ) consists of the elements fixed by all suchautomorphisms. Two representations of the algebra Y( g n ) are called similar if they differ by such an automorphism.There is an analogue for Y( g n ) of the homomorphism Y( gl n ) → U( gl n )defined by (1.4). Namely, one can define a homomorphism Y( g n ) → U( g n ) by S ij ( x ) δ ij + E ij − e E ij x ± (1.14)where e E ij is the conjugate of the matrix unit E ij ∈ gl n relative to the form h , i on C n . This can be proved by using the defining relations for the generators S (1) ij , S (2) ij , . . . which we do not reproduce here ; see [M, Proposition 2.1.2] forthe proof. Further, there is an embedding U( g n ) → Y( g n ) defined by mapping E ij − e E ij S (1) ij . Hence the twisted Yangian Y( g n ) contains the universal enveloping algebraU( g n ) as a subalgebra. The homomorphism Y( g n ) → U( g n ) defined by (1.14)is evidently identical on the subalgebra U( g n ) ⊂ Y( g n ) . rreducible representations of Yangians 7 The twisted Yangian Y( g n ) is not only a subalgebra of Y( gl n ) , it is alsoa right coideal of the coalgebra Y( gl n ) relative to the comultiplication (1.5).Indeed, by the definition of the series (1.12) we get ∆ ( S ij ( x ))) = n X k,l =1 S kl ( x ) ⊗ e T ik ( − x ) T lj ( x ) . Therefore ∆ ( Y( g n )) ⊂ Y( g n ) ⊗ Y( gl n ) . Hence by taking a tensor product of an Y( g n ) -module with an Y( gl n ) -modulewe get another Y( g n ) -module.The trivial representation of the algebra Y( g n ) is defined by restricting thecounit homomorphism Y( gl n ) → C to the subalgebra Y( g n ) ⊂ Y( gl n ) . Underthis representation S ij ( x ) δ ij . Note that restricting any representation ofY( gl n ) to the subalgebra Y( g n ) amounts to taking the tensor product of thatrepresentation of Y( gl n ) with the trivial representation of the algebra Y( g n ) .A parametrization of irreducible finite-dimensional representations of thealgebra Y( g n ) was given by Molev, see [M, Chapter 4]. In the present article wewill use another parametrization, introduced in [KN]. In the next subsectionwe will establish a correspondence between the two parametrizations. Let uscall an Y( so n ) -module integrable if the action of the Lie algebra so n ⊂ Y( so n )on it integrates to an action of the complex Lie group SO n . When working withthe algebra Y( so n ) , we will be considering the integrable representations only. First recall the parametrization from [M]. Write n = 2 l or n = 2 l + 1depending on whether n is even or odd. If g n = sp n then n has to be even.Up to equivalence and similarity, the irreducible finite-dimensional modulesof the algebra Y( sp n ) are parametrized by sequences of l monic polynomials Q ( x ) , . . . , Q l ( x ) with complex coefficients, where the last polynomial Q l ( x )is even. Further, if n is even then the integrable irreducible finite-dimensionalY( so n ) -modules are parametrized by the same sequences of polynomials, andby an extra parameter δ ∈ { +1 , − } in the case when Q l (0) = 0 .If n is odd then g n = so n . The integrable irreducible finite-dimensionalmodules of the algebra Y( so n ) with odd n are parametrized by sequences of l monic polynomials Q ( x ) , . . . , Q l ( x ) with complex coefficients, but withoutany further conditions on the polynomial Q l ( x ) . For the trivial module of thealgebra Y( g n ) with arbitrary g n we always have Q ( x ) = . . . = Q l ( x ) = 1 ,and there is no extra parameter δ then, even if g n = so n and n = 2 l .Now let f m = sp m if g n = sp n , and let f m = so m if g n = so n . Let h m bea Cartan subalgebra of the reductive Lie algebra f m . The Weyl group of sp m is isomorphic to the hyperoctahedral group H m = S m ⋉ Z m . The Weyl groupof so m is isomorphic to a subgroup of H m of index two. In the latter case theaction of the Weyl group on h m can be extended by a diagram automorphismof SO m of order two, so that the extended Weyl group is still isomorphicto H m . Instead of Q ( x ) , . . . , Q l ( x ) for any g n we will use certain orbits in h ∗ m × h ∗ m under the diagonal shifted action of the group H m . Khoroshkin, Nazarov and Papi
When working with the Lie algebra f m it will be convenient to label thestandard basis vectors of C m by the indices − m , . . . , − , , . . . , m . Let a , b be any pair of these indices. Let E ab ∈ gl m be the corresponding matrixunit. Choose the antisymmetric bilinear form on C m so that the subalgebra f m ⊂ gl m preserving this bilinear form is spanned by the elements F ab = E ab − sign ( a b ) · E − b, − a or F ab = E ab − E − b, − a for f m = sp m or f m = so m respectively. Choose the Cartan subalgebra h m of f m with the basis F , . . . , F mm . For any weight λ ∈ h ∗ m define the sequence λ , . . . , λ m of its labels by λ a = λ ( F aa ) for a = 1 , . . . , m . For the half-sum ρ ∈ h ∗ m of positive roots of f m we have ρ a = − a if f m = sp m , and ρ a = 1 − a if f m = so m . Here the positive roots are chosen as in [KN, Subsection 4.1].The group H m acts on the Cartan subalgebra h m by permuting the basisvectors chosen above, and by multiplying any of them by − H m also acts on the dual space h ∗ m . The shifted action of any element w ∈ H m is given by the universal formula (1.6). Let κ ∈ h ∗ m be the weight of f m suchthat κ a = n/ a = 1 , . . . , m . Instead of the sequences of polynomials Q ( x ) , . . . , Q l ( x ) for our parametrization of irreducible Y( g n ) -modules we usethe orbits of the pairs ( λ , µ ) ∈ h ∗ m × h ∗ m such that all labels of the weight ν = λ − µ + κ belong to the set { , . . . , n − } . Note that the definitions of ν here and in the case of the Yangian Y( gl n ) are different.Given any such a pair ( λ , µ ) define a sequence Q ( x ) , . . . , Q l ( x ) of monicpolynomials as follows. For any g n and each i = 1 , . . . , l put Q i ( x ) = Y ν a = i ( x + µ a + ρ a ) · Y ν a = n − i ( x − µ a − ρ a ) . (1.15)Here µ , . . . , µ m and ν , . . . , ν m are the labels of the weights µ and ν of f m .Note that if n is even then l = n − l , so that Q l ( x ) is an even polynomial then.The simultaneous shifted action of the subgroup S m ⊂ H m on the weights λ and µ gives a permutational action of S m on the labels of µ + ρ and of ν . Further, for any a = 1 , . . . , m multiplying the basis vector F aa ∈ h m by − µ a + ρ a and ν a to respectively − µ a − ρ a and n − ν a . Therefore each of polynomials (1.15) depends only on the H m -orbit of the pair ( λ , µ ) ∈ h ∗ m × h ∗ m . Moreover, the orbit is determined bythese polynomials uniquely. Furthermore, any sequence of monic polynomials Q ( x ) , . . . , Q l ( x ) of total degree m with complex coefficients arises in this way,provided that for an even n the last polynomial Q l ( x ) is also even.In the next subsection, to each of these orbits we will attach an irreducibleY( g n ) -module, unless g n = so n with even n and µ a + ρ a = 0 for some a . Inthe latter case, to such an orbit we will attach two irreducible Y( g n ) -modules,not equivalent to each other. For these two, we will have δ = 1 and δ = − Q ( x ) , . . . , Q l ( x ) of the attached modules willbe given by (1.15). For the definition of Q ( x ) , . . . , Q l ( x ) for any irreduciblefinite-dimensional Y( g n ) -module see [KN, Subsections 5.3, 5.4, 5.5]. rreducible representations of Yangians 9 For k = 0 , , . . . , n let us denote by Φ − kt the Y( gl n ) -module obtained bypulling the Y( gl n ) -module Φ kt back through the automorphism (1.11). Notethat due to Lemma 2.3 the Y( gl n ) -module Φ − kt is similar to the module Φ n − k − t .Take any ( λ , µ ) ∈ h ∗ m × h ∗ m such that all labels of the weight ν = λ − µ + κ belong to the set { , . . . , n − } . Let the weights λ and µ vary so that ν is fixed.For the Cartan subalgebra h m of f m the weight µ is generic if µ a − µ b / ∈ Z and µ a + µ b / ∈ Z whenever a = b , and 2 µ a / ∈ Z for any a .Consider the tensor product of Y( gl n ) -modules (1.8) where µ , ν and ρ arenow weights of f m , not of gl m as before. It is known that the restriction of theY( gl n ) -module (1.8) to the subalgebra Y( g n ) ⊂ Y( gl n ) is irreducible if (butnot only if) the weight µ of f m is generic, see [KN, Theorems 5.3, 5.4, 5.5].Moreover, if (but not only if) µ is generic then all the Y( g n ) -modules obtainedfrom (1.8) by permuting the tensor factors and by replacing any tensor factor Φ kt by Φ − kt , are equivalent to each other. In particular, for each generic weight µ of f m there is a unique non-zero Y( g n ) -intertwining operator Φ ν µ + ρ + ⊗ . . . ⊗ Φ ν m µ m + ρ m + → Φ − ν µ + ρ + ⊗ . . . ⊗ Φ − ν m µ m + ρ m + (1.16)corresponding to the element of the group H m of the maximal length. We willdenote this operator by J ( µ ) ; it is unique up to a multiplier from C \ { } .For all generic weights µ of f m , the source and the target vector spaces ofthe operators J ( µ ) are the same tensor product of exterior powers of C n Λ ν ( C n ) ⊗ . . . ⊗ Λ ν m ( C n ) . Hence J ( µ ) is a function of µ taking values in the space of linear operatorson this tensor product. The multipliers from C \{ } can be chosen so that J ( µ ) becomes a rational function of µ , see Section 3 for a particular choice.Any such a choice allows to determine intertwining operators (1.16) for thosenon-generic weights µ where the function J ( µ ) is regular. The next theoremsummarizes the results of [KN, Subsections 5.3, 5.4, 5.5]. It will be generalizedin Subsection 3.7 of the present article.For the Lie algebra f m the weight λ is dominant if λ a − λ b = − , − , . . . and λ a + λ b = 1 , , . . . for all a < b , with an extra condition that λ a = 1 , , . . . in the case f m = sp m . The pair ( λ , µ ) ∈ h ∗ m × h ∗ m is called good if the weight λ + ρ is dominant and ν a > ν b whenever λ a − λ b + ρ a − ρ b = 0 and a < b , (1.17) ν a + ν b n whenever λ a + λ b + ρ a + ρ b = 0 and a < b , (1.18)with an extra condition that in the case f m = sp m ν a n whenever λ a + ρ a = 0 . (1.19)The orbit of any ( λ , µ ) ∈ h ∗ m × h ∗ m under the shifted action of H m on h ∗ m doescontain a good pair. Here we assume only that ν , . . . , ν m ∈ { , . . . , n − } .Note that for f m = so m a good pair is contained already in the orbit of any( λ , µ ) under the shifted action of the Weyl group, which is a subgroup of H m of index two. However, this fact will not be used in the present article. Theorem 1.2.
For the fixed ν = λ − µ + κ the multipliers from C \{ } of theoperators (1.16) for all generic weights µ can be chosen so that the rationalfunction J ( µ ) is regular and non-zero whenever the pair ( λ , µ ) is good. Thenfor any good pair ( λ , µ ) the quotient of (1.8) by the kernel of the operator J ( µ ) is an irreducible Y( g n ) -module, unless g n = so n with even n and µ a + ρ a = 0 for some index a . In the latter case the quotient of (1.8) by the kernel of J ( µ ) is a direct sum of two irreducible Y( g n ) -modules, not equivalent to each other.For any g n the polynomials Q ( x ) , . . . , Q l ( x ) of the irreducible Y( g n ) -modulesoccurring as above are given by (1.15) .
