aa r X i v : . [ m a t h . QA ] A ug Parabolic Positive Representations of U q ( g R ) Ivan C.H. Ip ∗ August 21, 2020
Abstract
We construct a new family of irreducible representations of U q ( g R ) and itsmodular double by quantizing the classical parabolic induction correspondingto arbitrary parabolic subgroups, such that the generators of U q ( g R ) act bypositive self-adjoint operators on a Hilbert space. This generalizes the well-established positive representations which corresponds to induction by theminimal parabolic (i.e. Borel) subgroup. We also study in detail the specialcase of type A n acting on L ( R n ) with minimal functional dimension, andestablish the properties of its central characters and universal R operator. Weconstruct a positive version of the evaluation modules of the affine quantumgroup U q ( b sl n +1 ) modeled over this minimal positive representation of type A n . Keywords. quantum groups, positive representations, cluster algebra, parabolic sub-groups, evaluation modules
Primary 17B37, 13F60
Contents U q ( g ) and D q ( g ) . . . . . . . . . . . . . . . . . . . 92.3 Quantum torus algebra . . . . . . . . . . . . . . . . . . . . . . . . . 12 U q ( g ) ∗ Department of Mathematics, Hong Kong University of Science and TechnologyEmail: [email protected] author is supported by the Hong Kong RGC General Research Funds ECS Minimal positive representations of U q ( sl ( n + 1 , R )) U q ( sl (4 , R )) . . . . . . . . . . . . . . . . . . . 254.2 Minimal positive representations for U q ( sl ( n + 1 , R )) . . . . . . . . . 324.3 Non-simple generators . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Central characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Evaluation modules of U q ( b sl n +1 ) . . . . . . . . . . . . . . . . . . . . 38 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.2 Type B n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Positive representations of U q ( g R ) Positive representations was introduced in [9] to study the representation theory of split real quantum groups U q ( g R ) associated to semisimple Lie algebra g , as well asits modular double U qq ∨ ( g R ) introduced by [4, 5] in the regime where | q | = 1. Theserepresentations are natural generalizations of a special class of representations of U q ( sl (2 , R )) originally studied by Teschner et al. [2, 23, 24] from the physics point ofview of quantum Liouville theory, which is characterized by the actions of positive(essentally) self-adjoint operators on the Hilbert space L ( R ).Based on quantizing the regular action on smooth functions on the totally posi-tive flag variety G > /B > , we constructed, for the simply-laced cases in [9, 13] andnon-simply-laced cases in [14], a family of irreducible representations P λ of U qq ∨ ( g R )with the generators acting on certain Hilbert space as positive self-adjoint operators,parametrized by λ ∈ P R + in the positive real-span of the dominant weights.Recently a cluster algebraic realization of these representations were also con-structed, first for type A n in [25] and the general case in [17], where we establishedan embedding of U q ( g ) into certain quantum torus algebra X q associated to the ba-sic quiver D ( i ), such that a choice of polarization of X q coincides with the positiverepresentations. The positive representations and their cluster realizations are longknown to be closely related to quantum higher Teichm¨uller theory [6, 8, 20], andrecently a full geometric interpretation based on the modular space of decorated G -local system is given in the monumental work of Goncharov and Shen [10].2 ain results In this paper, we construct a large class of irreducible representations of U qq ∨ ( g R ),called the parabolic positive representations , by quantizing the regular action on thetotally positive part of the partial flag variety G/P for any parabolic subgroups.These representations are still characterized by the fact that the quantum groupgenerators act by positive self-adjoint operators on some Hilbert spaces. In par-ticular, the original positive representation is a special case of this new family ofrepresentations. To summarize, the main results of this paper are the following(Theorem 5.2, Corollary 5.5, Theorem 5.16).
Main Theorem.
We have a homomorphism of the Drinfeld’s double D q ( g ) ontoa quantum torus algebra X D ( i ) q associated to a subquiver D ( i ) of the basic quiver D ( i ) of the positive representations, such that a polarization of X D ( i ) q provides apositive representation P Jλ for U q ( g R ) and its modular double U qq ∨ ( g R ) , and it is atwisted quantization of the parabolic induction. The proof of the Main Theorem consists of several ingredients. First we reviewthe construction of the basic quivers D ( i ) in Section 3.2, which were first con-structed in [17] for a reduced word i of the longest element w of the Weyl group.In this paper, we adapt the more general construction due to [10] in order to give anexplicit realization of the extra vertices , which were artificially added to the quiverpreviously in [17]. Next, we define a new notion of generalized Heisenberg double (Definition 5.7), which decomposes the image of D q ( g ) in the cluster algebra X q satisfying a new set of algebraic relations, and finally we prove the DecompositionLemma (Lemma 5.13), which decomposes the original positive representations inorder to give us the parabolic ones. This requires an understanding of the combina-torics of the Coxeter moves of a reduced word (Lemma 5.15), which was previouslyknown to the author but never explicitly written down. We remark in Section 5.6that the construction naturally works for the modular double U qq ∨ ( g R ) as well, thusestablishing the Main Theorem. Minimal positive representations
A special case of the parabolic positive representations is worthy of special attention,which we call the minimal positive representations . These are the representations of U q ( sl ( n + 1 , R )) in type A n , where we take the parabolic subgroup P to be the max-imal one, such that its codimension in SL ( n + 1 , R ) is the minimum. Correspondingto this choice, the quantum group U q ( g R ) acts on the minimal positive representa-tion P min λ ≃ L ( R n ) as positive operators parametrized by a single number λ ∈ R ,and this Hilbert space has the smallest functional dimension possible.In fact, due to its simplicity, these representations served as our original mo-tivation to investigate a possibility of degenerating the well-established positiverepresentations, and we later realized a way to extend the construction to arbi-trary parabolic cases. The simplicity of such representations is expected to findapplications in mathematical physics and integrable systems. Therefore we study3his family in more detail and establish several important properties concerning theuniversal R operator and the central characters (Theorem 4.5, Theorem 4.8). Main Theorem.
