On explicit realization of algebra of complex powers of generators of U q (sl(3))
aa r X i v : . [ m a t h . QA ] O c t On explicit realization of algebra of complex powers ofgenerators of U q ( sl (3)) Pavel Sultanich , Moscow Center for Continuous Mathematical Education, 119002, Bolshoy Vlasyevsky Pereulok 11,Moscow, Russia
Abstract
In this note we prove an integral identity involving complex powers of generators of quantum group U q ( sl (3)) considered as certain positive operators in the setting of positive principal series representations.This identity represents a continuous analog of one of the Lusztig’s relations between divided powers ofgenerators of quantum groups, which play an important role in the study of irreducible modules [15].We also give definitions of arbitrary functions of U q ( sl (3)) generators and give another proofs for someof the known results concerning positive principal series representations of U q ( sl (3)). The notion of modular double of a quantum group U q ( g ) plays an important role in different areas ofmathematical physics such as Liouville theory [16],[6], relativistic Toda model [13] and others. It wasintroduced by Faddeev in [5] who noticed that certain representations of a quantum group U q ( sl (2)), q = e πıb have a remarkable duality under b ↔ b − and proposed to consider instead of single quantum group enlargedobject generated by two sets of generators K , E , F ∈ U q ( sl (2)) and ˜ K , ˜ E , ˜ F ∈ U ˜ q ( sl (2)), ˜ q = e πıb − . In[1] it was shown that in a special class of representations of modular double the rescaled generators definedby K , E = − ı ( q − q − ) E , F = − ı ( q − q − ) F of U q ( sl (2)) are positive operators. This allows one to usefunctional calculus and consider arbitrary functions of them. Moreover, the generators of dual group ˜ K ,˜ E = − ı (˜ q − ˜ q − ) ˜ E , ˜ F = − ı (˜ q − ˜ q − ) ˜ F are expressed as non-integer powers of the original generators˜ K = K b − , (1.1)˜ E = E b − , (1.2)˜ F = F b − . (1.3)These relations were called transcendental relations. This kind of representations, admitting the transcen-dental relations, has been generalized to higher ranks [7], [9] and has been called positive principal seriesrepresentations. Introduction of particular non-integer powers of generators of quantum group naturallyleads to consideration of arbitrary powers of generators. Thus, the modular double becomes a discretesubalgebra in the algebra generated by arbitrary powers of generators K ıp , E ıs , F ıt .In [15], eq.(4.1a)-eq.(4.1j) Lusztig summarized the relations between the divided powers of generators of U q ( g ) for simply-laced g . He used these identities in the study of finite-dimensional modules of U q ( g ) in thecase where q is a root of unity. So in the study of the algebra of arbitrary complex powers of quantum groupgenerators, the question of the generalization of these relations arises. Some of the relations were found in[8], eq.(6.16), eq.(6.17). Another integral relation which is a generalization of Kac’s identity [15], eq.(4.1a) E-mail: [email protected] U q ( sl (2)) in [19]. To write itdown explicitly, let G b ( x ) be quantum dilogarithm [4] which is a special function playing an important rolein the study of algebra of complex powers of generators of U q ( g ). Its properties will be outlined in Section 2.Let K j = q H j , E j , F j be U q ( g ) generators which are assumed to be positive operators so that the functionsof them are defined. Let continuous analogs of divided powers be defined by A ( ıs ) i = G b ( − ıbs ) A ıs . Explicitexpressions for the powers of operators under consideration will be given later. Then the generalized Kac’sidentity reads E ( ıs ) j F ( ıt ) j = Z C dτ e πbQτ F ( ıt + ıτ ) j K − ıτj G b ( ıbτ ) G b ( − bH j + ıb ( s + t + τ )) G b ( − bH j + ıb ( s + t + 2 τ )) E ( ıs + ıτ ) j , (1.4)where the contour C goes slightly above the real axis but passes below the pole at τ = 0. In this note weprove this identity for the case of positive principal series representations of U q ( sl (3)).The paper is organized as follows. In Section 2, we recall the definition of quantum group U q ( g ) andoutline the definition and basic properties of the quantum dilogarithm G b ( x ) and related function g b ( x ). InSection 3 we recall the construction of arbitrary functions of generators and generalized Kac’s identity inthe case of positive principal series representations of U q ( sl (2)).The main result of the paper is formulated inTheorem 4.1 in Section 4. We define arbitrary functions of U q ( sl (3)) generators in the positive principal seriesrepresentations. We prove the generalized Kac’s identity using unitary transform interwining the formulasof functions of generators of U q ( sl (2)) i subalgebra corresponding to simple root i , with the formulas for U q ( sl (2)), defined in Section 3. This calculation also represents another proof of Theorem 4.7 in [8] whichstates that positive principal series representation of U q ( sl (3)) decomposes into direct integral of positiveprincipal series representations of its U q ( sl (2)) subalgebra corresponding to each simple root. Acknowledgements:
The research was supported by RSF (project 16-11-10075). I am grateful toA.A.Gerasimov and D.R.Lebedev for helpful discussions and interest in this work.