2. Intertwining operators2.1.
In this subsection we develop the formalism of R -matrices ; it will be usedto produce explicit formulas for intertwining operators (1.9) and (1.16) overY( gl n ) and Y( g n ) respectively. Let P denote the linear operator on ( C n ) ⊗ exchanging the two tensor factors. The Yang R -matrix is the rational functionof a variable x R ( x ) = 1 − P x − taking values in End ( C n ) ⊗ = ( End C n ) ⊗ . It satisfies the
Yang-Baxter equation in ( End C n ) ⊗ R ( x ) R ( x + y ) R ( y ) = R ( y ) R ( x + y ) R ( x ) . (2.1)As usual, the subscripts in (2.1) indicate different embeddings of the algebra( End C n ) ⊗ to ( End C n ) ⊗ so that R ( x ) = R ( x ) ⊗ R ( y ) = 1 ⊗ R ( y ) .We have P = n X i,j =1 E ij ⊗ E ji . Denote P = n X i,j =1 E ij ⊗ E ij and e P = n X i,j =1 e E ij ⊗ E ji . (2.2)Then put R ( x ) = 1 − P x − and e R ( x ) = 1 − e P x − . The values of the function R ( x ) are obtained from those of R ( x ) by applyingthe matrix transposition to the first tensor factor of ( End C n ) ⊗ . The valuesof the function e R ( x ) are obtained from those of R ( x ) by applying to the firsttensor factor of ( End C n ) ⊗ the conjugation with respect to the form h , i .Now (2.1) and the relation P R ( x ) P = R ( x ) imply that R ( x + y ) R ( x ) R ( y ) = R ( y ) R ( x ) R ( x + y ) , (2.3) e R ( x + y ) e R ( x ) R ( y ) = R ( y ) e R ( x ) e R ( x + y ) . (2.4) rreducible representations of Yangians 11 Finally, denote b P = n X i,j =1 e E ij ⊗ E ij and put b R ( x ) = 1 − b P x − . The values of the function b R ( x ) are obtained from those of R ( x ) by applyingto the first tensor factor of ( End C n ) ⊗ the conjugation with respect to h , i .Therefore the relation (2.3) implies that b R ( x ) b R ( x + y ) R ( y ) = R ( y ) b R ( x + y ) b R ( x ) . (2.5)Note that P R ( x ) P = R ( x ) and P e R ( x ) P = e R ( x ) . Further, due to P = n P we have R ( x ) R ( n − x ) = 1 . By using the latter relation, (2.3) also implies that R ( n − x − y ) b R ( x ) e R ( y ) = e R ( y ) b R ( x ) R ( n − x − y ) . (2.6)Observe that R (1) = 1 − P . A direct calculation now shows that R ( x ) R ( x + 1) R (1) = (cid:0) − ( P + P ) x − (cid:1) ( 1 − P ) . (2.7)In particular, when y = 1, the rational function of x at either side of (2.1) hasno pole at x = − y = 1 , the rational function of x at eitherside of (2.3) or (2.4) has no pole at x = − e R ( x + 1) e R ( x ) R (1) = (cid:0) − ( e P + e P ) x − (cid:1) ( 1 − P ) . (2.8) Now consider the representation of Y( gl n ) obtained by pulling the definingrepresentation of gl n back through the homomorphism (1.4), and then backthrough the automorphism (1.2) of Y( gl n ) . The resulting Y( gl n ) -module hasbeen denoted by Φ t . Note that under this representation T ( x ) R ( t − x ) . (2.9)Here on the left we regard n × n matrices with entries from the algebra Y( gl n )as elements of End C n ⊗ Y( gl n ) ; we will always do so in this and in the nextsection. Our explicit formulas for intertwining operators (1.9) and (1.16) arebased on the following simple and well known lemma, first appeared in [KRS]. Lemma 2.1.
For any k = 1 , . . . , n and t ∈ C the Y( gl n ) -module Φ kt appearsas the submodule of Φ t + k − ⊗ . . . ⊗ Φ t +1 ⊗ Φ t (2.10) with underlying subspace Λ k ( C n ) ⊂ ( C n ) ⊗ k . (2.11) Proof.
First consider the standard action of the Lie algebra gl n on the vectorspace ( C n ) ⊗ k . Turn this vector space into an Y( gl n ) -module, by pulling theaction of gl n back through the homomorphism (1.4) and then back through theautomorphism (1.2) of Y( gl n ) . Under the resulting representation of Y( gl n ) T ( x ) P + . . . + P ,k − + P k ) ( x − t ) − , (2.12)see (2.2). Here we use the subscripts 0 , . . . , k − , k rather than 1 , . . . , k , k +1 tolabel the tensor factors of ( End C n ) ⊗ ( k +1) . The Y( gl n ) -module Φ kt is definedby restricting the above described action of Y( gl n ) to the subspace (2.11).Next consider the action of Y( gl n ) on (2.10). By (1.5) and (2.9), then T ( x ) R ( t + k − − x ) . . . R ,k − ( t + 1 − x ) R k ( t − x ) . The latter product is a rational function of x , valued in ( End C n ) ⊗ ( k +1) . Ittends to 1 when x → ∞ , and has a simple pole at x = t with the residue P + . . . + P ,k − + P k . Further, by an observation made after (2.7) for any i = 1 , . . . , k − R ( t + k − − x ) . . . R ,k − ( t + 1 − x ) R k ( t − x ) ( 1 − P i,i +1 )has no pole at x = t + k − i . Hence the restriction of the action of Y( gl n )on (2.10) to the subspace (2.11) coincides with the restriction of the action ofY( gl n ) on the vector space ( C n ) ⊗ k described by the assignment (2.12). ⊓⊔ The assignment (1.11) defines a coalgebra anti-automorphism of Y( gl n ) .This immediately implies another lemma, to be used when working with Y( g n ) . Lemma 2.2.
The Y( gl n ) -module obtained by pulling (2.10) back through theautomorphism (1.11) is equivalent to Φ − t ⊗ Φ − t +1 ⊗ . . . ⊗ Φ − t + k − . The linear operator on ( C n ) ⊗ k reversing the order of tensor factors intertwinesthe two equivalent modules. For k = 0 , , . . . , n denote respectively by ˙ Φ kt and ˙ Φ − kt the Y( gl n ) -modulesobtained by pulling Φ kt and Φ − kt back through the automorphism (1.3) where f ( x ) = x − tx − t + 1 . Lemma 2.3.
The Y( gl n ) -modules Φ − kt and ˙ Φ n − k − t are mutually equivalent.Proof. Let e , . . . , e n be the standard basis vectors of C n . Denote by x i theoperator of left multiplication by the element e i in the exterior algebra Λ ( C n ) .Let ∂ i be the corresponding operator of left derivation on Λ( C n ) . Note that rreducible representations of Yangians 13 x i ∂ j + ∂ j x i = δ ij . For each k = 0 , , . . . , n the action of the algebra Y( gl n ) on its module Φ kt isdefined by the homomorphism Y( gl n ) → End (Λ ( C n )) which maps T ij ( x ) δ ij + x i ∂ j x − t . (2.13)It suffices to prove Lemma 2.3 for any choices of the symmetric and of thealternating form h , i on C n . For the proof only, choose the form as follows. Put θ i = − g n = sp n and i > n/ θ i = 1 . Set h e i , e j i = θ i δ ˜ ı j where we write ˜ ı = n − i + 1 for short. Then e E ij = θ i θ j E ˜ ˜ ı . (2.14)Using (1.11) and (2.13), the action of Y( gl n ) on Φ − kt is then defined by T ij ( x ) δ ij − θ i θ j x ˜ ∂ ˜ ı x + t . (2.15)On the other hand, the action of the algebra Y( gl n ) on its module ˙ Φ n − k − t is defined by the homomorphism Y( gl n ) → End (Λ ( C n )) which maps T ij ( x ) x + t − x + t (cid:16) δ ij + x i ∂ j x + t − (cid:17) = δ ij − ∂ j x i x + t . (2.16)By comparing the right hand sides of the assignments (2.15) and (2.16), theequivalence of the Y( gl n ) -modules ˙ Φ n − k − t and Φ − kt can be realized by the linearmap of the underlying vector spaces Λ n − k ( C n ) → Λ k ( C n ) : e i ∧ . . . ∧ e i n − k ( θ j e ˜ ) ∧ . . . ∧ ( θ j k e ˜ k )where e i ∧ . . . ∧ e i n − k ∧ e j ∧ . . . ∧ e j k = e ∧ . . . ∧ e n . Indeed, for any indices i , j = 1 , . . . , n this map intertwines the operators ∂ j x i and θ i θ j x ˜ ∂ ˜ ı on the vector spaces Λ n − k ( C n ) and Λ k ( C n ) respectively. ⊓⊔ Once again take any weights λ and µ of gl m such that all the labels of theweight ν = λ − µ belong to the set { , . . . , n − } . Put N = ν + . . . + ν m andsplit the sequence 1 , . . . , N to the consecutive segments of lengths ν , . . . , ν m .Hence the a th segment is the sequence of numbers p = ν + . . . + ν a − + i where i = 1 , . . . , ν a . (2.17)Then put x p = µ a + ρ a + + ν a − i . (2.18)Let P ν : ( C n ) ⊗ ν ⊗ . . . ⊗ ( C n ) ⊗ ν m → ( C n ) ⊗ ν m ⊗ . . . ⊗ ( C n ) ⊗ ν be the linear operator on ( C n ) ⊗ N reversing the order of the tensor factors C n by segments of lengths ν , . . . , ν m in their sequence. Within any segment, theorder of tensor factors is not changed. Then P ν = P if m = 2 and ν = ν = 1 .Now let the weight µ ∈ t ∗ m vary while the weight ν is fixed. Let 1 ′ , . . . , N ′ be the sequence obtained from 1 , . . . , N by reversing the order of the terms bythe segments of lengths ν , . . . , ν m introduced above. Within every segment,the order of terms is not changed. Let 1 ′′ , . . . , N ′′ be the sequence obtainedfrom 1 , . . . , N by reversing the order of terms within the segments. The orderof the segments themselves is not changed now. Take the ordered product B ( µ ) = −→ Y ( p,q ) R pq ( x q − x p ) (2.19)where p < q and they belong to different segments of the sequence 1 , . . . , N .Here the pair ( p , q ) precedes ( r , s ) if p < r or if p = r and q precedes s in thesequence 1 ′′ , . . . , N ′′ . Note that B ( µ ) is a rational function of µ without polesat generic weights of gl m . Proposition 2.4.
Suppose that the weight µ of gl m is generic. Then P ν B ( µ ) is an intertwining operator of Y( gl n ) -modules Φ x ⊗ . . . ⊗ Φ x N → Φ x ′ ⊗ . . . ⊗ Φ x N ′ . (2.20) Proof.
Under the action of Y( gl n ) on the source module in (2.20), T ( x ) R ( x − x ) . . . R N ( x N − x ) . (2.21)Like in the proof of Lemma 2.1, here we use the subscripts 0 , , . . . , N ratherthan 1 , , . . . , N + 1 to label the tensor factors of ( End C n ) ⊗ ( N +1) . Similarly,under the action of Y( gl n ) on the target module in (2.20), T ( x ) R ( x ′ − x ) . . . R N ( x N ′ − x ) . (2.22)Denote by X and X ′ the right hand sides of the assignments (2.21) and (2.22)respectively. By using the relation (2.3) repeatedly, we get P ν B ( µ ) X = P ν R ′ ( x ′ − x ) . . . R N ′ ( x N ′ − x ) B ( µ ) = X ′ P ν B ( µ ) . The equality of the left and of the right hand sides here proves the claim. ⊓⊔ Let A k be the operator of antisymmetrization on ( C n ) ⊗ k , normalized sothat A k = A k . The subspace (2.11) is the image of A k . The ordered product −→ Y ( i,j ) R ij ( j − i ) = k ! A k (2.23)where 1 i < j k and the pairs ( i , j ) are ordered lexicographically. Theformula (2.23) has appeared in [KRS] but was already known to Jucys [J]. rreducible representations of Yangians 15 Let e , . . . , e n be the standard basis vectors of C n . For each k = 1 , . . . , n consider the vector ϕ k = e ∧ . . . ∧ e k ∈ Λ k ( C n ) . Using the embedding (2.11), ϕ k = A k ( e ⊗ . . . ⊗ e k ) . The next proposition is known ; see [N, Theorem 2] for a more general result.It is still instructive to give a proof here, as it will be used later on.