Acting on the minimal positive representations, • The universal R operator is well-defined as a unitary transformation on thetensor product P min λ ⊗ P min λ ′ of minimal positive representations. • The Casimir operators acts by real scalars on P min λ , and lies outside the spec-trum of the positive Casimirs. Finally, the simplicity of the minimal positive representations, interpreted as ahomomorphic image of the Drinfeld’s double D q ( g ) onto a quantum torus algebra X Q q associated to a very simple quiver Q , allows us to construct a new class ofpositive representations for the quantum affine algebra U q ( b sl ( n + 1 , R )) (althoughit is no longer parabolic). It is still an open question to appropriately define therepresentation theory of split real affine quantum groups U q ( b g R ), and this acciden-tal construction may allow for a first step. We show that the representations weobtained is isomorphic to the evaluation module of Jimbo [19] associated to theminimal positive representation of U q ( sl ( n + 1 , R )) constructed here. Geometric interpretation
In this paper, we follow our previous strategy in [9, 13, 17] to establish a posi-tive representations modeled over the totally positive part of the partial flag variety
G/P , and study its quantization through algebraic means, establishing a homomor-phism of the Drinfeld’s double onto the quantum torus algebra X D ( i ) q associated tothe (double of the) basic quiver Q ( i ). However, there are new difficulties that arenot present in the usual positive representation case. For example, in the generalparabolic case, we do not have a natural association of these basic quivers to trian-gles of the triangulation of punctured Riemann surfaces due to the degeneracy ofthe frozen degrees on the edges. Furthermore, several proofs in [9, 13, 14] and [10]utilized the freedom of the longest reduced word i (more precisely, the existenceof any reduced word of w starting with any letter) which is not available in theparabolic case as well.According to [10], the basic quiver Q ( i ) is naturally associated to the Poissonstructure of the partial configuration space Conf eu ( A ) of the principal affine space A for some element u ∈ W of the Weyl group. Therefore one should try to understandand possibly simplify the proofs of the parabolic positive representations by quan-tizing the geometrical methods using, perhaps, partial decorated G -local system,where the decorations are provided by partial flags, and study the combinatoricsof the its potential functions W i . In particular, we expect that the Lusztig’s braidgroup action can be established through the geometric point view of the parabolicpositive representations. 4 utline The outline of the paper is as follows. In Section 2, we establish our notations andgive preliminaries to the notion of total positivity, parabolic subgroups, quantumgroups, Drinfeld’s double, and quantum torus algebra. In Section 3 we summa-rize all the structural results of positive representations and its cluster realization,adapting to the notations used in this paper. In particular we recall the construc-tion of the basic quiver. In Section 4, we give our motivation with the toy model of U q ( sl (4 , R )), and construct the minimal positive representations for U q ( sl ( n + 1 , R ))in general, establishing some of its properties. We also generalize this constructionto the case of affine quantum group U q ( b sl ( n + 1 , R )) and compare with its evalu-ation modules. In Section 5, we construct the parabolic positive representation infull generality, by proving the Decomposition Lemma of the quantum group gen-erators. In Section 6 we provide two examples to demonstrate the construction ofthis paper, and finally in Section 7 we discuss several open research directions. Let g be a simple Lie algebra over C . Let G be the real simple Lie group cor-responding to the split real form g R of the Lie algebra g , and let B, B − be twoopposite Borel subgroups containing a split real maximal torus T = B ∩ B − . Let U + ⊂ B, U − ⊂ B − be the corresponding unipotent subgroups.Let I be the root index of the Dynkin diagram of g such that | I | = n = rank( g ) . (2.1)Let Φ be the set of roots of g , Φ + ⊂ Φ be the positive roots, and ∆ + = { α i } i ∈ I ⊂ Φ + be the positive simple roots. Let W = h s i i i ∈ I be the Weyl group of Φ generated bythe simple reflections s i := s α i . Definition 2.1.
Let ( − , − ) be a W -invariant inner product of the root lattice. Wedefine a ij := 2( α i , α j )( α i , α i ) , i, j ∈ I (2.2)such that A := ( a ij ) is the Cartan matrix .We normalize ( − , − ) as follows: we choose the symmetrization factors (alsocalled the multipliers ) such that for any i ∈ I , d i := 12 ( α i , α i ) = i is long root or in the simply-laced case, i is short root in type B, C, F , i is short root in type G , (2.3) The results of this paper obviously also work for the semisimple case by taking direct product. α i , α j ) = − i, j ∈ I are adjacent in the Dynkin diagram, such that d i a ij = d j a ji . (2.4) Definition 2.2.
Let w ∈ W . We call a sequence i = ( i , ..., i k ) , i k ∈ I (2.5)a reduced word of w if w = s i s i · · · s i k is a reduced expression, and let l ( w ) := | i | := k (2.6)be its length. We write i op := ( i k , ..., i ) (2.7)for the reduced word of w − .Let w be the longest element of W . Throughout the paper we let N := l ( w ) (2.8)and denote by i a reduced word of w .A useful fact is the following: Proposition 2.3. [10, Lemma 10.8] If i = ( i , ..., i k ) is a reduced word, then thereexists a reduced word i of w starting with i . Definition 2.4.
Let w ∈ W be the longest element of the Weyl group. TheDynkin involution I −→ Ii i ∗ (2.9)is defined by w s i w = s i ∗ . (2.10)Equivalently, we have w ( α i ) = α − i ∗ , α i ∈ ∆ + (2.11)where α − i are the negative simple roots.Next we recall the description of the Lusztig’s data for total positivity, given indetail in [22]. 6 efinition 2.5. For any i ∈ I , there exists a homomorphism SL ( R ) −→ G induced by (cid:18) a (cid:19) x i ( a ) ∈ U + i , (2.12) (cid:18) b b − (cid:19) h i ( b ) ∈ T, (2.13) (cid:18) c (cid:19) y i ( c ) ∈ U − i , (2.14)called a pinning of G , where U + i ⊂ U + and U − i ⊂ U − are the simple root subgroups of the unipotent subgroup U + and U − respectively. Lemma 2.6. [22, Proposition 2.7] Let i = ( i , ..., i N ) be a reduced word of thelongest element w . The map ι : R N> −→ U + ι : ( a , a , ..., a N ) x i ( a ) x i ( a ) ...x i N ( a N ) (2.15) is injective. The positive unipotent semigroup U + > is defined to be the image of ι : U + > := ι ( R N> ) . (2.16) We have the similar definition for U − > using y i instead. Definition 2.7.
The totally positive semigroup is defined to be G > := U − > T > U + > , (2.17)where U ± > is as above, and T > are generated by the images h i ( b ) with b ∈ R > , i ∈ I . Lemma 2.8.
We have the following identities in G > : for a, b, c ∈ R > and i, j ∈ I , x i ( a ) y j ( c ) = y j ( c ) x i ( a ) , if i = j, (2.18) h i ( b ) x j ( a ) = x j ( b a ij a ) h i ( b ) , (2.19) x i ( a ) h i ( b ) y i ( c ) = y i ( cac + b ) h i ( ac + b b ) x i ( aac + b ) , (2.20) x i ( a ) x j ( b ) = x j ( b ) x i ( a ) , a ij = 0 , (2.21) x i ( a ) x j ( b ) x i ( c ) = x j ( bca + c ) x i ( a + c ) x j ( aba + c ) , a ij = − , (2.22) x i ( a ) x j ( b ) x i ( c ) x j ( d ) = x j ( bc dS ) x i ( SR ) x j ( R S ) x i ( abcR ) , a ij = − , (2.23) We used h i instead of the more traditional χ i to distinguish it from the characters χ λ usedlater. here R = ab + ad + cd S = a b + d ( a + c ) . (2.24) We also have an explicit expression for the case a ij = − , see [1, Theorem 3.1],but we will not need them in this paper. The same relations (2.21) – (3.8) also holdfor y i . Next we recall some terminologies regarding parabolic subgroups that are es-sential in this paper.
Definition 2.9.