We start with the definition of quantum groups following [2],[14]. Let ( a ij ) ≤ i,j ≤ r be Cartan matrix ofsemisimple Lie algebra g of rank r . Let b ± ⊂ g be opposite Borel subalgebras. For simplicity let us restrictourselves to the simply-laced case a ii = 2, a ij = a ji = { , − } , i = j . Let U q ( g ) ( q = e πıb , b ∈ R \ Q ) bethe quantum group with generators E j , F j , K j = q H j , 1 ≤ j ≤ r and relations K i K j = K j K i , (2.1) K i E j = q a ij E j K i , (2.2) K i F j = q − a ij F j K i , (2.3) E i F j − F j E i = δ ij K i − K − i q − q − . (2.4)For a ij = 0 we have E i E j = E j E i , (2.5) F i F j = F j F i . (2.6)For a ij = − E i E j − ( q + q − ) E i E j E i + E j E i = 0 , (2.7) F i F j − ( q + q − ) F i F j F i + F j F i = 0 , (2.8)Coproduct is given by ∆ E j = E j ⊗ K − j ⊗ E j , (2.9)∆ F j = 1 ⊗ F j + F j ⊗ K j , (2.10)2 K j = K j ⊗ K j . (2.11)Non-compact quantum dilogarithm G b ( z ) is a special function introduced in [4] (see also [3], [6], [20],[11], [13], [1]). It is defined as followslog G b ( z ) = log ¯ ζ b − Z R + ı dtt e zt (1 − e bt )(1 − e b − t ) , (2.12)where Q = b + b − and ζ b = e πı + πı ( b b − . Note, that G b ( z ) is closely related to the double sine function S ( z | ω , ω ), see eq.(A.22) in [13].Below we outline some properties of G b ( z )1. The function G b ( z ) has simple poles and zeros at the points z = − n b − n b − , (2.13) z = Q + n b + n b − , (2.14)respectively, where n , n are nonnegative integer numbers.2. G b ( z ) has the following asymptotic behavior: G b ( z ) ∼ ( ¯ ζ b , Imz → + ∞ ,ζ b e πız ( z − Q ) , Imz → −∞ . (2.15)3. Functional equation: G b ( z + b ± ) = (1 − e πıb ± z ) G b ( z ) . (2.16)4. Reflection formula: G b ( z ) G b ( Q − z ) = e πız ( z − Q ) . (2.17)5. 4-5 integral identity, [20]: Z dτ e − πγτ G b ( α + ıτ ) G b ( β + ıτ ) G b ( α + β + γ + ıτ ) G b ( Q + ıτ ) = G b ( α ) G b ( β ) G b ( γ ) G b ( α + γ ) G b ( β + γ ) . (2.18)Define also the function g b ( x ) by g b ( x ) = ¯ ζ b G b ( Q + πıb log x ) . (2.19)It has the following properties:1. | g b ( x ) | = 1, if x ∈ R + . So if A is a positive self-adjoint operator, then g b ( A ) is unitary.2. Fourier transform: g b ( x ) = Z dτ x ıb − τ e πQτ G b ( − ıτ ) . (2.20)Let U , V be positive self-adjoint operators satisfying the relation U V = q V U . Then the following non-commutative identities hold:3. Quantum exponential relation, [5]: g b ( U ) g b ( V ) = g b ( U + V ) . (2.21)4. Quantum pentagon relation, [10]: g b ( V ) g b ( U ) = g b ( U ) g b ( q − U V ) g b ( V ) . (2.22)5. Another useful relation, [1]: U + V = g b ( qU − V ) U g ∗ b ( qU − V ) , (2.23)where the star means hermitian conjugation. 3 Algebra of complex powers of generators of U q ( sl (2)) Let q = e πıb , ( b ∈ R \ Q ) and let K = q H , E , F be the generators of U q ( sl (2)) subjected to the relations KE = q EK, (3.1) KF = q − F K, (3.2) EF − F E = K − K − q − q − . (3.3)Define the rescaled versions of generators E , F by E = − ı ( q − q − ) E, (3.4) F = − ı ( q − q − ) F. (3.5)Let ν be a positive real number. There is a well-known representation of U q ( sl (2)) (see e.g.[17]): H = − ıb − u, (3.6) K = q H = e πbu , (3.7) E = q − e πbν + πbu e − ıb∂ u + q e − πbν − πbu e − ıb∂ u , (3.8) F = q − e πbν − πbu e ıb∂ u + q e − πbν + πbu e ıb∂ u . (3.9)This representation is a particular example of positive principal series representations of U q ( g ) [9].The following lemma was proven for a slightly different representation of U q ( sl (2)) in [1] and for the represen-tation we use in this paper in [8]. It gives the expressions of the generators of U q ( sl (2)) in a form convenientfor the definition of functions of them. It is based on the formula eq.(B.2) in [1], stating that given positiveself-adjoint operators U , V satisfying U V = q V U , one can write U + V = g b ( qU − V ) U ( g b ( qU − V )) − . (3.10) Lemma 3.1
Let E , F be the rescaled positive generators of U q ( sl (2)) defined above. They can be written inthe following form: E = g b ( e − πbν − πbu ) e πbν + πbu − ıb∂ u g ∗ b ( e − πbν − πbu ) , (3.11) F = g b ( e − πbν +2 πbu ) e πbν − πbu + ıb∂ u g ∗ b ( e − πbν +2 πbu ) . (3.12) Proof . Let U , V be positive self-adjoint operators such that U V = q V U.
Then, [1]: U + V = g b ( qU − V ) U ( g b ( qU − V )) − . For E we have U = q − e πbν + πbu e − ıb∂ u , V = q e − πbν − πbu e − ıb∂ u and qU − V = qq e ıb∂ u e − πbν − πbu q e − πbν − πbu e − ıb∂ u = e − πbν − πbu , so E = g b ( e − πbν − πbu ) q − e πbν + πbu e − ıb∂ u ( g b ( e − πbν − πbu )) − . For F we have U = q − e πbν − πbu e ıb∂ u , V = q e − πbν + πbu e ıb∂ u , qU − V = qq e − ıb∂ u e − πbν + πbu q e − πbν + πbu e ıb∂ u = e − πbν +2 πbu , F = g b ( e − πbν +2 πbu ) q − e πbν − πbu e ıb∂ u ( g b ( e − πbν +2 πbu )) − . ✷ Multiplication by g b ( e − πbν − πbu ) and g b ( e − πbν +2 πbu ) is unitary transformation, since | g b ( x ) | = 1, for x ∈ R + . As a consequence, following [1], eq.(3.15), eq.