Proposition 2.5.
For any generic weight µ of gl m the vector ϕ ν ⊗ . . . ⊗ ϕ ν m is an eigenvector of the operator B ( µ ) on ( C n ) ⊗ N with the eigenvalue Y a ν b . (2.24) Proof.
This proposition immediately follows from its particular case of m = 2 .Let us consider this case only. Then we have B ( µ ) ( ϕ ν ⊗ ϕ ν ) = −→ Y i =1 ,...,ν (cid:16) ←− Y j =1 ,...,ν R i,ν + j ( x ν + j − x i ) (cid:17) × ( A ν ⊗ A ν ) ( e ⊗ . . . ⊗ e ν ⊗ e ⊗ . . . ⊗ e ν ) =( A ν ⊗ A ν ) ←− Y i =1 ,...,ν (cid:16) −→ Y j =1 ,...,ν R i,ν + j ( x ν + j − x i ) (cid:17) (2.25)( e ⊗ . . . ⊗ e ν ⊗ e ⊗ . . . ⊗ e ν ) (2.26)where the last equality is obtained by using the formula (2.23) and by applying(2.1) repeatedly. The reversed arrow over the product symbol indicates thatthe factors corresponding to the running index are arranged from right to left.First suppose that ν ν . Arguing like in the proof of Lemma 2.1 we canalways rewrite the product displayed in line (2.25) as( A ν ⊗ A ν ) ←− Y i =1 ,...,ν (cid:16) − ( x ν + ν − x i ) − ν X j =1 P i,ν + j (cid:17) where any sum over j = 1 , . . . , ν clearly commutes with the operator 1 ⊗ A ν on ( C n ) ⊗ ( ν + ν ) . But for ν ν the operators (1 ⊗ A ν ) P i,ν + j with i = j annihilate the vector (2.26) while the operators P i,ν + i do not change it. Henceapplying the operator (2.25) to (2.26) gives the vector ϕ ν ⊗ ϕ ν multiplied by Y i =1 ,...,ν (cid:0) − ( x ν + ν − x i ) − (cid:1) = λ − λ + ρ − ρ + ν µ − µ + ρ − ρ . Next suppose that ν > ν . We can always rewrite the product (2.25) as ( A ν ⊗ A ν ) −→ Y j =1 ,...,ν (cid:16) ←− Y i =1 ,...,ν R i,ν + j ( x ν + j − x i ) (cid:17) . Arguing like in the proof of Lemma 2.1 we can rewrite the latter product as( A ν ⊗ A ν ) −→ Y j =1 ,...,ν (cid:16) − ( x ν + j − x ) − ν X i =1 P i,ν + j (cid:17) where any sum over i = 1 , . . . , ν clearly commutes with the operator A ν ⊗ C n ) ⊗ ( ν + ν ) . But for ν > ν the operators ( A ν ⊗ P i,ν + j with i = j annihilate the vector (2.26) while the operators P i,ν + i do not change it. Henceapplying the operator (2.25) to (2.26) gives the vector ϕ ν ⊗ ϕ ν multiplied by Y j =1 ,...,ν (cid:0) − ( x ν + j − x ) − (cid:1) = λ − λ + ρ − ρ + ν λ − λ + ρ − ρ . This observation completes the proof of Proposition 2.5. ⊓⊔ Using the relations (2.1) and R (1) = 1 − P repeatedly, one demonstratesthat for any generic weight µ of gl m the operator B ( µ ) preserves the subspaceΛ ν ( C n ) ⊗ . . . ⊗ Λ ν m ( C n ) ⊂ ( C n ) ⊗ N , (2.27)see the proof of Proposition 2.5 above. Moreover, we have another proposition. Proposition 2.6.
For any generic weight µ of gl m the restriction of B ( µ ) tothe subspace (2.27) coincides with that of the operator −→ Y a X i ,...,i d j ,...,j d d Y k =1 P i k j k λ a − λ b + ρ a − ρ b + ν b − k (cid:17) (2.28) where we order the pairs ( a , b ) lexicographically while i , . . . , i d and j , . . . , j d are pairwise distinct numbers respectively from the a th and b th segments of thesequence , . . . , N taken so that different are all corresponding sets of d pairs ( i , j ) , . . . , ( i d , j d ) . (2.29) Proof.
Note that as the indices i , . . . , i d and j , . . . , j d are pairwise distinct,we have d ν a and d ν b for any non-zero summand in the brackets in (2.28).The proposition immediately follows from its particular case of m = 2 . Letus consider this case only. Then (2.28) is an operator on ( C n ) ⊗ ( ν + ν ) equal to X d > X i ,...,i d j ,...,j d d Y k =1 P i k j k x k − x ν + ν (2.30) rreducible representations of Yangians 17 where i , . . . , i d and j , . . . , j d are pairwise distinct numbers taken respectivelyfrom 1 , . . . , ν and ν + 1 , . . . , ν + ν so that different are all the correspondingsets of d pairs (2.29). Here we use the equalities x ν + ν = µ + ρ + and x k = µ + ρ + + ν − k for any k ν . We also assume that 1 is the only term in (2.30) with d = 0 .On the other hand, for m = 2 by definition we have B ( µ ) = −→ Y i =1 ,...,ν (cid:16) ←− Y j =1 ,...,ν R i,ν + j ( x ν + j − x i ) (cid:17) . (2.31)Let us relate two operators on the vector space ( C n ) ⊗ ( ν + ν ) by the symbol ≡ if their actions coincide on the subspaceΛ ν ( C n ) ⊗ Λ ν ( C n ) ⊂ ( C n ) ⊗ ( ν + ν ) . (2.32)We will establish the relation ≡ between (2.30) and (2.31) by induction on ν .If ν > ν − ν ;if ν = 1 then we are not making any assumption. Arguing like in the proof ofLemma 2.1 and then using the induction assumption, (2.31) is related by ≡ to −→ Y i =1 ,...,ν − (cid:16) ←− Y j =1 ,...,ν R i,ν + j ( x ν + j − x i ) (cid:17) × (cid:16) ν X j =1 P ν , ν + j x ν − x ν + ν (cid:17) ≡ (cid:16) X d > X i ,...,i d j ,...,j d d Y k =1 P i k j k x k − x ν + ν (cid:17) × (cid:16) ν X j =1 P ν , ν + j x ν − x ν + ν (cid:17) where i , . . . , i d and j , . . . , j d are distinct indices taken respectively from1 , . . . , ν − ν + 1 , . . . , ν + ν . We assume that all corresponding sets of d pairs (2.29) are different. The right hand side of the last relation equals X d > X i ,...,i d j ,...,j d d Y k =1 P i k j k x k − x ν + ν + (2.33) X d> X i ,...,i d j ,...,j d d X l =1 (cid:16) d Y k =1 P i k j k x k − x ν + ν (cid:17) P ν j l x ν − x ν + ν + (2.34) X d > X i ,...,i d j ,...,j d X j (cid:16) d Y k =1 P i k j k x k − x ν + ν (cid:17) P ν , ν + j x ν − x ν + ν (2.35)where the index j is taken from 1 , . . . , ν but ν + j = j , . . . , j d however.Consider the sum displayed in the line (2.34). Here we have (cid:16) d Y k =1 P i k j k (cid:17) P ν j l = (cid:16) Y k = l P i k j k (cid:17) P i l j l P ν j l = (cid:16) Y k = l P i k j k (cid:17) P ν j l P i k ν ≡ − (cid:16) Y k = l P i k j k (cid:17) P ν j l where the right hand side does not involve the index i l . Now let us fix a number j ∈ { , . . . , ν } and take any set of d pairs (2.29) such that one of the pairscontains the number ν + j . Then this number has the form of j l for someindex l . If the set of the other d − i k , j k ) with k = l is also fixed, then i l ranges over a set of cardinality ν − d , namely over the fixed set { , . . . , ν − } \ { i , . . . , i l − , i l +1 , . . . , i d } . Now let us perform the summation over the indices i l , j l and l in (2.34) firstof all the running indices. After that, rename the running indices i l +1 , . . . , i d and j l +1 , . . . , j d respectively by i l , . . . , i d − and j l , . . . , j d − . By the argumentsgiven in the previous paragraph, the sum (2.34) gets related by ≡ to the sum X d> X i ,...,i d − j ,...,j d − X j (cid:16) d − Y k =1 P i k j k x k − x ν + ν (cid:17) d − ν x d − x ν + ν P ν , ν + j x ν − x ν + ν (2.36)where i , . . . , i d − and j , . . . , j d are distinct indices taken respectively from1 , . . . , ν − ν + 1 , . . . , ν + ν so that different are all sets of d − i , j ) , . . . , ( i d − , j d − )while the index j is taken from 1 , . . . , ν but ν + j = j , . . . , j d − however.Replace the running index d > d − d > X d> X i ,...,i d − j ,...,j d − X j (cid:16) d − Y k =1 P i k j k x k − x ν + ν (cid:17) P ν , ν + j x ν − x ν + ν (2.37)with the same assumptions on the running indices as in (2.36). By adding uptogether the sums (2.36) and (2.37), we get X d> X i ,...,i d − j ,...,j d − X j (cid:16) d − Y k =1 P i k j k x k − x ν + ν (cid:17) P ν , ν + j x d − x ν + ν (2.38)by the equality x d + d = x ν + ν . The sum of (2.33) and (2.38) equals (2.30).Thus we have made the induction step. ⊓⊔ By Lemma 2.1 and by Proposition 2.4 the restriction of operator P ν B ( µ ) tothe subspace (2.27) is an Y( gl n ) -intertwining operator (1.9). Let I ( µ ) be thisrestriction divided by the rational function (2.24). Then by Proposition 2.5 I ( µ ) : ϕ ν ⊗ . . . ⊗ ϕ ν m ϕ ν m ⊗ . . . ⊗ ϕ ν . (2.39) rreducible representations of Yangians 19 Theorem 2.7.