Let J ⊂ I be a (possibly empty) subset of the Dynkin nodes,and W J ⊂ W the corresponding subgroups generated by s j , j ∈ J . The Levisubgroup L J is the subgroup of G generated by the maximal torus T and the rootsubgroups U + j , U − j where j ∈ J . The standard parabolic subgroup P J is defined as P J = B − L J .We write N J := l ( w J ) (2.25)to be the length of the longest element w J ∈ W J of the Weyl subgroup W J ⊂ W .Hence the codimension of P J in G is given bycodim G ( P J ) = N − N J . (2.26)It is well-known that any parabolic subgroup, i.e. subgroup containing a Borelsubgroup, is conjugated to P J for some J . Remark 2.10.
By convention, if J = ∅ , then P J := B − and W J = { e } is thetrivial subgroup. Example 2.11.
For G = SL ( R ), let J = { , , } . Then the standard parabolicsubgroup P J ⊂ G is the set of 5 × P J = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ . (2.27)If P J is a parabolic subgroup, we denote by P > J := G > ∩ P (2.28)its totally positive part.It follows from (2.17) that we have the Langlands decomposition as well in thetotally positive case. We used the opposite version instead of the usual P J = L J B to fit the calculations later, sothat in SL n the parabolic subgroups P J are represented by lower block triangular matrices. emma 2.12. Let w J be the longest element of the subgroup W J ⊂ W and i J =( j , ..., j N J ) , j k ∈ J ⊂ I be its reduced expression. Then we have P > J = U − > T > M > (2.29) where M > is the image of the embedding ι : R N J > −→ U + > ( a , ..., a N J ) x j ( a ) x j ( a ) · · · x j NJ ( a N J ) . (2.30)Since P J contains the full subgroup of SL under the pinning correspondingto the simple root α j where j ∈ J , we have the following characterization of the characters of P > J in the totally positive scenario: Lemma 2.13.
Let h ( a ) · · · h n ( a n ) ∈ h > , a i ∈ R > (2.31) be the coordinates of the abelian component T > of the Langlands decomposition of g ∈ P > J . Then any positive character χ : P > J −→ R > of the totally positive part P > J is parametrized by the scalars λ = ( λ k ) ∈ R | I \ J | ,given by χ λ ( g ) := Y k ∈ I \ J a λ k k , g ∈ P > J . (2.32) U q ( g ) and D q ( g ) For any finite dimensional complex semisimple Lie algebra g , Drinfeld [3] and Jimbo[18] associated to it a remarkable Hopf algebra U q ( g ) known as quantum group ,which is certain deformation of the universal enveloping algebra. We follow thenotations used in [17] for U q ( g ) as well as the Drinfeld’s double D q ( g ) of its Borelpart.In the following, we assume again that g is of simple Dynkin type. Definition 2.14.
Let d i be the multipliers (2.3). We define q i := q d i , (2.33)which we will also write as q l := q, (2.34) q s := (cid:26) q if g is of type B, C, F ,q if g is of type G, (2.35)for the q parameters corresponding to long and short roots respectively.9 efinition 2.15. We define D q ( g ) to be the C ( q s )-algebra generated by the ele-ments { E i , F i , K ± i , K ′± i } i ∈ I subject to the following relations (we will omit the obvious relations involving K − i and K ′ i − below for simplicity): K i E j = q a ij i E j K i , K i F j = q − a ij i F j K i , (2.36) K ′ i E j = q − a ij i E j K ′ i , K ′ i F j = q a ij i F j K ′ i , (2.37) K i K j = K j K i , K ′ i K ′ j = K ′ j K ′ i , K i K ′ j = K ′ j K i , (2.38)[ E i , F j ] = δ ij K i − K ′ i q i − q − i , (2.39)together with the Serre relations for i = j : − a ij X k =0 ( − k [1 − a ij ] q i ![1 − a ij − k ] q i ![ k ] q i ! E ki E j E − a ij − ki = 0 , (2.40) − a ij X k =0 ( − k [1 − a ij ] q i ![1 − a ij − k ] q i ![ k ] q i ! F ki F j F − a ij − ki = 0 , (2.41)where [ k ] q := q k − q − k q − q − is the q -number, and [ n ] q ! := n Y k =1 [ k ] q is the q -factorial.The algebra D q ( g ) is a Hopf algebra with comultiplication∆( E i ) = 1 ⊗ E i + E i ⊗ K i , ∆( K i ) = K i ⊗ K i , (2.42)∆( F i ) = F i ⊗ K ′ i ⊗ F i , ∆( K ′ i ) = K ′ i ⊗ K ′ i , (2.43)We will not need the counit and antipode in this paper. Definition 2.16.
The quantum group U q ( g ) is defined as the quotient U g ( g ) := D q ( g ) / h K i K ′ i = 1 i i ∈ I , (2.44)and it inherits a well-defined Hopf algebra structure from D q ( g ). Remark 2.17. D q ( g ) is the Drinfeld’s double of the quantum Borel subalgebra U q ( b ) generated by E i and K i . Definition 2.18.
We define the rescaled generators e i := (cid:18) √− q i − q − i (cid:19) − E i , f i := (cid:18) √− q i − q − i (cid:19) − F i . (2.45)10y abuse of notation , we will also denote by D q ( g ) the C ( q s )-algebra generated by { e i , f i , K i , K ′ i } i ∈ I (2.46)and the corresponding quotient by U q ( g ). The generators satisfy all the definingrelations above except (2.39) which is modified to be[ e i , f j ] = δ ij ( q i − q − i )( K ′ i − K i ) . (2.47)In the split real case, we require | q | = 1 and write q := e π √− b (2.48)where 0 < b < Definition 2.19.
We define U q ( g R ) to be the real form of U q ( g ) induced by thestar structure e ∗ i = e i , f ∗ i = f i , K ∗ i = K i , (2.49)with q ∗ = q = q − , making it a Hopf-* algebra.Finally, we recall the Lusztig’s transformations, which gives a braid group action B on U q ( g ) as automorphisms. Here we introduce the positive version constructedin [15]. We only need the simply-laced case in this paper. Definition 2.20.
In the simply-laced case, the positive Lusztig’s braid group action is given by T i : U q ( g ) −→ U q ( g ) , i ∈ I, such that T i ( K j ) := K j K − a ij i , (2.50) T i ( e i ) := q K i f i , (2.51) T i ( e j ) := e j , a ij = 0 , (2.52) T i ( e j ) := [ e j , e i ] q / q − q − , a ij = − , (2.53)where the q -commutator is defined as[ X, Y ] q := qXY − q − Y X, (2.54)and similarly for the f i generators, such that the image is still self-adjoint.A well known fact is that the Lusztig’s braid group action induces the inter-change of generators as follows: This is the convention adopted in e.g. [10, 26]. roposition 2.21. If i = ( i , ..., i N ) is a reduced word for the longest element w ∈ W , then the automorphism T i := T i · · · T i N (2.55) is given by T i ( e i ) = f i ,T i ( f i ) = e i ,T i ( K i ) = K − i . Example 2.22.