(3.21), one can define functions of generators E and F as follows: 4 efinition 3.1 Let ϕ ( x ) be a complex-valued function and let K , E , F be U q ( sl (2)) generators in the positiveprincipal series representation. The functions of these operators are defined as follows ϕ ( K ) = ϕ ( e πbu ) , (3.13) ϕ ( E ) = g b ( e − πbν − πbu ) ϕ ( e πbν + πbu − ıb∂ u ) g ∗ b ( e − πbν − πbu ) , (3.14) ϕ ( F ) = g b ( e − πbν +2 πbu ) ϕ ( e πbν − πbu + ıb∂ u ) g ∗ b ( e − πbν +2 πbu ) . (3.15)In particular, the powers of E and F are given by E ıs = g b ( e − πbν − πbu ) e πıbsν + πıbsu + bs∂ u g ∗ b ( e − πbν − πbu ) , (3.16) F ıt = g b ( e − πbν +2 πbu ) e πıbtν − πıbtu − bt∂ u g ∗ b ( e − πbν +2 πbu ) . (3.17)The formulas for the powers in this particular representation were obtained in [8].Define the arbitrary divided powers of A by A ( ıs ) = G b ( − ıbs ) A ıs . (3.18) Theorem 3.1 [19] The following generalized Kac’s identity holds E ( ıs ) F ( ıt ) = Z C dτ e πbQτ F ( ıt + ıτ ) K − ıτ G b ( ıbτ ) G b ( − bH + ıb ( s + t + τ )) G b ( − bH + ıb ( s + t + 2 τ )) E ( ıs + ıτ ) , (3.19) where the contour C goes slightly above the real axis but passes below the pole at τ = 0 . U q ( sl (3)) In this section we prove the Generalized Kac’s identity in the case of positive principal series representationsof U q ( sl (3)).Let q = e πıb , ( b ∈ R \ Q ). Let a ij , ( i , j = 1, 2) be the Cartan matrix corresponding to sl (3) Lie algebra,i.e. a = a = 2, a = a = − U q ( sl (3)) ( q = e πıb , b ∈ R \ Q ) is defined by generators E j , F j , K j = q H j , 1 ≤ j ≤ K i K j = K j K i , (4.1) K i E j = q a ij E j K i , (4.2) K i F j = q − a ij F j K i , (4.3) E i F j − F j E i = δ ij K i − K − i q − q − . (4.4)For i = j we have E i E j − ( q + q − ) E i E j E i + E j E i = 0 , (4.5) F i F j − ( q + q − ) F i F j F i + F j F i = 0 . (4.6)The general construction of the positive principal series representations of U q ( g ) in the simply-laced caseusing Lusztig’s data was given in [9]. Let w be the longest element of the Weyl group. There are differentrealizations of positive principal series representations corresponding to each reduced expressions of w . Inthe case of U q ( sl (3)) there are two options w = s s s and w = s s s . In the following we give the explicitformulas for both of this cases.Let E j = − ı ( q − q − ) E j , F j = − ı ( q − q − ) F j , j = 1, 2 be the rescaled versions of U q ( sl (3)) generators.5 roposition 4.1 [9]. Let K j , E j , F j be the rescaled generators of U q ( sl (3)) . Let w = s s s be reducedexpression of the longest Weyl element. Let ν , ν be positive real numbers. The positive principal seriesrepresentation of U q ( sl (3)) corresponding to these data is given by: K = e − πbν +2 πbu − πbv +2 πbw , (4.7) K = e − πbν − πbu +2 πbv − πbw , (4.8) E = e πbw − ıb∂ w + e − πbw − ıb∂ w , (4.9) E = e πbv − πbw − ıb∂ v + e πbu − ıb∂ u − ıb∂ v + ıb∂ w + e − πbu − ıb∂ u − ıb∂ v + ıb∂ w + e − πbv + πbw − ıb∂ v , (4.10) F = e πbν − πbu + πbv − πbw + ıb∂ w + e πbν − πbu + ıb∂ u + e − πbν + πbu + ıb∂ u + e − πbν +2 πbu − πbv + πbw + ıb∂ w , (4.11) F = e πbν + πbu − πbv + ıb∂ v + e − πbν − πbu + πbv + ıb∂ v . (4.12)Similar to U q ( sl (2)) case using eq.(B.2), [1] we represent the generators in a form convenient for thedefinition of functions of them Lemma 4.1
Let E i , F i , ( i = 1 , be the generators of U q ( sl (3)) in the positive principal series representationcorresponding to the reduced expression w = s s s . They can be represented in the following form E = g b ( e − πbw ) e πbw − ıb∂ w g ∗ b ( e − πbw ) , (4.13) E = g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) × e πbv − πbw − ıb∂ v × g ∗ b ( e − πbv +2 πbw ) g ∗ b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) , (4.14) F = g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) × e πbν − πbu + πbv − πbw + ıb∂ w × g ∗ b ( e − πbν +4 πbu − πbv +2 πbw ) g ∗ b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g ∗ b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) , (4.15) F = g b ( e − πbν − πbu +2 πbv ) e πbν + πbu − πbv + ıb∂ v g ∗ b ( e − πbν − πbu +2 πbv ) . (4.16) Proof . Let q = e πıb and let U , V be positive essentially self-adjoint operators subjected to the relation U V = q V U . We will need the following identity,[1]: U + V = g b ( qU − V ) U g ∗ b ( qU − V ) , and the quantum exponential relation, [5]: g b ( U + V ) = g b ( U ) g b ( V ) . Let us start with E . It has the following form E = U + V, where U = e πbw − ıb∂ w , V = e − πbw − ıb∂ w . Using the identities e A e B = e [ A,B ]2 e A + B and e A e B = e [ A,B ] e B e A inthe case when the commutator [ A, B ] commutes with both A and B , and also the identity [ x, ∂ x ] = −
1, onechecks that
U V = e πbw − ıb∂ w e − πbw − ıb∂ w = e [ πbw − ıb∂ w , − πbw − ıb∂ w ] e − πbw − ıb∂ w e πbw − ıb∂ w = e πıb e − πbw − ıb∂ w e πbw − ıb∂ w = q V U, qU − V = e πıb e − πbw + ıb∂ w e − πbw − ıb∂ w = e πıb e [ − πbw + ıb∂ w , − πbw − ıb∂ w ] = e − πbw , so we have E = g b ( qU − V ) U g ∗ b ( qU − V ) = g b ( e − πbw ) e πbw − ıb∂ w g ∗ b ( e − πbw ) . For E we have E = U + U + U + U , where U = e πbv − πbw − ıb∂ v U = e πbu − ıb∂ u − ıb∂ v + ıb∂ w , U = e − πbu − ıb∂ u − ıb∂ v + ıb∂ w , U = e − πbv + πbw − ıb∂ v .These operators satisfy the relations U i U j = q U j U i , if i < j . We obtain E = g b ( qU − ( U + U + U )) U g ∗ b ( qU − ( U + U + U )) = g b ( qU − U ) g b ( qU − U ) g b ( qU − U ) U g ∗ b ( qU − U ) g ∗ b ( qU − U ) g ∗ b ( qU − U ) , where in the second equality we have used the quantum exponential relation, provided that operators( qU − U i ) are positive and satisfy ( qU − U i )( qU − U j ) = q ( qU − U j )( qU − U i ) for 1 < i < j . qU − U = e