For any fixed weight ν = λ − µ the rational function I ( µ ) isregular at any point µ ∈ t ∗ m where the weight λ + ρ is dominant. The operator valued rational function I ( µ ) does not vanish at any point µ ∈ t ∗ m due to the normalization (2.39). The regularity of I ( µ ) was proved in[KN] for all µ where the pair ( λ , µ ) is good; our Theorem 2.7 is more general.In next two subsections, we give two proofs of Theorem 2.7. Each of themprovides an explicit formula for the operator I ( µ ) whenever λ + ρ is dominant.We give two proofs, because the resulting formulas for I ( µ ) are quite different.However, in both proofs we will use the following observation. Suppose thatthe weight λ + ρ of gl m is dominant. It means that λ a − λ b + ρ a − ρ b = − , − , . . . whenever a < b . (2.40)Then λ a − λ b + ρ a − ρ b + ν b = 0since ν b is a positive integer. So the rational function (2.24) does not vanishthen. Moreover, then any factor of the product (2.24) with ν a > ν b has a simplepole as a function of λ a − λ b + ρ a − ρ b at the zero point. In this subsection we will derive Theorem 2.7 from Proposition 2.6. Letus consider the factor of product (2.28) corresponding to any pair of indices a < b . There k = 1 , . . . , d . Since the indices i , . . . , i d and j , . . . , j d arepairwise distinct, we may assume that d ν a and d ν b in (2.28). If ν a < ν b then k < ν b . Therefore any factor of (2.28) corresponding to a < b with ν a < ν b is regular at any point µ ∈ t ∗ m where the weight λ + ρ is dominant.If ν a > ν b then the denominator in (2.28) is zero if and only if k = d = ν b and λ a + ρ a = λ b + ρ b . Hence any factor in the product (2.28) with ν a > ν b hasa simple pole as a function of λ a − λ b + ρ a − ρ b , at the zero point. Moreover,the residue of the latter function at the zero point equals1( ν b −
1) ! X i ,...,i νb P i . . . P i νb ν b where the indices i , . . . , i ν b are distinct and taken from 1 , . . . , ν a .Using the observation on (2.24) made at the end of the previous subsection,we now complete the proof of Theorem 2.7 for any dominant weight λ + ρ .Note that Proposition 2.5, like Theorem 2.7 above, can also be derived fromProposition 2.6; see the proof of [KN, Proposition 4.4]. Further, Proposition 2.6and Theorem 2.7 both can be proved by using [KN, Subsections 1.4 and 4.4]. In this subsection we will give another proof of Theorem 2.7. It providesa multiplicative formula for the operator I ( µ ) with dominant λ + ρ . Considerthe product (2.19). It is taken over the pairs ( p , q ) where p < q while p and q belong to different segments of the sequence 1 , . . . , N ; see (2.17). Let a and b be the numbers of these two segments, so that a < b . Let us now rearrangethe pairs ( p , q ) in the product (2.19) as follows. The new order on the pairs ( p , q ) will be an extension of the lexicographicalorder on the corresponding pairs ( a , b ) . To define the extension, we have toorder the pairs ( p , q ) corresponding to a given ( a , b ) . Take another pair ( r , s )such that the indices r and s belong to the segments a and b , that is to thesame segments as the indices p and q respectively. For ν a < ν b , the pair ( p , q )will precede ( r , s ) if p < r or if p = r and q > s . For ν a > ν b , the pair ( p , q ) willprecede ( r , s ) if q > s or if q = s and p < r . By exchanging commuting factorsin (2.19), this rearrangement does not alter the value of the ordered product.Let i and j be the numbers of the indices p and q within their segments,so that by definition we have the equalities (2.18) and x q = µ b + ρ b + + ν b − j . (2.41)Consider the factor R pq ( x q − x p ) in the reordered product (2.19). As a functionof µ , this factor has a pole at x p = x q . The latter equation can be written as λ a − λ b + ρ a − ρ b = i − j (2.42)using (2.18) and (2.41). Hence if x p = x q while λ + ρ is dominant, then i > j .First, suppose that ν a < ν b . If i > j , then j < ν b since i ν a . Then theindex q is not the last in its segment, so that the pair ( p , q + 1) immediatelyprecedes ( p , q ) in the new ordering. Consider the pairs which follow ( p , q ) inthe new ordering. Take the product of the factors in (2.19) corresponding tothe latter pairs, and multiply it on the right by R q,q +1 ( x q − x q +1 ) = R q,q +1 (1) = 1 − P q,q +1 . (2.43)Using (2.1), the resulting product is also divisible by (2.43) on the left. Dueto (2.7) we can therefore replace in (2.19) the product of two adjacent factors R p,q +1 ( x q +1 − x p ) R pq ( x q − x p ) by 1 − ( P pq + P p,q +1 ) / ( x q +1 − x p )without changing the restriction of the operator (2.19) to the subspace (2.27).But the replacement does not have a pole at x p = x q . So the factors in (2.19)corresponding to the pairs ( p , q ) with ν a < ν b do not increase the order of thepole of I ( µ ) at any point µ such that the weight λ + ρ is dominant.Next, suppose that ν a > ν b while i > p is not the firstin its segment. Then the pair ( p − , q ) immediately precedes ( p , q ) in the newordering. Consider the pairs following ( p , q ) . Take the product of the factorsin (2.19) corresponding to the latter pairs. Multiply it on the right by R p − ,p ( x p − − x p ) = R p − ,p (1) = 1 − P p − ,p . (2.44)Using (2.1), the resulting product is also divisible by (2.44) on the left. Dueto (2.7) we can now replace in (2.19) the product of two adjacent factors R p − ,q ( x q − x p − ) R pq ( x q − x p ) by 1 − ( P p − ,q + P pq ) / ( x q − x p − )without changing the restriction of (2.19) to (2.27). The replacement has nopole at x p = x q . So the factors in (2.19) corresponding to the pairs ( p , q ) with ν a > ν b and i > I ( µ ) at any point µ . rreducible representations of Yangians 21 Last, suppose that ν a > ν b and i = 1 . If x p = x q then λ a + ρ a = λ b + ρ b and j = 1 whenever the weight λ + ρ is dominant, due to (2.42). The observationon (2.24) made at the end of Subsection 2.5 now proves Theorem 2.7.Note that all the replacements in the product (2.19) described above canbe made simultaneously. Hence our argument provides an explicit formula forthe operator I ( µ ) whenever λ + ρ is dominant. See also [GNP, Subsection 2.3]. In this subsection we will generalize Theorem 1.1. Theorem 2.7 allowsus to determine the intertwining operator I ( µ ) of the Y( gl n ) -modules (1.9)for any µ ∈ t ∗ m , provided the weight λ + ρ is dominant. Our generalization ofTheorem 1.1 is based on the following lemma. For any index c = 1 , . . . , m − s c ∈ S m be the transposition of c and c + 1 . Here the symmetric group S m acts on the numbers 1 , . . . , m by their permutations. The latter correspond topermutations of the basis vectors E , . . . , E mm of t m . Lemma 2.8.
Fix c > and suppose that both λ + ρ and s c ( λ + ρ ) are dominant.Then the images of the intertwining operators I ( µ ) and I ( s c ◦ µ ) correspondingto the pairs ( λ , µ ) and ( s c ◦ λ , s c ◦ µ ) are equivalent as Y( gl n ) -modules.Proof. Note that in this lemma the intertwining operator I ( s c ◦ µ ) correspondsto the weight s c ◦ λ , not to λ . Moreover, the source and target Y( gl n ) -modulesof this operator are different from those of the operator I ( µ ) in general. Thisdifference should not cause any confusion however.Let ˇ λ and ˇ µ and ˇ ρ be the weights of gl with the labels λ c , λ c +1 and µ c , µ c +1 and ρ c , ρ c +1 respectively. The weights ˇ λ + ˇ ρ and s (ˇ λ + ˇ ρ ) of gl are dominant.By using Theorem 2.7 with m = 2 we get the Y( gl n ) -intertwining operators Φ ν c µ c + ρ c + ⊗ Φ ν c +1 µ c +1 + ρ c +1 + → Φ ν c +1 µ c +1 + ρ c +1 + ⊗ Φ ν c µ c + ρ c + , (2.45) Φ ν c +1 µ c +1 + ρ c +1 + ⊗ Φ ν c µ c + ρ c + → Φ ν c µ c + ρ c + ⊗ Φ ν c +1 µ c +1 + ρ c +1 + . (2.46)These two operators are inverse to each other. This assertion can be proved firstfor any generic weight µ of gl m , either by a direct calculation employing thedefinition (2.19), or by observing that for any generic µ all the double tensorproducts above are Y( gl n ) -irreducible while (2.45) and (2.46) respectively map ϕ ν c ⊗ ϕ ν c +1 ϕ ν c +1 ⊗ ϕ ν c and ϕ ν c +1 ⊗ ϕ ν c ϕ ν c ⊗ ϕ ν c +1 . Then the assertion extends to µ such that λ + ρ and s c ( λ + ρ ) are dominant.Now denote by I the operator which acts on the tensor product of c th and( c + 1) th factors of (1.8) as the intertwining operator (2.45), and which actstrivially on all other m − gl n ) -modules in (1.9) for any generic µ , we get the relation I ( µ ) = I I ( s c ◦ µ ) I .
It proves the lemma, since I is invertible and intertwines Y( gl n ) -modules. ⊓⊔ For any λ ∈ t ∗ m denote by S λ the subgroup of S m consisting of all elements w such that w ◦ λ = λ . Let O be any orbit of the shifted action of the subgroup S λ ⊂ S m on t ∗ m . If ν , . . . , ν m ∈ { , . . . , n − } for at least one weight µ ∈ O ,then every µ ∈ O satisfies the same condition. Suppose this is the case for O .If λ + ρ is dominant, then there is at least one weight µ ∈ O such that the pair( λ , µ ) is good. Theorem 1.1 generalizes due to the following proposition. Proposition 2.9. If λ + ρ is dominant, then for all µ ∈ O the images of thecorresponding operators I ( µ ) are equivalent to each other as Y( gl n ) -modules.Proof. Take any w ∈ S λ and any reduced decomposition w = s c l . . . s c . Itcan be derived from [B, Corollary VI.1.2] that the weight s c k . . . s c ( λ + ρ ) of gl m is dominant for each k = 1 , . . . , l . Proposition 2.9 now follows by applyingLemma 2.8 repeatedly. Note that this proposition can also be proved by usingthe results of Zelevinsky [Z, Theorem 6.1] together with those of [AS, D1]. ⊓⊔ Thus all assertions of Theorem 1.1 will remain valid if we replace the goodpair there by any pair ( λ , µ ) such that the weight λ + ρ of gl m is dominant.However, we still assume that ν , . . . , ν m ∈ { , . . . , n − } for the latter pair.
3. More intertwining operators3.1.
Now let λ and µ be any weights of the Lie algebra f m = sp m or f m = so m such that all labels of the weight ν = λ − µ + κ are in { , . . . , n − } . We willkeep using the notation (2.17),(2.18),(2.19). But now λ a , µ a , ν a and ρ a with a = 1 , . . . , m are labels of weights of f m . Recall that κ a = n/ µ ∈ h ∗ m vary while ν is fixed. Determine the rational function B ( µ ) by the same formula (2.19) as for Y( gl n ) . Also take the ordered product C ( µ ) = ←− Y ( p,q ) e R pq ( n − x p − x q ) (3.1)where 1 p < q N and the pairs ( p , q ) are ordered lexicographically. Herethe reversed arrow indicates that the factors corresponding to these pairs arearranged from right to left. Note that C ( µ ) is a rational function of µ ∈ h ∗ m without poles at the generic weights µ of f m .Take the sequence 1 ′′ , . . . , N ′′ introduced in the previous subsection. Let Q ν be the linear operator on ( C n ) ⊗ N which for each p = 1 , . . . , N exchangesthe tensor factors C n labeled by p and p ′′ . Then Q ν = P if m = 1 and ν = 2 . Proposition 3.1.
Suppose the weight µ of f m is generic. Then Q ν B ( µ ) C ( µ ) is an intertwining operator of Y( g n ) -modules Φ x ⊗ . . . ⊗ Φ x N → Φ − x ′′ ⊗ . . . ⊗ Φ − x N ′′ . (3.2) rreducible representations of Yangians 23 Proof.