In the special case when g = sl n +1 , we define for 1 ≤ i < j ≤ n , e i,j := T i T i +1 · · · T j − ( e j ) , f i,j := T i T i +1 · · · T j − ( f j ) . The index (shifted by one appropriately) coincides with the corresponding entries inthe ( n +1)-dimensional fundamental representation of U q ( sl n +1 ). Under the naturalorder of positive roots γ k := s i ...s i k − α i k ∈ Φ + , k = 1 , ..., N (2.56)corresponding to the standard reduced word i = (1 , , , ..., n, , , ..., n − , ..., , ,
1) (2.57)of the longest element w , the (ordered) monomials of these generators form the positive PBW basis of U q ( sl n +1 ) [15]. In this subsection we recall some definitions and properties concerning quantumtorus algebra and their cluster realizations.
Definition 2.23. A cluster seed is a datum Q = ( Q, Q , B, D ) (2.58)where Q is a finite set, Q ⊂ Q is a subset called the frozen subset , B = ( b ij ) i,j ∈ Q a skew-symmetrizable Z -valued matrix called the exchange matrix , and D = diag ( d j ) j ∈ Q is a diagonal Q -matrix called the multiplier with d − j ∈ Z , such that W := DB = − B T D (2.59)is skew-symmetric. 12n the following, we will consider only the case where there exists a decoration η : Q −→ I (2.60)to the root index of a simple Dynkin diagram, such that D = diag( d η ( j ) ) j ∈ Q where( d i ) i ∈ I are the multipliers given in (2.3).Let Λ Q be a Z -lattice with basis {−→ e i } i ∈ Q , and let d = min j ∈ Q ( d j ). Also let w ij = d i b ij = − w ij . (2.61)We define a skew symmetric d Z -valued form ( − , − ) on Λ Q by( −→ e i , −→ e j ) := w ij . (2.62) Definition 2.24.
Let q be a formal parameter. We define the quantum torusalgebra X Q q associated to a cluster seed Q to be an associative algebra over C [ q d ]generated by { X ± i } i ∈ Q subject to the relations X i X j = q − w ij X j X i , i, j ∈ Q. (2.63)The generators X i ∈ X Q q are called the quantum cluster variables , and they are frozen if i ∈ Q . We denote by T Q q the non-commutative field of fractions of X Q q .Alternatively, X Q q is generated by { X λ } λ ∈ Λ Q with X := 1 subject to the rela-tions q ( λ,µ ) X λ X µ = X λ + µ , µ, λ ∈ Λ Q . (2.64) Notation 2.25.
Under this realization, we shall write X i = X −→ e i , (2.65)and define the notation X i ,...,i k := X −→ e i + ... + −→ e ik , (2.66)or more generally for n , ..., n k ∈ Z , X i n ,...,i nkk := X n −→ e i + ... + n k −→ e ik . (2.67) Definition 2.26.
We associate to each cluster seed Q = ( Q, Q , B, D ) with dec-oration η a quiver, denoted again by Q , with vertices labeled by Q and adjacencymatrix C = ( c ij ) i,j ∈ Q , where c ij = (cid:26) b ij if d i = d j ,w ij otherwise. (2.68)We call i ∈ Q a short (resp. long ) node if q i := q d i = q s (resp. q i = q l = q ). Anarrow i −→ j represents the algebraic relation X i X j = q − ∗ X j X i (2.69)where q ∗ = q s if both i, j are short nodes, or q ∗ = q otherwise.13bviously one can recover the cluster seed from the quiver and the multipliers. Notation 2.27.
We will use squares to denote frozen nodes i ∈ Q and circlesotherwise. We will also use dashed arrows if c ij = , which only occur betweenfrozen nodes.We will represent the algebraic relations (2.63) by thick or thin arrows (seeFigure 1) for display conveniences (thickness is not part of the data of the quiver).Thin arrows only occur in the non-simply-laced case between two short nodes.Finally, we may omit the superscript Q in X Q q or T Q q if the datum / quiver isclear from the context. i j X i X j = q − X j X i i j X i X j = q − X j X i i j X i X j = q − s X j X i i j X i X j = q − s X j X i Figure 1: Arrows between nodes and their algebraic meaning.
Definition 2.28. A polarization π of the quantum torus algebra X Q q on a Hilbertspace H = L ( R M ) is an assignment X i e πbL i , i ∈ Q, (2.70)where L i := L i ( u k , p k , λ k ) is a linear combination of the position and momentumoperators { u k , p k } Mk =1 satisfying the Heisenberg relations[ u j , p k ] := δ jk π √− λ k ∈ R , such that they satisfy algebraically[ L i , L j ] = w ij π √− . (2.72)Each generator X i acts as a positive essentially self-adjoint operator on H and givesa representation of X Q q on H . Remark 2.29.
The domains of these positive unbounded operators are taken tobe the largest subspace W of the Schwartz space of L ( R M ) invariant under theexponential actions e πbL i . They are essentially self-adjoint over the dense subspace { e − x T Ax + β · x P ( x ) | A positive definite , β ∈ C M , P ( x ) polynomial } ⊂ W . (2.73)These spaces are discussed in detail in e.g. [8, 11, 24] and in this paper we assumethe algebraic relations among these operators are well-defined on W .14 otation 2.30. We will simplify notations and write e ( L ) := e πbL (2.74)for L a linear combination of position, momentum operators and scalars as above.Observe that we have e ( X ) e ( Y ) = q e ( Y ) e ( X ) (2.75)on W whenever [ X, Y ] = π √− .We also denote by[ L ] e ( L ) := e ( L + L ) + e ( − L + L ) . (2.76)It is straightforward to check that Proposition 2.31.
Let Q be a cluster seed and X Q q the quantum torus algebra.Then the following assignment X i := e ( − p i + X j ∈ Q w ij u j ) acting on L ( R | Q | ) is a polarization. We call the assignment in Proposition 2.31 the standard polarization of X Q q .Note that the position and momentum parts commute, so the action on the densesubspace W can be explicitly written down as X i · f ( ..., u i , ... ) = e πb P w ij u j f ( ..., u i + √− b, ... ) (2.77)where the shift is in the complex direction. Lemma 2.32.
Assume the rank of the skew-symmetric form of the lattice Λ Q is M . Then there is a polarization of X Q q on H = L ( R M ) and any polarization of X Q q on H is unitary equivalent by an Sp (2 M ) action on the lattice Λ Q (known asthe Weil representation [10]). Next we recall the notion of quantum cluster mutations.
Definition 2.33.
Given a cluster seed Q = ( Q, Q , B, D ) and an element k ∈ Q \ Q , a cluster mutation in direction k is another seed Q ′ = ( Q ′ , Q ′ , B ′ , D ′ ) with Q = Q ′ , Q = Q ′ and b ′ ij = (cid:26) − b ij if i = k or j = k,b ij + b ik | b kj | + | b ik | b kj otherwise , (2.78) d ′ i = d i . (2.79)15he cluster mutation in direction k induces an isomorphism µ qk : T Q ′ q −→ T Q q called the quantum cluster mutation , defined by µ qk ( X ′ i ) = X − k if i = k,X i | b ki | Y r =1 (1 + q r − i X k ) if i = k and b ki ≤ ,X i b ki Y r =1 (1 + q r − i X − k ) − if i = k and b ki ≥ , (2.80)where we denote by X ′ i the quantum cluster variables of X Q ′ q .Finally we recall the notion of amalgamation of two quantum torus algebras [7]. Definition 2.34.