πıb e − πbv + πbw + ıb∂ v e πbu − ıb∂ u − ıb∂ v + ıb∂ w = e πıb e [ − πbv + πbw + ıb∂ v ,πbu − ıb∂ u − ıb∂ v + ıb∂ w ] e πbu − πbv + πbw − ıb∂ u + ıb∂ w = e πıb e − πıb e πbu − πbv + πbw − ıb∂ u + ıb∂ w = e πbu − πbv + πbw − ıb∂ u + ıb∂ w . Analogously we obtain qU − U = e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ,qU − U = e − πbv +2 πbw . Substituting these expressions into the formula for E we obtain E = g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) e πbv − πbw − ıb∂ v × g ∗ b ( e − πbv +2 πbw ) g ∗ b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) . For the generator F we have F = U + U + U + U , where we used the notations U = e πbν − πbu + πbv − πbw + ıb∂ w ,U = e πbν − πbu + ıb∂ u ,U = e − πbν + πbu + ıb∂ u ,U = e − πbν +2 πbu − πbv + πbw + ıb∂ w . Again, for i < j we have the relations U i U j = q U j U i , and for 1 < i < j ( qU − U i )( qU − U j ) = q ( qU − U j )( qU − U i ) . Explicit expressions for the operators qU − U i are given by qU − U = e πbu − πbv + πbw + ıb∂ u − ıb∂ w ,qU − U = e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ,qU − U = e − πbν +4 πbu − πbv +2 πbw . U + V = g b ( qU − V ) U g ∗ b ( qU − V ) and g b ( U + V ) = g b ( U ) g b ( V ) for positive operatorssatisfying the relation U V = q V U we obtain F = g b ( qU − ( U + U + U )) U g ∗ b ( qU − ( U + U + U )) = g b ( qU − U ) g b ( qU − U ) g b ( qU − U ) U g ∗ b ( qU − U ) g ∗ b ( qU − U ) g ∗ b ( qU − U ) = g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) × e πbν − πbu + πbv − πbw + ıb∂ w × g ∗ b ( e − πbν +4 πbu − πbv +2 πbw ) g ∗ b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g ∗ b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) . Generator F . F = A + A , where A = e πbν + πbu − πbv + ıb∂ v ,A = e − πbν − πbu + πbv + ıb∂ v . Then qA − A = e − πbν − πbu +2 πbv ,A A = q A A , and using the identity A + A = g b ( qA − A ) A g ∗ b ( qA − A ) , we obtain F = g b ( e − πbν − πbu +2 πbv ) e πbν + πbu − πbv + ıb∂ v g ∗ b ( e − πbν − πbu +2 πbv ) . ✷ Similar to eq.(3.15), eq.(3.21) [1], we define the functions of the operators in the following way:
Definition 4.1
Let ϕ ( x ) be a complex-valued function of one variable and let K i , E i , F i , ( i = 1 , be thegenerators of U q ( sl (3)) in the positive principal series representation corresponding to the reduced expression w = s s s . ϕ ( K ) = ϕ ( e − πbν +2 πbu − πbv +2 πbw ) , (4.17) ϕ ( K ) = ϕ ( e − πbν − πbu +2 πbv − πbw ) , (4.18) ϕ ( E ) = g b ( e − πbw ) ϕ ( e πbw − ıb∂ w ) g ∗ b ( e − πbw ) , (4.19) ϕ ( E ) = g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) × ϕ ( e πbv − πbw − ıb∂ v ) × g ∗ b ( e − πbv +2 πbw ) g ∗ b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) , (4.20) ϕ ( F ) = g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) × ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) × g ∗ b ( e − πbν +4 πbu − πbv +2 πbw ) g ∗ b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g ∗ b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) , (4.21) ϕ ( F ) = g b ( e − πbν − πbu +2 πbv ) ϕ ( e πbν + πbu − πbv + ıb∂ v ) g ∗ b ( e − πbν − πbu +2 πbv ) . (4.22)8hoosing in the definition the function to be ϕ ( x ) = x ıs we obtain the expressions for arbitrary powers ofgenerators.In the following we repeat the same steps for representations corresponding to another choice of reducedexpression w = s s s . Proposition 4.2 [9]. The positive principal series representation of U q ( sl (3)) corresponding to the reducedexpression of the longest Weyl element w = s s s and positive real parameters ν , ν is given by K = e − πbν − πbu +2 πbv − πbw , (4.23) K = e − πbν +2 πbu − πbv +2 πbw , (4.24) E = e πbv − πbw − ıb∂ v + e πbu − ıb∂ u − ıb∂ v + ıb∂ w + e − πbu − ıb∂ u − ıb∂ v + ıb∂ w + e − πbv + πbw − ıb∂ v , (4.25) E = e πbw − ıb∂ w + e − πbw − ıb∂ w , (4.26) F = e πbν + πbu − πbv + ıb∂ v + e − πbν − πbu + πbv + ıb∂ v , (4.27) F = e πbν − πbu + πbv − πbw + ıb∂ w + e πbν − πbu + ıb∂ u + e − πbν + πbu + ıb∂ u + e − πbν +2 πbu − πbv + πbw + ıb∂ w . (4.28) Lemma 4.2
Let E i , F i , ( i = 1 , be the generators of U q ( sl (3)) in the positive principal series representationcorresponding to the reduced expression w = s s s . They can be represented in the following form E = g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) × e πbv − πbw − ıb∂ v × g ∗ b ( e − πbv +2 πbw ) g ∗ b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) , (4.29) E = g b ( e − πbw ) e πbw − ıb∂ w g ∗ b ( e − πbw ) , (4.30) F = g b ( e − πbν − πbu +2 πbv ) e πbν + πbu − πbv + ıb∂ v g ∗ b ( e − πbν − πbu +2 πbv ) , (4.31) F = g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) × e πbν − πbu + πbv − πbw + ıb∂ w × g ∗ b ( e − πbν +4 πbu − πbv +2 πbw ) g ∗ b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g ∗ b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) . (4.32) Definition 4.2