Under the action of Y( gl n ) on the source module in (3.2), S ( x ) b R N ( x N + x ) . . . b R ( x + x ) R ( x − x ) . . . R N ( x N − x ) . Let us denote by Y the right hand side of this assignment. Here we use thesubscripts 0 , , . . . , N rather than 1 , , . . . , N + 1 to label the tensor factors of( End C n ) ⊗ ( N +1) , like we did in the proof of Lemma 2.1. Further, let us denote Y ′ = R ( x − x ) . . . R N ( x N − x ) b R N ( x N + x ) . . . b R ( x + x ) . By using the relation (2.6) repeatedly, we get the equality C ( µ ) Y = Y ′ C ( µ ) .Here we also use the relation P e R ( x ) P = e R ( x ) .Under the action of Y( g n ) on the target module in (3.2), S ( x ) R N ( x N ′′ − x ) . . . R ( x ′′ − x ) b R ( x ′′ + x ) . . . b R N ( x N ′′ + x ) . Denote by Y ′′ the right hand side of the latter assignment. Observe that k ′′ =( N − k + 1) ′ for each k = 1 , . . . , N . Therefore we can write Y ′′ = R N ( x ′ − x ) . . . b R ( x N ′ − x ) b R ( x N ′ + x ) . . . b R N ( x ′ + x ) . But by using the relations (2.3) and (2.5) repeatedly, we get Q ν B ( µ ) Y ′ = Q ν R ′ ( x ′ − x ) . . . R N ′ ( x N ′ − x ) b R N ′ ( x N ′ + x ) . . . b R ′ ( x ′ + x ) B ( µ )= Y ′′ Q ν B ( µ ) ;see also the proof of Proposition 2.4. Thus we get the equality Q ν B ( µ ) C ( µ ) Y = Y ′′ Q ν B ( µ ) C ( µ ) , which proves the claim. ⊓⊔ Observe that e P P = ± e P . Thus e P R (1) = 0 in the case g n = so n . Hencein this case the restriction of the operator e P pq to the subspace (2.27) is zero forany two distinct indices p , q from the same segment of the sequence 1 , . . . , N .Here we mean the segments of lengths ν , . . . , ν m as defined in Subsection 2.3.In the case g n = so n the relation (2.4) now implies that when consideringthe restriction of the operator C ( µ ) to the subspace (2.27), we can skip thosefactors in the product (3.1) which correspond to the pairs ( p, q ) where both p and q belong to the same segment: skipping does not change the restriction.In particular, in the case g n = so n the restriction of the operator C ( µ ) to thesubspace (2.27) does not have a pole if µ a − µ b / ∈ Z and µ a + µ b / ∈ Z whenever a = b ; the condition 2 µ a / ∈ Z is not needed here.Let us give an analogue of Proposition 2.5 for C ( µ ) . Except in the proofof Lemma 2.3, we worked with any symmetric or alternating non-degeneratebilinear form h , i on C n so far. Choose the form as in the proof, so that (2.14)holds. The elements E ij − e E ij with i j span a Borel subalgebra of g n ⊂ gl n then; the elements E ii − e E ii span the corresponding Cartan subalgebra of g n . Proposition 3.2.
For any generic weight µ of f m the vector ϕ ν ⊗ . . . ⊗ ϕ ν m isan eigenvector of the operator C ( µ ) on ( C n ) ⊗ N . The eigenvalue is the product Y a n ;1 if ν a + ν b n ; (3.3) multiplied in the case of f m = sp m by the product Y a m λ a + ρ a µ a + ρ a if ν a > n ;1 if ν a n . (3.4) Proof.
This proposition immediately follows from its particular cases of m = 1and m = 2 . We will consider these two cases only. First suppose that m = 1 .In this case, for g n = so n the restriction of the operator C ( µ ) to the subspace(2.27) is the identity operator ; see the observation made in the very beginningof the present subsection. Suppose that g n = sp n . Put ε = ( − [ ν / . Then C ( µ ) ϕ ν = ←− Y i =1 ,...,ν − (cid:16) ←− Y j = i +1 ,...,ν e R ij ( n − x i − x j ) (cid:17) × ε A ν ( e ν ⊗ . . . ⊗ e ) = ε A ν −→ Y i =1 ,...,ν − (cid:16) −→ Y j = i +1 ,...,ν e R ij ( n − x i − x j ) (cid:17) ( e ν ⊗ . . . ⊗ e ) (3.5)where the latter equality is obtained by using (2.4) and (2.23). We will prove byinduction on ν that the vector (3.5) equals ϕ ν multiplied by the scalar (3.4)where m = 1 . Recall that ν = λ − µ + n/ ν n/ e P ij ( e ν ⊗ . . . ⊗ e ) = 0by our choice of the form h , i on C n . Hence then the vector (3.5) equals ϕ ν as required. We will also use that equality for ν n/ ν > n/ ν > n is even. The inductionassumption then implies that(1 ⊗ A ν − ) −→ Y i =2 ,...,ν − (cid:16) −→ Y j = i +1 ,...,ν e R ij ( n − x i − x j ) (cid:17) ( e ν ⊗ e ν − ⊗ . . . ⊗ e ) = u e ν ⊗ A ν − ( e ν − ⊗ . . . ⊗ e )where u = λ + ρ − µ + ρ . rreducible representations of Yangians 25 We use the inequality 2 ( ν − > n which follows from ν > n/ n iseven. Arguing like in the proof of Lemma 2.1 and using the relation (2.8) we get(1 ⊗ A ν − ) −→ Y j =2 ,...,ν e R j ( n − x − x j ) =(1 ⊗ A ν − ) (cid:16) − ( n − x − x ) − ν X j =2 e P j (cid:17) . Further, by our choice of the form h , i on C n the vector A ν e P j ( e ν ⊗ . . . ⊗ e )is equal to 2 A ν ( e ν ⊗ . . . ⊗ e )if j = 2 ν − n , and is equal to zero for any other j > ε A ν −→ Y j =2 ,...,ν e R j ( n − x − x j ) × −→ Y i =2 ,...,ν − (cid:16) −→ Y j = i +1 ,...,ν e R ij ( n − x i − x j ) (cid:17) ( e ν ⊗ . . . ⊗ e )we now see that the vector (3.5) is equal to ϕ ν multiplied by the scalar n − x − x − n − x − x u = λ + ρ µ + ρ . Thus we have finished the proof of Proposition 3.2 in the case m = 1 ,and will now suppose that m = 2 . Then by the definition (3.1) the operator C ( µ ) is the ordered product of e R pq ( n − x p − x q ) over the pairs ( p , q ) where1 p < q ν + ν ; the arrangement of the pairs is reversed lexicographical.Without changing the product, we can rearrange these pairs as follows. Fromleft to right, first come the pairs ( p , q ) where ν < p < q ν + ν , secondcome the pairs where 1 p ν and ν < q ν + ν ; third come the pairswhere 1 p < q ν . Within each of the three groups, the arrangement ofthe pairs ( p , q ) is still reversed lexicographical.Consider the two products of e R pq ( n − x p − x q ) , over the first and the thirdgroup of pairs ( p , q ) . The already settled case m = 1 implies that ϕ ν ⊗ ϕ ν is an eigenvector for each of these two products. If g n = so n then each of thetwo corresponding eigenvalues is 1 . If g n = sp n then the product of the twoeigenvalues is (3.4) where m = 2 . For g n = so n , sp n we will show that ϕ ν ⊗ ϕ ν is an eigenvector of the product of e R pq ( n − x p − x q ) over the second group ofpairs, with the eigenvalue (3.3) where m = 2 . This will settle the case m = 2 .Use the induction on ν . Denote the last mentioned product by Z , so that Z = ←− Y i =1 ,...,ν (cid:16) ←− Y j =1 ,...,ν e R i,ν + j ( n − x i − x ν + j ) (cid:17) (3.6)by definition. By using (2.4) and (2.23), we get the equality Z ( A ν ⊗ A ν ) ( e ν ⊗ . . . ⊗ e ⊗ e ν ⊗ . . . ⊗ e ) =( A ν ⊗ A ν ) −→ Y i =1 ,...,ν (cid:16) −→ Y j =1 ,...,ν e R i,ν + j ( n − x i − x ν + j ) (cid:17) (3.7)( e ν ⊗ . . . ⊗ e ⊗ e ν ⊗ . . . ⊗ e ) . If ν + ν n then by our choice of the form h , i on C n e P i,ν + j ( e ν ⊗ . . . ⊗ e ⊗ e ν ⊗ . . . ⊗ e ) = 0for i , j as above. Hence then the product (3.6) acts on the vector ϕ ν ⊗ ϕ ν asthe identity. We will also use this result for ν + ν n as the induction base.Let ν + ν > n . Then ν > ν < n . By the induction assumption(1 ⊗ A ν − ⊗ A ν ) −→ Y i =2 ,...,ν (cid:16) −→ Y j =1 ,...,ν e R i,ν + j ( n − x i − x ν + j ) (cid:17) ( e ν ⊗ e ν − ⊗ . . . ⊗ e ⊗ e ν ⊗ . . . ⊗ e ) = v e ν ⊗ ( A ν − ⊗ A ν ) ( e ν − ⊗ . . . ⊗ e ⊗ e ν ⊗ . . . ⊗ e )where v = λ + λ + ρ + ρ − µ + µ + ρ + ρ . Arguing like in the proof of Lemma 2.1 and using the relation (2.8) we get theequality in the algebra ( End C n ) ⊗ ( ν + ν ) (1 ⊗ A ν ) −→ Y j =1 ,...,ν e R ,ν + j ( n − x − x ν + j ) =(1 ⊗ A ν ) (cid:16) − ( n − x − x ν +1 ) − ν X j =1 e P ,ν + j (cid:17) . Further, by our choice of the form h , i on C n the vector( A ν ⊗ A ν ) e P q ( e ν ⊗ . . . ⊗ e ⊗ e ν ⊗ . . . ⊗ e )is equal to ( A ν ⊗ A ν ) ( e ν ⊗ . . . ⊗ e ⊗ e ν ⊗ . . . ⊗ e ) rreducible representations of Yangians 27 if j = ν + ν − n , and is equal to zero for any other j . Writing (3.7) as( A ν ⊗ A ν ) −→ Y j =1 ,...,ν e R ,ν + j ( n − x − x ν + j ) × −→ Y i =2 ,...,ν (cid:16) −→ Y j =1 ,...,ν e R i,ν + j ( n − x − x ν + j ) (cid:17) we now see that ϕ ν ⊗ ϕ ν is an eigenvector for (3.6) with the eigenvalue n − x − x ν +1 − n − x − x ν +1 v = λ + λ + ρ + ρ µ + µ + ρ + ρ . This observation completes the proof of Proposition 3.2. ⊓⊔ Using the relations (2.1),(2.4) and R (1) = 1 − P one shows that for anygeneric weight µ of f m the operator C ( µ ) preserves the subspace (2.27), see theproof of Proposition 3.2. Moreover, we have another proposition. It can be usedto give another proof of Proposition 3.2, see the proof of [KN, Proposition 4.6]. Proposition 3.3.
For any generic weight µ of f m the restriction of C ( µ ) tothe subspace (2.27) coincides with that of the operator ←− Y a b m X d> X i ,...,i d j ,...,j d d Y k =1 e P i k j k λ a + λ b + ρ a + ρ b − k if a < b ;1 + X d> X i ,...,i d j ,...,j d d Y k =1 e P i k j k λ a + ρ a − k ) if a = b , f m = sp m ;1 if a = b , f m = so m ; here the pairs ( a , b ) are ordered lexicographically. If a < b then i , . . . , i d and j , . . . , j d are distinct numbers from the a th and b th segments of the sequence , . . . , N respectively, taken so that different are all the sets (2.29) . If a = b then i , j , . . . , i d , j d are pairwise distinct numbers from the a th segment of thesequence , . . . , N taken so that different are all the sets of d unordered pairs { i , j } , . . . , { i d , j d } . (3.8) Proof.