Let Q , Q ′ be two cluster seeds and let X Q q , X Q ′ q be the corre-sponding quantum torus algebras. Let S ⊂ Q and S ′ ⊂ Q ′ be subsets of thefrozen vertices with a bijection φ : S −→ S ′ such that d ′ φ ( i ) = d i for i ∈ S . Thenthe amalgamation of X Q q and X Q ′ q along φ is identified with the subalgebra e X q ⊂ X Q q ⊗ X Q ′ q (2.81)generated by the variables { e X i } i ∈ Q ∪ Q ′ where e X i := (cid:26) X i ⊗ i ∈ Q \ S, ⊗ X ′ i if i ∈ Q ′ \ S ′ , (2.82) e X i = e X φ ( i ) := X i ⊗ X ′ φ ( i ) i ∈ S. Equivalently, the amalgamation of the corresponding quivers Q , Q ′ is a newquiver e Q := Q ∗ φ Q ′ (2.83)constructed by gluing the frozen vertices S along φ , defrozening those vertices thatare glued, and removing any resulting 2-cycles. We obviously have e X q ≃ X e Q q . (2.84) U q ( g ) The original positive representations of U q ( g R ) is constructed in [9, 13, 14] where thegenerators of the quantum group are represented by positive essentilly self-adjointoperators on the Hilbert space L ( R N ) where N = l ( w ). The representationsare constructed by certain quantization of the regular representation of G on thefunctions of the flag variety B := B − \ G (where we used the right cosets) bymultiplication on the right. 16 emark 3.1. In later sections, we sometimes call this the maximal positive repre-sentations, referring to the codimension of B − ⊂ G , to distinguish it from the moregeneral parabolic positive representations. The maximal positive representation isa special case of the parabolic positive representation where we take the parabolicsubgroup to be P J = B − , i.e. we take the subset J ⊂ I to be the empty set J = ∅ . Although the positive representations can be canonically defined using an embed-ding of D q ( g ) into a quantum torus algebra [10, 17] (see Section 3.3 for a summary),it is instructive to recall the original motivation of the construction in order to drawparallel with the construction of the parabolic positive representations later.Recall that for G a split real simple Lie group, the positive representations areoriginally constructed by certain quantization of the principal series representationsof G following the steps below:1. First, take the regular representation of G > acting on C ∞ ( B − > \ G > ) byright multiplication, together with the action of a character χ λ on the maximalsplit torus parametrized by λ ∈ P R > in the R > -span of the dominant weightslattice.2. Obtain the infinitesimal action of g as differential operators on C ∞ ( B − > \ G > )using the Lusztig coordinates (2.15) of the totally positive part.3. Apply the formal Mellin transformation and convert the differential operatorsinto finite difference operators, acting on the same coordinates.4. Finally, we perform a twisted quantization to define positive essentially self-adjoint operators acting on L ( B − > \ G > ) ≃ L ( R N ) where N = l ( w ). Theseoperators represent the split real quantum group U q ( g R ).Here the twisted quantization is done by first quantizing the operators, and then in-troduce an analytic continuation by rescaling and shifting the variables with certainmultiples of √− b + b − . In particular, due to the involvement of b − , the positiverepresentations do not have the usual classical limit, even though one can talk aboutits semi-classical limit by taking log to the finite difference operators. Remark 3.2.
As remarked in [10], the classical principal series representation cannaturally be viewed as action on the larger space L ( U − \ G )for some unipotent radical U − ⊂ B − , together with a Cartan action H on the left,such that the classical principal series representations are the irreducible compo-nents of the decomposition of the H -action by its central characters. L ( U − \ G ) = Z ⊕ P R > P λ dλ. (3.1)17his decomposition naturally lifts to the quantum case and gives a natural identi-fication of the Hilbert space H underlying the positive representations of U q ( g R ).Let { u k , p k } k =1 ,...,N be the standard position and momentum operators actingon the Hilbert space L ( R N , du ...du N ) with the same coordinates.We summarize the results of the construction as follows: Theorem 3.3. [9, 13, 14] Let i = ( i , ..., i N ) be a fixed reduced word of w . Thenthere exists a family of irreducible representations π i λ of U q ( g R ) on P i λ ≃ L ( R N , du ...du N ) (3.2) parametrized by λ ∈ P R > , or equivalently by λ = ( λ , ..., λ n ) ∈ R rank ( g ) > , such that • The generators e i , f i , K i are represented by positive essentially self-adjointoperators acting on L ( R N ) . • For any reduced words i and i ′ of w , we have unitary equivalence P λ : = P i λ ≃ P i ′ λ (3.3) where each Coxeter move of the reduced words induces a unitary transforma-tion (by quantum cluster mutations via the quantum dilogarithm function). • The generators are explicitly given as follows : π i λ ( K i ) = e − λ i − d i N X j =1 a i,i j u j , (3.4) π i λ ( f i ) = π ( f − i ) + π ( f + i ) := X k : i k = i π ( f k, − ) + X k : i k = i π ( f k, + ) , (3.5) where π ( f k, ± ) := e ± d i k k − X j =1 a i k ,i j u j + d i k u k + 2 λ i k + 2 p k . (3.6) • The E i generators corresponding to the right most index i N of i is givenexplicitly by π i λ ( e i N ) = π ( e + i N ) + π ( e − i N ) := e ( d i N u N − p N ) + e ( − d i N u N − p N ) , (3.7) while for the other generators with i = i N we have in general π i λ ( e i ) = π ( e + i ) + π ( e − i ) , (3.8) where π ( e ± i ) are obtained from e ( ± d i N u N − p N ) by a sequence of quantumcluster mutations. Compared with [14] in the non-simply-laced case, we rescaled the variables u k by p d i k . .2 Basic quivers Following the suggestion of [25], in [17] we gave a cluster realization of the positiverepresentations, where we construct an explicit embedding of the Drinfeld’s double D q ( g ) into the quantum torus algebra X D q of certain quiver D . The polarization of X D q then recover the positive representations up to unitary equivalence.In this subsection, we recall the construction of D , which is the double of theso-called basic quiver [17]. Here we follow (and modify) the extended approachdescribed in [10] which systematically includes the extra vertices { e i } that wereoriginally added manually in [17]. Definition 3.4.
Let i, k ∈ I . The elementary quiver J k ( i ) consists of • The vertex set Q = Q = ( I \ { i } ) ∪ { i l } ∪ { i r } ∪ { k e } (3.9)which are all frozen; • The multipliers D = ( d j ) j ∈ Q which is the pull-back of the multipliers (2.3)from I under the natural projection Q −→ I sending { i l , i r } to i and k e to k ;and • The adjancy matrix C = ( c ij ) which is defined to be c i l ,j = c j,i r = d i a ij , j ∈ I \ { i } , (3.10) c i l ,i r = c i r ,k e = c k e ,i l = 1 . (3.11)The vertices are organized in levels , such that the vertex j ∈ I \ { i } is placed atlevel j , { i l , i r } are placed on the left and right of level i , and k e is placed on anextra level labeled by k ′ . We have dashed arrows between the vertex j and { i l , i r } .Intuitively, we call the set ( I \{ i } ) ∪{ i l } the left frozen vertices , and ( I \{ i } ) ∪{ i r } the right frozen vertices . Definition 3.5.