Let ϕ ( x ) be a complex-valued function of one variable and let K i , E i , F i , ( i = 1 , be thegenerators of U q ( sl (3)) in the positive principal series representation corresponding to the reduced expression w = s s s .The functions of the generators are defined as follows: ϕ ( K ) = ϕ ( e − πbν − πbu +2 πbv − πbw ) , (4.33)9 ( K ) = ϕ ( e − πbν +2 πbu − πbv +2 πbw ) , (4.34) ϕ ( E ) = g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) × ϕ ( e πbv − πbw − ıb∂ v ) × g ∗ b ( e − πbv +2 πbw ) g ∗ b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) , (4.35) ϕ ( E ) = g b ( e − πbw ) ϕ ( e πbw − ıb∂ w ) g ∗ b ( e − πbw ) , (4.36) ϕ ( F ) = g b ( e − πbν − πbu +2 πbv ) ϕ ( e πbν + πbu − πbv + ıb∂ v ) g ∗ b ( e − πbν − πbu +2 πbv ) , (4.37) ϕ ( F ) = g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) × ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) × g ∗ b ( e − πbν +4 πbu − πbv +2 πbw ) g ∗ b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g ∗ b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) . (4.38)Assuming ϕ ( x ) = x ıs , we obtain the expressions for arbitrary powers of generators.Recall the definition of the divided powers of AA ( ıs ) = G b ( − ıbs ) A ıs . (4.39)Now, after we have defined arbitrary devided powers of the generators of U q ( sl (3)) in the positive principalseries representations corresponding to both reduced expressions of the Weyl element, we can state the maintheorem Theorem 4.1
Let q = e πıb , ( b ∈ R \ Q ) and let K j = q H j , E j = − ı ( q − q − ) E j , F j = − ı ( q − q − ) F j , ≤ j ≤ be U q ( sl (3)) generators in the positive principal series representation corresponding to any reducedexpression of the Weyl element. Then the following generalized Kac’s identity holds: E ( ıs ) j F ( ıt ) j = Z C dτ e πbQτ F ( ıt + ıτ ) j K − ıτj G b ( ıbτ ) G b ( − bH j + ıb ( s + t + τ )) G b ( − bH j + ıb ( s + t + 2 τ )) E ( ıs + ıτ ) j , (4.40) where the contour C goes slightly above the real axis but passes below the pole at τ = 0 .Proof . The proof follows from the results stated in Proposition 4.3, Lemma 4.3, Proposition 4.4, Corollary4.1, Corollary 4.2, Corollary 4.3.The next statement (Theorem 5.7 in [9]) establishes the unitary equivalence of positive principal seriesrepresentations corresponding to different expressions of the longest Weyl element. We give here anotherproof for the case of U q ( sl (3)) of this result which allows explicitly illustrate its validness for arbitraryfunctions of generators. In this proof the pentagon identity [10] is extensively used which states that forpositive self-adjoint operators U , V satisfying the relation U V = q V U we have g b ( V ) g b ( U ) = g b ( U ) g b ( q − U V ) g b ( V ) . (4.41)10 roposition 4.3 Let X s s s be any generator of U q ( sl (3)) in the positive principal series representationcorresponding to the reduced expression w = s s s . Let X s s s be the same generator in the positiveprincipal series representation corresponding to the reduced expression w = s s s and let ϕ ( x ) be a complex-valued function. The unitary transformation defined by U = ( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) . (4.42) relates the functions of generators in these two representations by ϕ ( X s s s ) = U ϕ ( X s s s ) U ∗ . Proof . Let E be the generator in s s s representation. Its unitary transform is given by U ϕ ( E ) U ∗ = ( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) × g b ( e − πbw ) ϕ ( e πbw − ıb∂ w )( h.c. ) =( uv )( vw ) e − w∂ u e w∂ v g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) × g b ( e − πbw ) ϕ ( e πbw − ıb∂ w )( h.c. ) . We have used the commutation of the factors g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) and g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ).Now, according to the identity g b ( x ) g b ( x ) = e πı π b log x we have g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) = e − πı π b ( πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) . Let A = e − πbw ,A = e − πbu + πbv − πbw − ıb∂ u + ıb∂ w , Then q − A A = e − πbu + πbv − πbw − ıb∂ u + ıb∂ w and A A = q A A . Using the pentagon identity, [10]: g b ( A ) g b ( A ) = g b ( A ) g b ( q − A A ) g b ( A ) , we have g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g b ( e − πbw ) = g b ( e − πbw ) g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) . Substituting all these into the expression for
U ϕ ( E ) U ∗ we obtain U ϕ ( E ) U ∗ =( uv )( vw ) e − w∂ u e w∂ v e − πı π b ( πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) × g b ( e − πbw ) g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) ϕ ( e πbw − ıb∂ w )( h.c. ) . Note, that g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) and ϕ ( e πbw − ıb∂ w ) commute, so the quantum dilogarithm passesthrough and cancels with its hermitian conjugate: U ϕ ( E ) U ∗ =( uv )( vw ) e − w∂ u e w∂ v e − πı π b ( πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) × g b ( e − πbw ) g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) ϕ ( e πbw − ıb∂ w )( h.c. ) . Let A , B be self-adjoint operators satisfying the relation [ A, B ] = c , where c is a number. Let f ( x ) be afunction and α a number. Then e αB f ( A ) = f ( A − cαB ) e αB . e − πı π b ( πbu − πbv + πbw − ıb∂ u + ıb∂ w ) to the right we obtain U ϕ ( E ) U ∗ = ( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) × g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) g b ( e − πbu +2 πbv − πbw ) ϕ ( e πbu − πbv +2 πbw − ıb∂ u )( h.c. )Now use the relation e αx∂ y f ( y, ∂ x ) = f ( y + αx, ∂ x − α∂ y ) e αx∂ y , to push the exponents e − w∂ u and e w∂ v to the right: U ϕ ( E ) U ∗ = ( uv )( vw ) g b ( e − πbu + πbv + πbw + ıb∂ v − ıb∂ w ) × g b ( e − πbu + πbv − πbw + ıb∂ v − ıb∂ w ) g b ( e − πbu +2 πbv ) ϕ ( e πbu − πbv − ıb∂ u )( h.c. ) = g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) ϕ ( e πbv − πbw − ıb∂ v )( h.c. )Let E be the generator in s s s representation. Then U ϕ ( E ) U ∗ = ( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) × g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) ϕ ( e πbv − πbw − ıb∂ v ) × ( h.c. ) , where by ( h.c. ) we denoted the hermitian conjugate operator of everything that stands before ϕ ( e πbv − πbw − ıb∂ v ).Noticing that g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) = 1 , we obtain U ϕ ( E ) U ∗ = ( uv )( vw ) e − w∂ u e w∂ v × g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbv +2 πbw ) ϕ ( e πbv − πbw − ıb∂ v ) × ( h.c. ) . Let A , A be as follows A = e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ,A = e − πbv +2 πbw , Then q − A A = e − πbu − πbv + πbw − ıb∂ u + ıb∂ w , moreover A A = q A A , and we can apply the pentagon identity [10]: g b ( A ) g b ( q − A A ) g b ( A ) = g b ( A ) g b ( A ) , which leads to the following result U ϕ ( E ) U ∗ = ( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbv +2 πbw ) g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) ϕ ( e πbv − πbw − ıb∂ v ) × ( h.c. ) . Operators g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) and ϕ ( e πbv − πbw − ıb∂ v ) commute, so the quantum dilogarithm passesthrough and cancels with its conjugate. We obtain U ϕ ( E ) U ∗ = ( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbv +2 πbw ) ϕ ( e πbv − πbw − ıb∂ v ) × ( h.c. ) =( uv )( vw ) g b ( e − πbv ) ϕ ( e πbv − ıb∂ v ) g ∗ b ( e − πbv )( vw )( uv ) = g b ( e − πbw ) ϕ ( e πbw − ıb∂ w ) g ∗ b ( e − πbw ) . The case of F U ϕ ( F ) U ∗ =( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) × b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w )( h.c. )The factors g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) and g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) commute so we can change theirorder. After that we observe that g b ( e − πbu + πbv − πbw − ıb∂ u + ıb∂ w ) g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) = e πı π b ( πbu − πbv + πbw + ıb∂ u − ıb∂ w ) , which follows from the identity g b ( x ) g b ( x ) = e πı π b log x . Doing these we obtain U ϕ ( F ) U ∗ =( uv )( vw ) e − w∂ u e w∂ v e πı π b ( πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) × g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w )( h.c. ) . Again using the identity g b ( x ) g b ( x ) = e πı π b log x to rewrite g ∗ b ( e πbu − πbv + πbw − ıb∂ u + ıb∂ w ) = e − πı π b ( πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) , we have U ϕ ( F ) U ∗ =( uv )( vw ) e − w∂ u e w∂ v e πı π b ( πbu − πbv + πbw + ıb∂ u − ıb∂ w ) e − πı π b ( πbu − πbv + πbw − ıb∂ u + ıb∂ w ) g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) × g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w )( h.c. ) . Let A = e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ,A = e − πbν +4 πbu − πbv +2 πbw . Then q − A A = e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ,A A = q A A , and we can apply the pentagon identity g b ( A ) g b ( q − A A ) g b ( A ) = g b ( A ) g b ( A ): g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) = g b ( e − πbν +4 πbu − πbv +2 πbw ) g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w )Note also that g b ( e − πbu + πbv − πbw + ıb∂ u − ıb∂ w ) commutes with ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) so we obtain U ϕ ( F ) U ∗ =( uv )( vw ) e − w∂ u e w∂ v e πı π b ( πbu − πbv + πbw + ıb∂ u − ıb∂ w ) e − πı π b ( πbu − πbv + πbw − ıb∂ u + ıb∂ w ) × g b ( e − πbν +4 πbu − πbv +2 πbw ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w )( h.c. )To push the quadratic exponents to the right we use the formula e αB f ( A ) = f ( A − cαB ) e αB for self-adjoint A and B satisfying the relation [ A, B ] = c , where c , α are numbers: U ϕ ( F ) U ∗ = ( uv )( vw ) e − w∂ u e w∂ v e πı π b ( πbu − πbv + πbw + ıb∂ u − ıb∂ w ) × g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w )( h.c. ) =( uv )( vw ) e − w∂ u e w∂ v g b ( e − πbν +2 πbu ) ϕ ( e πbν − πbu + ıb∂ u )( h.c. ) =( uv )( vw ) g b ( e − πbν +2 πbu − πbw ) ϕ ( e πbν − πbu + πbw + ıb∂ u )( h.c. ) = g b ( e − πbν − πbu +2 πbv ) ϕ ( e πbν + πbu − πbv + ıb∂ v )( h.c. ) . ✷ K , E , F be the subset of U q ( sl (3)) generators in s s s principal series representation. Recall thatfor a complex-valued function ϕ ( x ) we have ϕ ( K ) = ϕ ( e − πbν +2 πbu − πbv +2 πbw ) ,ϕ ( E ) = g b ( e − πbw ) ϕ ( e πbw − ıb∂ w ) g ∗ b ( e − πbw ) ,ϕ ( F ) = g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) × ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) × g ∗ b ( e − πbν +4 πbu − πbv +2 πbw ) g ∗ b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g ∗ b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) , This subset generates U q ( sl (2)) subalgebra.The unitary transform from the following proposition was given in the proof of Theorem 4.7 in [8]. It wasused as the first step in mapping the generators K i , E i , F i of U q ( sl (2)) i subalgebra of U q ( g ) to the formulascorresponding to positive principal series representations of U q ( sl (2)). We explicitly check its action on thefunctions of generators. Lemma 4.3