This proposition immediately follows from its particular cases of m = 1and m = 2 . We will consider these two cases only. First suppose that m = 1 .We have already observed that then for g n = so n the restriction of the operator C ( µ ) to the subspace (2.27) is the identity operator. Suppose g n = sp n . Thenin the second displayed line in Proposition 3.3 we have the operator on ( C n ) ⊗ ν X d > X i ,...,i d j ,...,j d d Y k =1 e P i k j k x k − n − i , j , . . . , i d , j d are pairwise distinct and taken from 1 , . . . , ν so thatdifferent are all the sets of d unordered pairs (3.8). Here we use the equality x k = λ + ρ + ( n + 1) / − k (3.10)for any k ν . We also assume that 1 is the only term in (3.9) with d = 0 .On the other hand, for m = 1 we can write C ( µ ) = ←− Y j =2 ,...,ν (cid:16) ←− Y i =1 ,...,j − e R ij ( n − x i − x j ) (cid:17) . (3.11)Let us now relate two operators on the vector space ( C n ) ⊗ ν by the symbol ≡ if their actions coincide on the subspaceΛ ν ( C n ) ⊂ ( C n ) ⊗ ν . (3.12)We will establish the relation ≡ between (3.9) and (3.11) by induction on ν .If ν > ν − ν ;if ν = 1 then we are not making any assumption. Arguing like in the proof ofLemma 2.1 and then using the induction assumption, (3.11) is related by ≡ to (cid:16) ν − X i =1 e P i ν x + x ν − n (cid:17) × ←− Y j =2 ,...,ν − (cid:16) ←− Y i =1 ,...,j − e R ij ( n − x i − x j ) (cid:17) ≡ (cid:16) ν − X i =1 e P i ν x + x ν − n (cid:17) × (cid:16) X d > X i ,...,i d j ,...,j d d Y k =1 e P i k j k x k − n − (cid:17) where i , j , . . . , i d , j d are distinct and taken from 1 , . . . , ν − X d > X i ,...,i d j ,...,j d d Y k =1 e P i k j k x k − n − X d> d X l =1 X i ,...,i d j ,...,j d e P i l ν + e P j l ν x + x ν − n d Y k =1 e P i k j k x k − n − X d > X i ,...,i d j ,...,j d X i e P i ν x + x ν − n d Y k =1 e P i k j k x k − n − i is taken from 1 , . . . , ν − i , j , . . . , i d , j d . rreducible representations of Yangians 29 Consider the sum displayed in the line (3.14). Here we have e P i l ν d Y k =1 e P i k j k = (cid:16) Y k = l e P i k j k (cid:17) e P i l ν e P i l j l = (cid:16) Y k = l e P i k j k (cid:17) e P i l ν P j l ν ≡ − (cid:16) Y k = l e P i k j k (cid:17) e P i l ν where the right hand side does not involve j l . Similarly, in (3.14) we have e P j l ν d Y k =1 e P i k j k ≡ − (cid:16) Y k = l e P i k j k (cid:17) e P j l ν where the right hand side does not involve the index i l .Now fix a number i ∈ { , . . . , ν − } and take any set of d pairs (3.8) suchthat one of the pairs contains the number i . Then i = i l or i = j l for some l . Let j be the element of the pair { i l , j l } different from i , so that j = j l or j = i l respectively. If the set of the d − { i k , j k } with k = l is also fixed,then j ranges over a set of cardinality ν − d , namely over the fixed set { , . . . , ν − } \ { i , j , . . . , i l − , j l − , i , i l +1 , j l +1 , . . . , i d , j d } . Let us perform the summation over the indices i l , j l and l in (3.14) firstof all the running indices. After that rename the running indices i l +1 , . . . , i d and j l +1 , . . . , j d respectively by i l , . . . , i d − and j l , . . . , j d − . By the argumentsgiven in the previous two paragraphs, the sum (3.14) gets related by ≡ to X d> X i ,...,i d − j ,...,j d − X i d − ν x d − n − e P i ν x + x ν − n d − Y k =1 e P i k j k x k − n − i and i , j , . . . , i d − , j d − are distinct indices taken from 1 , . . . , ν − d − { i , j } , . . . , { i d − , j d − } . Replace the running index d > d − d > X d> X i ,...,i d − j ,...,j d − X i e P i ν x + x ν − n d − Y k =1 e P i k j k x k − n − X d> X i ,...,i d − j ,...,j d − X i e P i ν x d − n − d − Y k =1 e P i k j k x k − n − by the equality 2 x d + 2 d = x + 1 + x ν + ν . The sum of (3.13) and (3.18) equals (3.9). This makes the induction step. Thuswe have finished the proof of Proposition 3.3 in the case m = 1 .Now let m = 2 . We will begin considering this case in the same way aswe did it in the proof of Proposition 3.3. Namely, by the definition (3.1) theoperator C ( µ ) is the ordered product of e R pq ( n − x p − x q ) over the pairs( p , q ) where 1 p < q ν + ν ; the arrangement of the pairs is reversedlexicographical. Without changing the product, we can rearrange these pairsas follows. From left to right, first come the pairs ( p , q ) where ν < p < q ν + ν , second come the pairs where 1 p ν and ν < q ν + ν ; thirdcome the pairs where 1 p < q ν . Within each of the three groups, thearrangement of the pairs ( p , q ) is still reversed lexicographical.Consider the two products of e R pq ( n − x p − x q ) , taken over the first and thethird group of pairs ( p , q ) . If g n = so n then the restriction of each of the twoproducts to the subspace (2.32) is 1 . If g n = sp n then the already settled caseof m = 1 implies that the restrictions of the two products to (2.32) coincidewith that of the sum in the second displayed line in Proposition 3.3, where a = 1 and a = 2 respectively. Now consider the product of e R pq ( n − x p − x q )over the second group of pairs. We have denoted this product by Z , see (3.6).For g n = so n , sp n we will show that the operator Z has the same restrictionto (2.32) as the sum in the first displayed line in Proposition 3.3, where a = 1and b = 2 . This will settle the case of m = 2 .Recall (2.31). There for any fixed i we arrange the factors correspondingto the indices j = 1 , . . . , ν from right to left. That is, we arrange from left toright the factors corresponding to j = ν , . . . , x ν + j in (2.31)with j = ν , . . . , x ν + ν , . . . , x ν +1 . (3.19)It was only the increasing by 1 property of (3.19) that we used to prove that therestrictions of the operators (2.30) and (2.31) to the subspace (2.32) are equal.Hence we can replace the sequence (3.19) in this equality by any other sequenceof length ν that is increasing by 1 . As a replacement, let us use the sequence n − x ν +1 , . . . , n − x ν + ν . Since k ν in (2.30), then we get the equality in ( End C n ) ⊗ ( ν + ν ) (cid:16) X d > X i ,...,i d j ,...,j d d Y k =1 P i k j k x k + x ν +1 − n (cid:17) ( A ν ⊗ A ν ) = (3.20) −→ Y i =1 ,...,ν (cid:16) ←− Y j =1 ,...,ν R i,ν + j ( n − x i − x ν + ν − j +1 ) (cid:17) × ( A ν ⊗ A ν ) . (3.21) rreducible representations of Yangians 31 The indices i , . . . , i d and j , . . . , j d in (3.20) have the same range as inthe first displayed line in Proposition 3.3. By applying to (3.20) the operatorconjugation relative to the form h , i in each of the first ν tensor factors of( End C n ) ⊗ ( ν + ν ) we get the product( A ν ⊗ (cid:16) X d > X i ,...,i d j ,...,j d d Y k =1 e P i k j k x k + x ν +1 − n (cid:17) (1 ⊗ A ν ) = (cid:16) X d > X i ,...,i d j ,...,j d d Y k =1 e P i k j k x k + x ν +1 − n (cid:17) ( A ν ⊗ A ν ) . (3.22)The sum over d > a = 1 and b = 2, by the equalities (3.10) for k ν and x ν +1 = λ + ρ + ( n − / . Note that the product (3.20) commutes with the operator on ( C n ) ⊗ ( ν + ν ) reversing the order of the last ν tensor factors. Let us conjugate the product(3.21) by this operator. The result is the product −→ Y i =1 ,...,ν (cid:16) ←− Y j =1 ,...,ν R i,ν + ν − j +1 ( n − x i − x ν + ν − j +1 ) (cid:17) × ( A ν ⊗ A ν )= −→ Y i =1 ,...,ν (cid:16) −→ Y j =1 ,...,ν R i,ν + j ( n − x i − x ν + j ) (cid:17) × ( A ν ⊗ A ν ) =( A ν ⊗ × ←− Y i =1 ,...,ν (cid:16) −→ Y j =1 ,...,ν R i,ν + j ( n − x i − x ν + j ) (cid:17) × (1 ⊗ A ν ) . By applying to the last displayed line the operator conjugation relative to h , i in each of the first ν tensor factors of ( End C n ) ⊗ ( ν + ν ) we get the product Z × ( A ν ⊗ A ν ) . (3.23)The equality of (3.20) and (3.21) implies the equality of (3.22) and (3.23). Thelatter equality settles the case of m = 2 . ⊓⊔ The operators B ( µ ) and C ( µ ) preserve the subspace (2.27). Due to Lemmas2.1,2.2 and Propositions 2.4,3.1 the restriction of operator B ( µ ) C ( µ ) to thissubspace is an Y( g n ) -intertwining operator (1.16). Let J ( µ ) be this restrictiondivided by the rational functions (2.24) and (3.3), and in the case of g n = sp n also divided by the rational function (3.4). Then by Propositions 2.5 and 3.2 J ( µ ) : ϕ ν ⊗ . . . ⊗ ϕ ν m ϕ ν ⊗ . . . ⊗ ϕ ν m . (3.24) Theorem 3.4.
For a fixed weight ν = λ − µ + κ the rational function J ( µ ) isregular at any point µ ∈ h ∗ m where the weight λ + ρ is dominant. The operator valued rational function J ( µ ) does not vanish at any point µ ∈ h ∗ m due to the normalization (3.24). The regularity of J ( µ ) was proved in[KN] for all µ where the pair ( λ , µ ) is good; our Theorem 3.4 is more general.Below we will give explicit formulas for the operator J ( µ ) with dominant λ + ρ . Theorem 3.4 will follow from these formulas. For related results see thework of Isaev and Molev [IM]. Note that both Proposition 3.3 and Theorem 3.4can also be proved by using the arguments from [KN, Subsections 1.4 and 4.4]. Let the weight λ + ρ of f m = so m or of f m = sp m be dominant. Then λ , . . . , λ m obey the inequalities (2.40) where ρ , . . . , ρ m are now the labels ofthe half-sum of positive roots of f m . But for any given a < b the difference ρ a − ρ b = b − a is now the same as it was for gl m . Moreover, for f m we havethe same relation λ a − λ b = µ a − µ b + ν a − ν b as we had for gl m . Any of our two proofs of Theorem 2.7 now shows that theoperator B ( µ ) divided by (2.24) has a regular restriction to the subspace (2.27).Moreover, each of the two proofs gives an explicit formula for the restriction.We will give two parallel proofs of the regularity of restriction to (2.27) ofthe operator C ( µ ) divided by (3.3), and in the case f m = sp m also divided by(3.4). We will keep assuming that the weight λ + ρ is dominant. In particular, λ a + λ b + ρ a + ρ b = 1 , , . . . whenever a < b (3.25) λ a + ρ a = 1 , , . . . if f m = sp m . (3.26)Then the operator C ( µ ) has a regular restriction to (2.27) by Proposition 3.3.Let us now prove the last fact directly, that is without using Proposition 3.3.Take any pair of indices ( p , q ) with p < q and consider the correspondingfactor e R pq ( n − x p − x q ) of the product (3.1). Let a and b be the numbers of thesegments of the sequence 1 , . . . , N containing the indices p and q respectively.Let i and j be the numbers of the indices p and q within their segments. Then x p + x q − n = λ a + ρ a + λ b + ρ b − i − j + 1 . (3.27)First, suppose that p and q belong to different segments, so that a < b .Then the right hand side of the equality (3.27) is not zero by (3.25). Thereforethe factor e R pq ( n − x p − x q ) of the product (3.1) with a < b is regular.Next, suppose that p and q belong to the same segment, that is a = b . If f m = so m then the factor e R pq ( n − x p − x q ) can be skipped without changingthe restriction of (3.1) to the subspace (2.27). If f m = sp m then by (3.26) theright hand side of the equality (3.27) is not zero whenever the sum i + j is odd.In particular, if f m = sp m then the factor e R pq ( n − x p − x q ) with q = p + 1 isregular, because for this factor j = i + 1 . rreducible representations of Yangians 33 Last, suppose that q > p + 1 while a = b . Then the pair following ( p , q )in the reversed lexicographical ordering is ( p , q −
1) . Moreover, then the index q − p and q . Take the product of the factorsin (3.1) corresponding the pairs following ( p , q −
1) . Multiply this product by R q − ,q ( x q − − x q ) = R q − ,q (1) = 1 − P q − ,q (3.28)on the right. By using the relation (2.4) repeatedly, one shows that the resultingproduct is also divisible by (3.28) on the left. Due to (2.8) we can then replacein (3.1) the product of two adjacent factors e R pq ( n − x p − x q ) e R p,q − ( n − x p − x q − )by 1 − ( e P pq + e P p,q − ) / ( n − x p − x q − )without changing the restriction of the operator (3.1) to the subspace (2.27).But the replacement is regular at x p + x q = n . This observation completes oursecond proof of the regularity of the restriction of C ( µ ) to the subspace (2.27),whenever the weight λ + ρ of f m is dominant.Now recall that defining the operator J ( µ ) involves dividing C ( µ ) by (3.3),and also dividing by (3.4) in the case f m = sp m . So we have to consider thezeroes of the rational functions (3.3) and (3.4). In Subsection 3.5 for m = 1 and g n = sp n we will prove that (3.11) annihilates the subspace (3.12) whenever λ + ρ = 0 and 2 ν > n . (3.29)In Subsection 3.6 for m = 2 and both g n = so n , sp n we will prove that theoperator (3.6) annihilates the subspace (2.32) whenever λ + λ + ρ + ρ = 0 and ν + ν > n . (3.30)Theorem 3.4 for any m > For m = 1 and g n = sp n consider the operator (3.11) on the vector space( C n ) ⊗ ν . Here the positive integer n is even. For p = 1 , . . . , ν introduce therational function of x ∈ C taking values in the operator algebra ( End C n ) ⊗ ν D ( x , p ) = ←− Y j =2 ,...,p (cid:16) ←− Y i =1 ,...,j − e R ij ( i + j − x − (cid:17) . (3.31)By (3.10) we have C ( µ ) = D ( λ + ρ , ν ) (3.32)while D ( x ,
1) = 1 by definition. Put D ( x ,
0) = 1 . Like we did at the beginningof the proof of Proposition 3.3, let us relate two operators on the vector space( C n ) ⊗ ν by the symbol ≡ if their actions on the subspace (3.12) coincide. Lemma 3.5.