We define J ( i ) to be the full subquiver of J k ( i ) obtained by re-moving the vertex { k e } . Definition 3.6.
Let i = ( i , ..., i m ) be a reduced word. Let β j := s i m s i m − · · · s i j +1 ( α i j ) , α i ∈ ∆ + , j = 1 , ..., m (3.12)be a chain of positive roots.We define the auxiliary quiver H ( i ) to have frozen vertex set I , labelled by { i e } i ∈ I and placed on level i ′ , and the same multipliers (2.3). The adjancy matrix C = ( c ij ) is given by c ij := s ij d i a ij , (3.13)19here s ij := (cid:26) sgn( r − s ) β s = α i and β r = α j , Definition 3.7.
Let i = ( i , ..., i m ) be a reduced word. The basic quiver Q ( i ) isconstructed by amalgamating the elementary quivers Q ( i ) := J i ( i ) ∗ J i ( i ) ∗ · · · ∗ J i ( i m ) ∗ H ( i ) , (3.15)where J i ( i j ) := (cid:26) J k ( i j ) if β j = α k , J ( i j ) otherwise, (3.16)and successively the right frozen vertices of J i ( i k − ) are amalgamated to the leftfrozen vertices of J i ( i k ) on the same level. The vertices of H ( i ) are amalgamatedto the corresponding extra nodes { k e } of J k ( i j ).Finally without loss of generality we remove all the vertices that are disjointfrom the quiver. We redefine Q so that the frozen vertices in the resulting quiverconsists of the left- and right-most vertices of each level, as well as the extra vertices { i e } . Proposition 3.8.
The quiver Q ( i ) coincides with the basic quiver described in[17] for a specific choice of a reduced word i of w .Proof. The elementary part J ( i ) (the full subquiver without the extra vertices { k e } )is identical to the one used in [17]. The arrows concerning the extra vertices { k e } isgoverned by the construction of positive representations, and is equivalent to howthe Coxeter moves of reduced words modify the chain of positive roots (3.12). Definition 3.9.
The symplectic double of Q ( i ) is defined by amalgamating thebasic quiver corresponding to the opposite words along the frozen vertices on theleft side of Q ( i ) and the right side of Q ( i op ), namely D ( i ) := Q ( i op ) ∗ Q ( i ) . (3.17) Remark 3.10.
In [17] we call Q ( i op ) the mirror quiver . It is obtained from Q ( i )by a horizontal mirror reflection, followed by reversing all the arrows. Notation 3.11.
Following [17] (mainly for typesetting purpose ), we label thevertices successively from left to right on the same level by { f ji : i ∈ I, j = 0 , ..., n i } (3.18) In [10], the quiver Q ( i ) here is denoted by J ( i ) and the corresponding labeling is given by f ji := (cid:18) ij (cid:19) and e i := (cid:18) i −∞ (cid:19) instead. i is the level of the vertex, and n i is the number of occurrences of i in thereduced word i . Note that the labels f ji (except the right-most one) are naturallyordered according to the reduced word i .The extra vertices { i e } are labeled by { e i } i ∈ I .We label the vertices of the quiver Q ( i op ) by { f − ji } , and { e i } , (3.19)and write | f − ji | := f ji . (3.20)Note that Q ( i ) and Q ( i op ) share the same vertices { f i } and { e i } in the doublequiver D ( i ). Example 3.12.
Consider g = sl and let i = (3 , ,
1) be a reduced word. Thenonly β := α is a simple root and H ( i ) is trivial. Hence the basic quiver is theamalgamation of Q ( i ) = J (3) ∗ J (2) ∗ J (1) , (3.21)see Figure 2.3 l r J (3) ←→ l r J (2) ←→ l r e J (1) = ⇒ f f f f f f e Q ( i )Figure 2: The basic quiver for i = (3 , , D ( i ) = Q ( i op ) ∗ Q ( i ) is shown in Figure 4.Let th verties of Q ( i ) be indexed by Q , and let those of Q ( i op ) be indexed by Q op . Then there is a standard polarization of the double acting on L ( R | i | ) asfollows. Proposition 3.13.
The symplectic double polarization for the quantum torus al-gebra X D ( i ) q is given by X i := e ( P j w ij u j − p i ) , i ∈ Q,e ( P j w | i | j u j + 2 p i ) , i ∈ Q op ,e (2 P j w ij u j ) , i ∈ Q ∩ Q op , (3.22)21 here ( w ij ) i,j ∈ Q is given by (2.61) obtained from the exchange matrix ( b ij ) i,j ∈ Q ofthe cluster seed associated to the basic quiver Q ( i ) . Corollary 3.14.
The quiver D ( i ) has | i | + k vertices and of rank | i | where k = | Q ∩ Q op | . In particular, the central characters are generated by k elements.Proof. It suffices to show that the Weyl algebra generators e ( u i ) , e (2 p i ) can beobtained as monomials of the cluster variables. (Equivalently, elements X L for L ∈ Λ R in the real span of the defining lattice.) The operator e ( p i ) can be obtainedeasily by taking the ratios of the opposite pair of cluster variables X f ji and X f − ji .Therefore we can focus on the position operators. In fact we can focus on the fullsubquiver of D ( i ) by removing the vertices { e i } .If i starts with i , then the cluster variable X f i consists of a single variable. Henceit follows by induction that one can obtain e ( u i ) by successively taking ratios of thecluster variables corresponding to the i k and i k − -th letter of i .Hence we can introduce the eigenvalues of the central characters e ( λ i ) ∈ R , i ∈| Q ∩ Q op | to the polarization of X e i . (If we introduce them in the unfrozen variable,then we can do an appropriate unitary transformation p i λ and get rid of them.) Finally, we summarize the cluster realization of positive representations using thedouble quiver D ( i ) for a longest reduced word i defined in the previous subsection. Theorem 3.15. [17] There exists an embedding of the Drinfeld’s double ι : D q ( g ) ֒ → X D ( i ) q { e i , f i , K i , K ′ i } 7→ { e i , f i , K i , K ′ i } , (3.23) such that K i K ′ i lies in the center of X D ( i ) q . In particular we have an embedding ι : U q ( g ) ֒ → X D ( i ) q / h K i K ′ i = 1 i . (3.24) There exists a polarization π λ of X D ( i ) q where π λ ( K i K ′ i ) = 1 and the other n centralcharacters acting by e ( λ i ) ∈ R > , such that the composition with the embedding (3.24) coincides with the expression of the positive representations P λ . Definition 3.16.