Let K , E , F be as above. Let ϕ ( x ) be a complex-valued function. Let V = e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u e − u∂ w e − πıw g b ( e πbw ) g b ( e πbν − πbu ) (4.43) be a unitary transform. Then V ϕ ( K ) V ∗ = ϕ ( e πbw ) , (4.44) V ϕ ( E ) V ∗ = ϕ ( e − ıb∂ w ) , (4.45) V ϕ ( F ) V ∗ = g b ( e πbw e − πbu ) g b ( e πbw e ıb∂ u ) g b ( e πbw e πbu ) × ϕ ( e − πbw + ıb∂ w ) × g ∗ b ( e πbw e πbu ) g ∗ b ( e πbw e ıb∂ u ) g ∗ b ( e πbw e − πbu ) . (4.46) Proof . V ϕ ( F ) V ∗ = e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u e − u∂ w e − πıw g b ( e πbw ) × g b ( e πbν − πbu ) g b ( e πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) × ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) × ( h.c. ) . Let U = e πbν − πbu ,U = e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w . Then q − U U = e πbu − πbv + πbw + ıb∂ u − ıb∂ w , and the following relation holds U U = q U U , g b ( U ) g b ( q − U U ) g b ( U ) = g b ( U ) g b ( U ) . We obtain
V ϕ ( F ) V ∗ = e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u e − u∂ w e − πıw g b ( e πbw ) × g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e πbν − πbu ) g b ( e − πbν +4 πbu − πbv +2 πbw ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) × ( h.c. )Note that the operator g b ( e πbν − πbu ) commutes with g b ( e − πbν +4 πbu − πbv +2 πbw ) and ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ),so it goes through and cancels with its hermitian conjugate. V ϕ ( F ) V ∗ = e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u e − u∂ w e − πıw g b ( e πbw ) × g b ( e − πbν +3 πbu − πbv + πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) × ( h.c. ) . Using the following operator relations e αx + βx f ( ∂ x ) = f ( ∂ x − αx − β ) e αx + βx ,e αx∂ y f ( y, ∂ x ) = f ( y + αx, ∂ x − α∂ y ) e αx∂ y , we move all the exponents to the right V ϕ ( F ) V ∗ = e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u e − u∂ w g b ( e πbw ) × g b ( e − πbν +3 πbu − πbv +2 πbw + ıb∂ u − ıb∂ w ) g b ( e − πbν +4 πbu − πbv +2 πbw ) ϕ ( e πbν − πbu + πbv − πbw + ıb∂ w ) × ( h.c. ) = e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u g b ( e πbw − πbu ) × g b ( e − πbν + πbu − πbv +2 πbw + ıb∂ u ) g b ( e − πbν +2 πbu − πbv +2 πbw ) ϕ ( e πbν + πbv − πbw + ıb∂ w ) × ( h.c. ) = e ( ν + v ) ∂ w e ( ν + v ) ∂ u g b ( e πbw − πbu ) g b ( e − πbν − πbv +2 πbw + ıb∂ u ) × g b ( e − πbν +2 πbu − πbv +2 πbw ) ϕ ( e πbν + πbv − πbw + ıb∂ w ) × ( h.c. ) = g b ( e πbw − πbu ) g b ( e πbw + ıb∂ u ) g b ( e πbw +2 πbu ) ϕ ( e − πbw + ıb∂ w ) g ∗ b ( e πbw +2 πbu ) g ∗ b ( e πbw + ıb∂ u ) g ∗ b ( e πbw − πbu ) . ✷ To finish the mapping ϕ ( K ), ϕ ( E ), ϕ ( F ) → ϕ ( K ), ϕ ( E ), ϕ ( F ), where the second set of operators isdefined by the equations (3.14)-(3.15), we need to perform a certain integral transformation which will bedefined shortly.Let λ be a positive real number. Define the following set of functions, [12], [13]Φ λ ( u ) = e πıu + πQu G b ( − ıu + ıλ ) G b ( − ıu − ıλ ) . (4.47)The integral transform Φ is defined by Φ : L ( R ) → L ( R + , dµ ( λ )) , Φ : f ( u ) → F ( λ ) = Z R − ı duf ( u )Φ ∗ λ ( u ) , (4.48)This transform is an isometry, see [12]. The inverse is given byΦ − : L ( R + , dµ ( λ )) → L ( R ) , Φ − : F ( λ ) → f ( u ) = lim ǫ → ∞ Z F ( λ )Φ λ ( u + ıǫ ) e − πǫu dµ ( λ ) , (4.49)with the measure given by dµ ( λ ) = 4 sinh( πbλ ) sinh( πb − λ ).15 roposition 4.4 The function Φ λ ( u ) is an eigenfunction of the operator g b ( e πbw e − πbu ) g b ( e πbw e ıb∂ u ) g b ( e πbw e πbu ) : g b ( e πbw e − πbu ) g b ( e πbw e ıb∂ u ) g b ( e πbw e πbu )Φ λ ( u ) = g b ( e πbλ +2 πbw ) g b ( e − πbλ +2 πbw )Φ λ ( u ) . (4.50) Proof . Recalling the definition of g b ( x ) g b ( x ) = ¯ ζ b G b ( Q + πıb log x ) , and the Fourier transform g b ( x ) = Z dτ x ıb − τ e πQτ G b ( − ıτ ) , we obtain g b ( e πbw e − πbu ) = ¯ ζ b G b ( Q − ıw + ıu ) ,g b ( e πbw e πbu ) = ¯ ζ b G b ( Q − ıw − ıu ) ,g b ( e πbw e ıb∂ u ) = Z dτ e πQτ +2 πıwτ G b ( − ıτ ) e − τ∂ u . Substituting these expressions into the left-hand side of the eigenvalue equation we obtain g b ( e πbw e − πbu ) g b ( e πbw e ıb∂ u ) g b ( e πbw e πbu )Φ λ ( u ) =1 G b ( Q − ıw + ıu ) Z dτ e πQτ +2 πıwτ G b ( − ıτ ) e − τ∂ u e πıu + πQu G b ( − ıu + ıλ ) G b ( − ıu − ıλ ) G b ( Q − ıw − ıu ) =1 G b ( Q − ıw + ıu ) Z dτ e πQτ +2 πıwτ + πı ( u − τ ) + πQ ( u − τ ) G b ( − ıτ ) G b ( − ıu + ıλ + ıτ ) G b ( − ıu − ıλ + ıτ ) G b ( Q − ıw − ıu + ıτ )Applying the reflection formula G b ( − ıτ ) = e − πıτ − πQτ G b ( Q + ıτ ) , we get g b ( e πbw e − πbu ) g b ( e πbw e ıb∂ u ) g b ( e πbw e πbu )Φ λ ( u ) = e πıu + πQu G b ( Q − ıw + ıu ) Z dτ e − π ( Q + ıu − ıw ) τ G b ( − ıu + ıλ + ıτ ) G b ( − ıu − ıλ + ıτ ) G b ( Q − ıw − ıu + ıτ ) G b ( Q + ıτ ) . Let α = − ıu + ıλ , β = − ıu − ıλ , γ = Q + ıu − ıw . Then g b ( e πbw e − πbu ) g b ( e πbw e ıb∂ u ) g b ( e πbw e πbu )Φ λ ( u ) = e πıu + πQu G b ( Q − ıw + ıu ) Z dτ e − πγτ G b ( α + ıτ ) G b ( β + ıτ ) G b ( α + β + γ + ıτ ) G b ( Q + ıτ ) = e πıu + πQu G b ( Q − ıw + ıu ) G b ( α ) G b ( β ) G b ( γ ) G b ( α + γ ) G b ( β + γ ) , here 4 − α , β , γ we obtain g b ( e πbw e − πbu ) g b ( e πbw e ıb∂ u ) g b ( e πbw e πbu )Φ λ ( u ) = e πıu + πQu G b ( − ıu + ıλ ) G b ( − ıu − ıλ ) G b ( Q + ıλ − ıw ) G b ( Q − ıλ − ıw ) =1 G b ( Q + ıλ − ıw ) G b ( Q − ıλ − ıw ) Φ λ ( u ) = g b ( e πbλ +2 πbw ) g b ( e − πbλ +2 πbw )Φ λ ( u ) . ✷ orollary 4.1 Let ϕ ( K ) , ϕ ( E ) , ϕ ( F ) be the functions of the subset of generators of U q ( sl (3)) in the positiveprincipal series representation corresponding to the reduced expression w = s s s of the longest Weylelement. Let Φ , Φ ∗ be the integral transform and its inverse defined in (4.48)-(4.49). Let Ω be the unitarytransform defined by Ω = e − πı ( λ + w ) g b ( e − πbλ − πbw ) ◦ Φ ◦ e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u e − u∂ w e − πıw g b ( e πbw ) g b ( e πbν − πbu ) . (4.51) Then Ω ϕ ( K )Ω ∗ = ϕ ( e πbw ) , (4.52)Ω ϕ ( E )Ω ∗ = g b ( e − πbλ − πbw ) ϕ ( e πbλ + πbw − ıb∂ w ) g ∗ b ( e − πbλ − πbw ) , (4.53)Ω ϕ ( F )Ω ∗ = g b ( e − πbλ +2 πbw ) ϕ ( e πbλ − πbw + ıb∂ w ) g ∗ b ( e − πbλ +2 πbw ) . (4.54) Corollary 4.2