For each p = 2 , . . . , ν we have the relation in ( End C n ) ⊗ ν e P p − ,p D ( x , p ) ≡ x + n/ − p + 1 x − e P p − ,p D ( x − , p − . (3.33) Proof.
Using Proposition 3.3 where m = 1 whereas the numbers λ + ρ and ν are replaced by x and p respectively, we obtain the relation D ( x , p ) ≡ X d > X i ,...,i d j ,...,j d d Y k =1 e P i k j k x − k ) (3.34)where i , j , . . . , i d , j d are pairwise distinct indices from the sequence 1 , . . . , p taken so that different are all the sets of d unordered pairs (3.8). Like in theproof of Proposition 3.3 we assume that 1 is the only term in (3.34) with d = 0 .For any d > d pairs (3.8) made as aboveconsider the corresponding product e P i j . . . e P i d j d showing in (3.34). Multiplythe product by e P p − ,p on the left. If neither of the indices p − , p occurs in thepairs (3.8) then leave the result of multiplication as it is, e P p − ,p e P i j . . . e P i d j d . Next, suppose that exactly one of the indices p − , p occurs in (3.8). Wecan assume that then j d = p − j d = p without further loss of generality.Put j = p or j = p − e P p − ,p e P i j . . . e P i d j d = e P p − ,p e P i j . . . e P i d − j d − P i d j ≡− e P p − ,p e P i j . . . e P i d − j d − . (3.35)Note that in either case, that is j d = p − j d = p , for any given distinctindices i , j , . . . , i d − , j d − taken from 1 , . . . , p − p − d choices of the index i d yielding the same term (3.35) where i d does not occur.Counting both cases, we will get the term (3.35) with multiplicity 2 ( p − d ) .Finally, suppose that both of the indices p − , p occur in (3.8). If theyoccur in the same pair, then without further loss of generality we may assumethat i d = p − j d = p . Then e P p − ,p e P i j . . . e P i d j d = n e P p − ,p e P i j . . . e P i d − j d − . (3.36)If p − , p occur in different pairs in (3.8) then without further loss of generalitywe may assume that j d − = p − i d = p . Then e P p − ,p e P i j . . . e P i d j d = e P p − ,p e P i j . . . e P i d − j d − P i d − ,p e P p,j d ≡− e P p − ,p e P i j . . . e P i d − j d − e P i d − j d . (3.37)Without altering the value of the term (3.37), we can either exchange the pair( i d − , j d ) with any of the pairs ( i , j ) , . . . , ( i d − , j d − ) or swap the two indices i d − , j d between each other. Counting these together with the initial choice of i d − , j d we will get the term (3.37) with multiplicity 2 ( d −
1) . Note that herethe term (3.37) arises only when d >
2. But for d = 1 we have 2 ( d −
1) = 0 . rreducible representations of Yangians 35
Let us multiply the relation (3.34) by e P p − ,p on the left, and then performthe summation over the indices d and i , j , . . . , i d , j d . The terms (3.35),(3.36)occur only when d > d − d > p − d − j d in (3.37) by j d − and then replace d − d . After this, the corresponding multiplicity is 2 d . Then d > d > d = 0 is zero.After these replacements, the product e P p − ,p D ( x , p ) gets related by ≡ to X d > X i ,...,i d j ,...,j d e P p − ,p (cid:18) n − p − d − − d x − d − (cid:19) d Y k =1 e P i k j k x − k ) = X d > X i ,...,i d j ,...,j d e P p − ,p x + n/ − p + 1 x − d − d Y k =1 e P i k j k x − k ) = X d > X i ,...,i d j ,...,j d e P p − ,p x + n/ − p + 1 x − d Y k =1 e P i k j k x − k − i , j , . . . , i d , j d are distinct indices taken from the sequence 1 , . . . , p − d unordered pairs (3.8). But the sum inthe last displayed line is related by ≡ to the right hand side of (3.33). Here weuse the relation (3.34) with x − p − x and p respectively. ⊓⊔ From now until the end of this subsection we assume that 2 ν > n . Denote l = ν − n/ . It is a classical fact [W, Section VI.3] that then the subspace (3.12) is containedin the span of the images of all operators on ( C n ) ⊗ ν of the form e P i j . . . e P i l j l where i , j , . . . , i l , j l are any pairwise distinct indices taken from 1 , . . . , ν .To show that C ( µ ) ≡ e P i j . . . e P i l j l C ( µ ) ≡ λ + ρ = 0 (3.38)and for all those i , j , . . . , i l , j l . Here we also use the equality e P = n e P andthe fact that the operator C ( µ ) with generic µ preserves the subspace (3.12).Furthermore, due to the latter fact, it suffices to prove the relation (3.38) forany single choice of the indices i , j , . . . , i l , j l . Let us choose i = ν − l + 1 , j = ν − l + 2 , . . . , i l = ν − , j l = ν . By using (3.32) and then applying Lemma 3.5 repeatedly, namely applying it l times, we get the following relation between rational functions of µ ∈ h ∗ : e P ν − l +1 , ν − l +2 . . . e P ν − ,ν C ( µ ) ≡ λ + ρ µ + ρ × e P ν − l +1 , ν − l +2 . . . e P ν − ,ν D ( µ + ρ , n − ν ) . To get the latter relation, we also used the equality ν − l = n − ν . If λ + ρ = 0 then µ + ρ = − l , and the fraction in the above display equalszero. The last factor in that display is regular at λ + ρ = 0 by the definition(3.31). This proves (3.38) for our choice of i , j , . . . , i l , j l .As a rational function of µ , the last displayed factor can be replaced bythe sum at the right hand side of the relation (3.34) where x = µ + ρ and p = n − ν . Arguing like in Subsection 3.4, we can also provide a multiplicativeformula for the value of that factor whenever µ + ρ = 1 , , . . . .Multiplying the last displayed relation on the left by operators on ( C n ) ⊗ ν which permute the ν tensor factors, we obtain analogues of that relation forall other choices of i , j , . . . , i l , j l . Therefore in the case m = 1 and 2 ν > n ,the last displayed relation determines the action on the subspace (3.12) for anyvalue of the function C ( µ ) divided by (3.4), whenever µ + ρ = 1 , , . . . . For m = 2 and g n = so n , sp n consider the operator (3.6) on ( C n ) ⊗ ( ν + ν ) .For p = 1 , . . . , ν and q = ν + 1 , . . . , ν + ν introduce the rational functionof x ∈ C taking values in the algebra ( End C n ) ⊗ ( ν + ν ) D ( x , p , q ) = ←− Y i =1 ,...,p (cid:16) ←− Y j =1 ,...,q − ν e R ij ( i + j − x − (cid:17) . (3.39)By (3.10) Z = D ( λ + λ + ρ + ρ , ν , ν + ν ) . (3.40)Put D ( x , p , ν ) = D ( x , , q ) = 1 . Let us relate operators on ( C n ) ⊗ ( ν + ν ) bythe symbol ≡ if their actions coincide on the subspace (2.32). Lemma 3.6.
For any p = 1 , . . . , ν and q = ν + 1 , . . . , ν + ν we have e P pq D ( x , p , q ) ≡ x + n − p − q + ν + 1 x − e P pq D ( x − , p − , q − . (3.41) Proof.