We call π λ the group-like polarization of X D ( i ) q .One can construct the group-like polarization π λ explicitly by taking successiveratios of the terms f k, ± in (3.5). It is then clear that the rank of this polarization is2 N (using the same argument as in the proof of Corollary 3.14), hence by Lemma2.32 it is unitarily equivalent to the standard polarization of the symplectic doubleinduced by an Sp (2 N ) action on the lattice. Notation 3.17.
To clarify the formula in the rest of the paper, we will always use bold face to denote elements of the quantum groups U q or D q , and the correspond-ing Roman script to denote elements in a quantum torus algebra X q .22rom the explicit construction (3.4)–(3.7) we can rephrase the embedding asfollows. Corollary 3.18. [17] The embedding (3.23) is such that (cf. Notation 2.25): • f i are represented by a telescoping sum, and K ′ i are monomials, explicitlygiven by f i = n i − X k = − n i X f i − ni ,...,f mi , (3.25) K ′ i = X f − nii ,...,f nii . (3.26) • If i is chosen such that i N = i , then e i = X f nii + X f nii ,e i , (3.27) K i = X f nii ,e i ,f − nii . (3.28) • K i are always monomial. • When i is appropriately chosen, the generators e i can also be represented bytelescopic sums . It is known that there is a sequence of cluster transformations that sends e i or f i into a single monomial of X q such that the vertices corresponding to the variables aresinks. In particular the embedding (3.23) is into the universally Laurent polynomial of the cluster algebra generated by the mutation class of X q [10, Proposition 13.11].A conjecture in [17] is that these Laurent polynomials are in fact always polynomialswith no negative powers of the variables involved in any cluster, cf. Conjecture 7.1. Remark 3.19.
In [10, Proposition 11.2], they generalize the same formula of theembedding (3.25)–(3.26) to give a homomorphism (not necessarily injective) of theBorel part U q ( b − ) to X Q ( i ) q for arbitrary reduced word i , even in the Kac-Moodycase.Finally, combining Theorem 3.3 and Corollary 3.18, for any root index i ∈ I ,we can choose i with i N = i again in order to observe that Corollary 3.20.
We have the following decomposition in X Q ( i op ) q ⊗ X Q ( i ) q e i = e + i + K + i e − i , (3.29) f i = f − i + K ′ i − f + i , (3.30) K i = K − i K + i , (3.31) K ′ i = K ′ i − K ′ i + , (3.32) With slight modification in type C n , F and E . The notations used here differ slightly from (3.5)–(3.8). here the (+) generators belong to ⊗ X Q ( i ) q and ( − ) generators belong to X Q ( i op ) q ⊗ , so that they mutually commute, [ e + i , f + j ] q i − q − i = δ ij K ′ i + , [ e − i , f − j ] q i − q − i = − δ ij K − i , (3.33) and { e ± i , f ± i , K ± i , K ′ i ± } satisfy all other quantum groups relations (2.36) – (2.41) (ex-cept (2.39) ). The triple { e ± i , f ± i , K ± i } forms the commutation relation of the Heisenberg dou-ble H ± q ( g ) [20] of the Borel part of U q ( g ). This is an important observation for usto construct a representation of D q ( g ) on the double quiver D ( i ) by generalizingthe Heisenberg double realization (cf. Section 5.2). Example 3.21.
Consider g = sl of type A and choose the reduced expression i = (1 , , , , , , , , , , , , , , . The representations of e i and f i are given by a telescopic sum of polynomials,represented by paths on the quiver presented in Figure 3. The e i -paths alwaysstart with the node f n i i and the f i -paths start with f − n i i . In this notation we havefor example f = X f − + X f − ,f − + · · · + X f − ,...,f ,K ′ = X f − ,f − ,...,f ,f . Note that the embedding of the f i generators never include the last node of the f i -paths, which only exists in K ′ i .On the other hand, since the word i ends with i N = 1, we have a simpleexpression forthe embedding of e given by e = X f + X f ,e ,K = X f ,e ,f − , again with the same rules as the f i generators. U q ( sl ( n +1 , R )) Following the strategy for the positive representations of U q ( g R ), given a parabolicsubgroup P J ⊂ G containing B − , we construct a representation of U q ( g R ) by quan-tizing the infinitesimal action of g on the functions of totally positive partial flagvarieties C ∞ ( P > J \ G > ) induced by the regular action g · f ( h ) = χ λ ([ hg ] ) f ([ hg ]) (4.1) Formally we should draw the quiver such that the blue arrows are horizontal on their respectivelevel. The quiver we present here first appeared in [25] which is more aesthetically pleasing. − f f − f f − f f − f f − f f − f f − f f − f f − f f − f f − f f − f f − f f − f f − f f f f f f e e e e e Figure 3: A -quiver, with the e i -paths colored in red and f i -paths colored in blue.where χ λ is some character on P J parametrized by the weights λ ∈ R | I \ J | (cf.Lemma 2.13).In this section, we illustrate the above procedure explicitly for the maximalparabolic subgroup and construct a family of positive representations of U q ( sl ( n +1 , R )), acting as positive essentially self-adjoint operators on the Hilbert space L ( R n ). Since this Hilbert space has the minimal functional dimension possible, wecall this family the minimal positive representations . The simplicity of such repre-sentations is expected to find applications in mathematical physics and integrablesystems. U q ( sl (4 , R )) Let us consider the standard maximal parabolic subgroup P J = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (4.2)25orresponding to J = { , } . It admits a Langlands decomposition of the form P > J = U − > T > M > : P > J = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
00 0 0 ∗ ∗ ∗
00 1 ∗
00 0 1 00 0 0 1 (4.3)The positive characters on P > J is parametrized by a single scalar λ ∈ R given onthe abelian component T > : h ( t , t , t ) := h ( t ) h ( t ) h ( t ) (4.4)by χ λ ( h ( t , t , t )) = t λ (4.5)which only depends on the last coordinate.Consider the action on P > J \ G > by right multiplication, here the right cosetcan be expressed as ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ a b
00 0 1 c for a, b, c ∈ R > , where in fact we can rewrite the representative in terms of theLusztig coordinates as a b
00 0 1 c = c b