Let ϕ ( K ) , ϕ ( E ) , ϕ ( F ) be the functions of the subset of generators of U q ( sl (3)) in the positiveprincipal series representation corresponding to the reduced expression w = s s s of the longest Weylelement. Let Ω be a unitary transform defined by Ω = e − πı ( λ + w ) g b ( e − πbλ − πbw ) ◦ Φ ◦ e ( ν + v ) ∂ w e ( ν + v ) ∂ u e − πıu +2 πıν u e − u∂ w e − πıw g b ( e πbw ) g b ( e πbν − πbu ) . (4.55) Then Ω ϕ ( K )Ω ∗ = ϕ ( e πbw ) , (4.56)Ω ϕ ( E )Ω ∗ = g b ( e − πbλ − πbw ) ϕ ( e πbλ + πbw − ıb∂ w ) g ∗ b ( e − πbλ − πbw ) , (4.57)Ω ϕ ( F )Ω ∗ = g b ( e − πbλ +2 πbw ) ϕ ( e πbλ − πbw + ıb∂ w ) g ∗ b ( e − πbλ +2 πbw ) . (4.58) Proof . Note, that swapping the indices K ↔ K , E ↔ E , F ↔ F , ν ↔ ν , of generators and parametersin a representation of U q ( sl (3)) corresponding to a particular choice of reduced expression of the longestWeyl element gives representation for another choice of reduced expression. So, from the statement, thatΩ transforms the action of operators ϕ ( K ), ϕ ( E ), ϕ ( F ) in s s s representation to the U q ( sl (2)) formulas(3.14)-(3.15), it automatically follows that Ω which is obtained from Ω by the replacement of the parameter ν by ν , transforms the action of operators ϕ ( K ), ϕ ( E ), ϕ ( F ) in s s s representation to U q ( sl (2)) formulas. ✷ Corollary 4.3
In the positive principal series representation corresponding to any reduced expression of theWeyl element the generalized Kac’s identity holds.Proof . As follows from the corollaries 4.2, 4.3, there is a unitary transformation which transforms the opera-tors ϕ ( K i ), ϕ ( E i ), ϕ ( F i ) defined in the positive principal series of U q ( sl (3)) to the operators ϕ ( K ), ϕ ( E ), ϕ ( F )defined in positive principal series representation of U q ( sl (2)). Since the generalized Kac’s identity is validin U q ( sl (2)) case, it follows that it is as well valid in the case of U q ( sl (3)). ✷ This completes the proof of the Theorem 4.1.Note, that we have also given another proof in U q ( sl (3)) for the Theorem 4.7 in [8] which states thatthe positive principal series representation of U q ( sl (3)) decomposes into direct integral of positive principalseries representations of its U q ( sl (2)) subalgebra corresponding to any simple root.17 eferences [1] A.Bytsko, J.Teschner, R-operator, co-product and Haar-measure for the modular double of U q ( sl (2; R )),arXiv:math/0208191v2.[2] V.Chari, A.Pressley, A guide to quantum groups, Cambridge University Press, 1994.[3] L.Faddeev, Current-Like Variables in Massive and Massless Integrable Models, arXiv:hep-th/9408041v1[4] L.Faddeev, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. v.3 ,(1995), 249.[5] L.Faddeev, Modular Double of Quantum Group, arXiv:math/9912078v1.[6] L.Faddeev, R.Kashaev, A.Volkov, Strongly coupled quantum discrete Liouville theory I: Algebraic ap-proach and duality, hep-th/0006156[7] I.Frenkel, I.Ip, Positive representations of split real quantum groups and future perspectives,arXiv:1111.1033v1[math.RT][8] I.Ip, Positive Representations of Split Real Quantum Groups: The Universal R Operator,arXiv:1212.5149v1[9] I.Ip, Positive Representations of Split Real simply-laced Quantum Groups, arXiv:1203.2018v4[10] R.M.Kashaev, On the spectrum of Dehn twists in quantum Teichmuller theory, arXiv:math/0008148v1[11] R.M.Kashaev, The non-compact quantum dilogarithm and the Baxter equations, J. Stat. Phys. 102(2001) 923–936.[12] R.M.Kashaev, The quantum dilogarithm and Dehn twist in quantum Teichmuller theory, IntegrableStructures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, Ukraine,September 25-30, 2000), NATO Sci. Ser. II Math. Phys. Chem., vol. 35, Kluwer, Dordrecht, 211-221(2001)[13] S.Kharchev, D.Lebedev, M.Semenov-Tian-Shansky, Unitary representations of U q ( sl (2 , R )), the modulardouble, and the multiparticle q -deformed Toda chains, Communication Math. Phys. v.225 (2002) 573–609.[14] G.Lusztig, Introduction to quantum groups, Progress in Mathematics,110, Boston, MA,1993.[15] G.Lusztig, Modular representations and quantum groups, Contemporary Mathematics, v. 82 (1989)59–77.[16] B.Ponsot, J.Teschner, Liouville bootstrap via harmonic analysis on a noncompact quantum group,arXiv:hep-th/9911110v2[17] B.Ponsot, J.Teschner, Clebsch-Gordan and Racah-Wigner coefficients for a continuous series of repre-sentations of U q ( sl (2 , R )), arXiv:math/0007097v2[18] P.Sultanich, On modular double of semisimple quantum groups, arXiv:1811.10934v1[19] P.Sultanich, On explicit realization of algebra of complex divided powers of U q ( sl (2)) ,,