At the end of the proof of Proposition 3.3 we established the equalityof the expressions (3.22) and (3.23). Replacing the numbers λ + λ + ρ + ρ and ν , ν in that equality by x and p , q − ν respectively, we get the relation D ( x , p , q ) ≡ X d > X i ,...,i d j ,...,j d d Y k =1 e P i k j k x − k (3.42)where i , . . . , i d and j , . . . , j d are distinct indices taken respectively from thesequences 1 , . . . , p and ν + 1 , . . . , q so that different are the corresponding setsof d pairs (2.29). We assume that 1 is the only summand in (3.42) with d = 0 .For any d > d pairs (2.29) made as aboveconsider the corresponding product e P i j . . . e P i d j d showing in (3.42). Multiplythe product by e P pq on the left. If neither of the indices p , q occurs in the pairs(2.29) then leave the result of multiplication as it is, that is e P pq e P i j . . . e P i d j d . rreducible representations of Yangians 37 Next, suppose that exactly one of the indices p , q occurs in (2.29). We canassume that then i d = p or j d = q without further loss of generality. Then e P pq e P i j . . . e P i d j d ≡ − e P pq e P i j . . . e P i d − j d − . (3.43)In the first case, that is if i d = p , for any given i , j , . . . , i d − , j d − there areexactly q − ν − d choices of the index j d yielding the same right hand sideof (3.43), where j d does not occur. In the second case, that is if j d = q , forany given i , j , . . . , i d − , j d − there are exactly p − d choices of the index i d yielding the same right hand side of (3.43), where i d does not occur. Countingboth cases, we will get the term (3.43) with multiplicity p + q − ν − d .Last suppose p , q both occur in (2.29). If they occur in the same pair, thenwithout further loss of generality we may assume that i d = p and j d = q . Then e P pq e P i j . . . e P i d j d = n e P pq e P i j . . . e P i d − j d − . (3.44)If p , q occur in different pairs in (2.29) then without further loss of generalitywe may assume that j d − = q and i d = p . Then e P pq e P i j . . . e P i d j d ≡ − e P pq e P i j . . . e P i d − j d − e P i d − j d . (3.45)Without altering the product at the right hand side of (3.45), we can exchangethe pair ( i d − , j d ) with any of the pairs ( i , j ) , . . . , ( i d − , j d − ) . Countingthese together with the initial choice of ( i d − , j d ) we will get the term (3.37)with multiplicity d − d > e P pq on the left, and then perform thesummation over the indices d and i , j , . . . , i d , j d . The terms (3.43),(3.44)occur only when d > d − d > p + q − ν − d − j d at the right hand side of (3.45)by j d − , and then replace d − d there. Having done this, the correspondingmultiplicity becomes d . Now d > d > d = 0 is zero.After these replacements, the product e P pq D ( x , p , q ) gets related by ≡ to X d > X i ,...,i d j ,...,j d e P pq (cid:18) n − ( p + q − ν − d − − dx − d − (cid:19) d Y k =1 e P i k j k x − k = X d > X i ,...,i d j ,...,j d e P pq x + n − p − q + ν + 1 x − d Y k =1 e P i k j k x − k − i , . . . , i d and j , . . . , j d are pairwise distinct indices taken respectivelyfrom 1 , . . . , p − ν + 1 , . . . , q − d pairs (2.29). The sum in the last displayed line is related by ≡ to theright hand side of (3.41). Here we use the relation (3.42) with x − , p − , q − x , p , q respectively. ⊓⊔ From now until the end of this subsection assume that ν + ν > n . Denote l = ν + ν − n . Then the subspace (2.32) is contained in the span of the images of all operatorson ( C n ) ⊗ ( ν + ν ) of the form e P i j . . . e P i l j l where i , . . . , i l and j , . . . , j l arepairwise distinct indices from the sequences 1 , . . . , ν and ν + 1 , . . . , ν + ν respectively; see for instance the proof of [W, Lemma V.7.B]. To show that Z ≡ e P i j . . . e P i l j l Z ≡ λ + λ + ρ + ρ = 0 (3.46)and for all those i , j , . . . , i l , j l . Here we also use the fact that for any generic µ ∈ h ∗ the operator Z preserves the subspace (2.32). Due to the latter fact, itsuffices to prove (3.46) for any single choice of i , j , . . . , i l , j l . Let us choose i = ν − l + 1 , j = ν + ν − l + 1 , . . . , i l = ν , j l = ν + ν . By using (3.40) and then applying Lemma 3.6 repeatedly, namely applying it l times, we get the following relation between rational functions of µ ∈ h ∗ : e P ν − l +1 ,ν + ν − l +1 . . . e P ν ,ν + ν Z ≡ λ + λ + ρ + ρ µ + µ + ρ + ρ × e P ν − l +1 ,ν + ν − l +1 . . . e P ν ,ν + ν D ( µ + µ + ρ + ρ , n − ν , n ) . If λ + λ + ρ + ρ = 0 then µ + µ + ρ + ρ = − l , and the fraction inthe above display equals zero. But the last factor in that display is regular at λ + λ + ρ + ρ = 0 by the definition (3.39). This proves the relation (3.46)for our particular choice of the indices i , j , . . . , i l , j l .As rational function of µ , the last factor can be replaced by the sum at righthand side of the relation (3.42) where x = µ + µ + ρ + ρ while p = n − ν and q = n . Arguing like in Subsection 3.4, we can also provide a multiplicativeformula for the value of that factor whenever µ + µ + ρ + ρ = 1 , , . . . .Further, multiplying the last displayed relation on the left by operators on( C n ) ⊗ ( ν + ν ) which permute the first ν tensor factors between themselves,and also permute the last ν tensor factors, we get analogues of that relationfor all other choices of i , j , . . . , i l , j l . So in the case m = 2 and ν + ν > n ,the last displayed relation determines the action on the subspace (2.32) for anyvalue of the function Z divided by (3.3), when µ + µ + ρ + ρ = 1 , , . . . . In this subsection we will generalize Theorem 1.2. Theorem 3.4 allows usto determine the intertwining operator J ( µ ) of the Y( g n ) -modules (1.16) forany µ ∈ h ∗ m , provided λ + ρ is dominant. Our generalization of Theorem 1.2 isbased on the next two lemmas. For each index c = 1 , . . . , m − s c as anelement of the group H m . The action of s c on h m exchanges the basis vectors F cc and F c +1 ,c +1 , leaving all other basis vectors of h m fixed. The first of thetwo lemmas is an analogue of Lemma 2.8 for the twisted Yangian Y( g n ) . rreducible representations of Yangians 39 Lemma 3.7.
Fix c > . Suppose that both λ + ρ and s c ( λ + ρ ) are dominant.Then the images of the intertwining operators J ( µ ) and J ( s c ◦ µ ) correspondingto the pairs ( λ , µ ) and ( s c ◦ λ , s c ◦ µ ) are equivalent as Y( g n ) -modules.Proof. Let ˇ µ and ˇ ρ be the weights of gl with the labels µ c , µ c +1 and ρ c , ρ c +1 respectively. Let ˇ λ be the weight of gl with labels µ c + ν c , µ c +1 + ν c +1 . Here µ c + ν c = λ c + n/ µ c +1 + ν c +1 = λ c +1 + n/ . The dominance of the weights λ + ρ and s c ( λ + ρ ) of f m implies that the weightsˇ λ + ˇ ρ and s (ˇ λ + ˇ ρ ) of gl are also dominant. By Theorem 2.7 with m = 2 , weget an intertwining operator of Y( gl n ) -modules (2.45). It is invertible, see theproof of Lemma 2.8. Like in that proof, denote by I the operator which actson the tensor product of c th and ( c + 1) th factors of (1.8) as this intertwiningoperator (2.45), and which acts trivially on other m − m = 2 once again, we get an invertibleintertwining operator of Y( gl n ) -modules Φ − ν c µ c + ρ c + ⊗ Φ − ν c +1 µ c +1 + ρ c +1 + → Φ − ν c +1 µ c +1 + ρ c +1 + ⊗ Φ − ν c µ c + ρ c + . (3.47)Now denote by J the operator which acts as (3.47) on the tensor product of c th and ( c + 1) th factors of the target Y( g n ) -module in (1.16), and which actstrivially on other m − g n ) -modules in (1.16)for any generic weight µ of f m , we obtain the relation J J ( µ ) = J ( s c ◦ µ ) I whenever λ + ρ and s c ( λ + µ ) are dominant. It proves Lemma 3.7, since I and J are invertible and intertwine Y( g n ) -modules by restriction from Y( gl n ). ⊓⊔ Let s ∈ H m be the element which acts on h m by mapping F to − F ,and leaves all other basis vectors fixed. Note that in the case of f m = so m theelement s belongs to the extended Weyl group, not to the Weyl group proper.Further, in this case the dominance of s ( λ + ρ ) is equivalent to that of λ + ρ . Lemma 3.8.
Suppose that both the weights λ + ρ and s ( λ + ρ ) are dominant.Then the images of the intertwining operators J ( µ ) and J ( s ◦ µ ) correspondingto the pairs ( λ , µ ) and ( s ◦ λ , s ◦ µ ) are similar as Y( g n ) -modules.Proof. Let ˇ λ , ˇ µ , ˇ ρ be the weights of f with labels λ , µ , ρ respectively. Theweights ˇ λ + ˇ ρ and s (ˇ λ + ˇ ρ ) of f are dominant. For f = sp this means that λ / ∈ Z \ { } . For f = so any weight is dominant. By using Theorem 3.4 with m = 1 we get the intertwining operators of Y( g n ) -modules Φ ν µ + ρ + → Φ − ν µ + ρ + , (3.48) Φ n − ν − µ − ρ → Φ ν − n − µ − ρ . (3.49) For f = so each of these two operators acts as the identity, see remarks atthe beginning of Subsection 3.2. For f = sp none of these two operators actsas the identity in general, but they are still invertible. The latter assertion canbe proved either by direct calculation, or by using the irreducibility of all fourY( g n ) -modules in (3.48) and (3.49) for generic ˇ µ , that is for µ / ∈ Z / gl n ) -modules in (3.49) we get an intertwiningoperator Φ − ν µ + ρ + → Φ ν µ + ρ + (3.50)of Y( sp n ) -modules. This operator maps ϕ ν ϕ ν as does the operator (3.48).Hence the operators (3.48) and (3.50) are inverse to each other for µ / ∈ Z / µ ∈ C such that λ / ∈ Z \ { } .By using Lemma 2.3 with k = ν and t = µ + ρ + we get an invertibleintertwining operator Φ − ν µ + ρ + → ˙ Φ n − ν − µ − ρ (3.51)of Y( gl n ) -modules. It is also an intertwiner of Y( g n ) -modules by restriction.Denote by I the operator which acts as the composition of (3.48) with (3.51)on the first tensor factor the source Y( g n ) -module in (1.16), and which actstrivially on other m − g n ) -modules (3.49) can also be regarded as that of theY( g n ) -modules ˙ Φ n − ν − µ − ρ → ˙ Φ ν − n − µ − ρ (3.52)where we use the notation introduced immediately before stating Lemma 2.3.Denote by J the operator which acts as the composition of (3.51) with (3.52)on the first tensor factor the target Y( g n ) -module in (1.16), and which actstrivially on other m − J ( s ◦ µ ) is an intertwining operator of Y( g n ) -modules Φ n − ν − µ − ρ ⊗ Φ ν µ + ρ + ⊗ . . . ⊗ Φ ν m µ m + ρ m + ↓ Φ ν − n − µ − ρ ⊗ Φ − ν µ + ρ + ⊗ . . . ⊗ Φ − ν m µ m + ρ m + . Let us now replace the first tensor factors of the above two Y( g n ) -modules by˙ Φ n − ν − µ − ρ and ˙ Φ ν − n − µ − ρ respectively. The operator J ( s ◦ µ ) also intertwines the resulting two tensorproducts as Y( g n ) -modules. Take J ( s ◦ µ ) in its latter capacity. Then arguinglike in the end of the proof of Lemma 2.8, that is either performing a directcalculation, or using the irreducibility of the source and target Y( g n ) -modulesin (1.16) for any generic weight µ of f m , we obtain the relation J J ( µ ) = J ( s ◦ µ ) I for any dominant weights λ + ρ and s ( λ + µ ) of f m . This proves Lemma 3.8,because both I and J are invertible and intertwine Y( g n ) -modules. ⊓⊔ rreducible representations of Yangians 41 For any λ ∈ h ∗ m let H λ be the subgroup of H m consisting of all elements w such that w ◦ λ = λ . Let O be an orbit of the shifted action of the subgroup H λ ⊂ H m on h ∗ m . If ν , . . . , ν m ∈ { , . . . , n − } for at least one weight µ ∈ O ,then every µ ∈ O satisfies the same condition. Suppose this is the case for O .If λ + ρ is dominant, then there is at least one weight µ ∈ O such that the pair( λ , µ ) is good. Theorem 1.2 generalizes due to the following proposition. Proposition 3.9. If λ + ρ is dominant, then for all µ ∈ O the images of thecorresponding operators J ( µ ) are similar to each other as Y( g n ) -modules.Proof. Take any w ∈ H λ and any reduced decomposition w = s c l . . . s c . Here c , . . . , c l > s c k . . . s c ( λ + ρ ) of f m is dominant for each k = 1 , . . . , l . Applying Lemmas3.7 and 3.8 now completes the proof of Proposition 3.9. ⊓⊔ Thus all assertions of Theorem 1.2 will remain valid if we replace the goodpair there by any pair ( λ , µ ) such that the weight λ + ρ of f m is dominant.However, we still assume that ν , . . . , ν m ∈ { , . . . , n − } for the latter pair. Acknowledgements
We are grateful to Ivan Cherednik for helpful discussions. The first author wassupported by the RFBR grant 11-01-00962, by the joint RFBR-Royal Societygrant 11-01-92612, and by the Federal Agency for Science and Innovations ofthe Russian Federation under the contract 14.740.11.0347.
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