00 0 1 00 0 0 1 a = x ( c ) x ( b ) x ( a )The group elements e tX , where X ∈ g , acts by multiplying on the right. Moreprecisely, we only need to consider the action of x i ( t ) = e tE i , y i ( t ) = e tF i h i ( t ) = e tH i (4.6)Hence using the Coxeter moves (2.18)–(3.8), we can rearrange the right multiplica-tion in the form x ( c ) x ( b ) x ( a ) e tX = n · h · x ( f ′ ) x ( e ′ ) x ( d ′ ) x ( c ′ ) x ( b ′ ) x ( a ′ ) , n ∈ U − , h ∈ T. (4.7)To obtain a regular representation on C ∞ ( P > J \ G > ), we project onto the coset P − \ G by ignoring the coordinates d ′ , e ′ , f ′ , and apply the character χ λ to h ∈ T > .26or example, x ( c ) x ( b ) x ( a ) x ( t ) = x ( c ) x ( b ) x ( a + t ) , and hence the regular action is given by e tE : f ( a, b, c ) f ( a + t, b, c ) . On the other hand x ( c ) x ( b ) x ( a ) y ( t ) = y ( t ct ) h (1 + ct ) x ( c ct ) x ( b ) x ( a )and hence the regular action is given by e tF : f ( a, b, c ) (1 + ct ) λ f ( a, b, c ct ) . We compute the rest of the actions to be e tE : f ( a, b, c ) f ( a + t, b, c ) ,e tE : f ( a, b, c ) f ( abb + t , b + t, c ) ,e tE : f ( a, b, c ) f ( a, bcc + t , c + t ) ,e tF : f ( a, b, c ) f ( aat + 1 , b ( at + 1) , c ) ,e tF : f ( a, b, c ) f ( a, bbt + 1 , c ( bt + 1)) ,e tF : f ( a, b, c ) ( ct + 1) λ f ( a, b, cct + 1 ) ,e tH : f ( a, b, c ) f ( ae − t , be t , c ) ,e tH : f ( a, b, c ) f ( ae t , be − t , ce t ) ,e tH : f ( a, b, c ) e tλ f ( a, be t , ce − t ) . The infinitesimal action X · f := ddt ( e tX · f ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 (4.8)27an now be computed to be E = ∂∂a ,E = ∂∂b − ab ∂∂a ,E = ∂∂c − bc ∂∂b ,F = − a ∂∂a + ab ∂∂b ,F = − b ∂∂b + bc ∂∂c ,F = 2 cλ − c ∂∂b ,H = − a ∂∂a + b ∂∂b ,H = a ∂∂a − b ∂∂b + c ∂∂c ,H = 2 λ + b ∂∂b − c ∂∂c . Now applying the formal Mellin transform [13] f ( a, b, c )
7→ F ( u, v, w ) := Z a u b v c w f ( a, b, c ) dadbdc, (4.9)we can turn the differential operators into finite difference operators as follows E : F ( u, v, w ) ( u + 1) F ( u + 1 , v, w ) ,E : F ( u, v, w ) ( − u + v + 1) F ( u, v + 1 , w ) ,E : F ( u, v, w ) ( − v + w + 1) F ( u, v, w + 1) ,F : F ( u, v, w ) ( − u + v + 1) F ( u − , v, w ) ,F : F ( u, v, w ) ( − v + w + 1) F ( u, v − , w ) ,F : F ( u, v, w ) (2 λ − w + 1) F ( u, v, w − ,H : F ( u, v, w ) ( − u + v ) F ( u, v, w ) ,H : F ( u, v, w ) ( u − v + w ) F ( u, v, w ) ,H : F ( u, v, w ) (2 λ + v − w ) F ( u, v, w ) . Remark 4.1.
By the same argument in [12], by introducing appropriate measure(a shift with complex parameters), one can identify L ( R > , dµ ( a, b, c )) ≃ L ( R , dudvdw ) (4.10)with the standard Lebesgue measure under the Mellin transformation.28inally, following [13] to perform the twisted quantization, and using Notation2.30, the quantized action of U q ( sl (4 , R )) is given (on the rescaled generators) by π Jλ ( e ) = [ u ] e ( − p u ) ,π Jλ ( e ) = [ − u + v ] e ( − p v ) ,π Jλ ( e ) = [ − v + w ] e ( − p w ) ,π Jλ ( f ) = [ − u + v ] e (2 p u ) ,π Jλ ( f ) = [ − v + w ] e (2 p v ) ,π Jλ ( f ) = [2 λ − w ] e (2 p w ) ,π Jλ ( K ) = e ( − u + v ) ,π Jλ ( K ) = e ( u − v + w ) ,π Jλ ( K ) = e ( v − w + 2 λ ) . One checks explicitly that this gives a representation of U q ( sl (4 , R )) as positiveessentially self-adjoint operators on L ( R ), in the sense of Remark 2.29. Remark 4.2.
Recall that the operator e (2 p u ) etc. acts on the dense subspace W by shifting the variable u u − √− b , and thus it is the analytic continuation (or Wick’s rotation ) of the classical finite difference operator.Furthermore, following [17], the above representations can also be represented asthe homomorphism from the Drinfeld’s double D q ( g ) together with a polarizationof the quantum torus algebra associated to a quiver with 10 vertices, as shown inFigure 4, which one can verify to be the same as the double quiver D ( i ) for thereduced word i = (3 , , e = X + X , , K = X , , ,e = X + X , , K = X , , ,e = X + X , , K = X , , ,f = X + X , , K ′ = X , , ,f = X + X , , K ′ = X , , ,f = X + X , , K ′ = X , , , where the polarization is explicitly given by X = e ( − u + v + 2 p u ) , X = e ( − u + v − p v ) ,X = e (2 u − v ) , X = e (2 λ − w + 2 p w ) ,X = e ( u − p u ) , X = e ( − λ + 2 w ) ,X = e ( − v + w + 2 p v ) , X = e ( − v + w − p w ) ,X = e (2 v − w ) , X = e ( − u ) .
29 0 34 2 67 5 98Figure 4: The double quiver D ( i ) for the word i = (3 , , e i -paths are colored in red, f i -paths in blue.By Lemma 2.32, under the unitary transformation of multiplication by e πi ( u + v + w ) ,which induces the Weil action2 p u p u + u, p v p v + v, p w p w + w (4.11)we recover (up to the parameter λ ) the standard polarization of the symplecticdouble.In fact the rank of the underlying lattice is easily verified to be 6 = 10 − L ( R ). Hence the center of the quantum torusalgebra X D ( i ) q is generated by 4 elements, namely K K ′ = X , , , ,K K ′ = X , , , ,K K ′ = X , , , , and the central character C := X , , , = e ( − λ ) . (4.12)The simplicity of this quiver allows us to generalize the construction easily tothe higher rank case in the next subsection.By obvious symmetry, by choosing another standard parabolic subgroup P = ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗ (4.13)30nd following the same procedure, we obtain an equivalent representation, whereall the arrows of the quiver are reversed.However, we note that the minimality of the functional dimension of H dependson the codimension of the parabolic subgroup chosen, not the fact that P J is max-imal. As we see from the projection P > J \ G > , the functional dimension is givenby the codimension N − N J , and in fact only for type A n we can achieve the min-imal possible functional dimension of L ( R n ) by choosing J = { , , ..., n − } or { , , ..., n } for the maximal parabolic subgroup.For example, if we choose the standard parabolic subgroup corresponding to theroot index { , } : P = ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ (4.14)we obtain a representation of U q ( sl (4 , R )) on L ( R ), represented by the doublequiver D ( i ) associated to the word i = (2 , , ,
2) with rank 12 − D q ( g ) −→ X D ( i ) q is given by e X + X , + X , , + X , , , , K X , , , , , e X + X , , K X , , , e X + X , + X , , + X , , , , K X , , , , , f X + X , , K ′ X , , , f X + X , + X , , + X , , , , K ′ X , , , , , f X + X , , K ′ X , , . D ( i ) for i = (2 , , , e i -paths are colored in red andorange, f i paths colored in blue.We will prove in Section 5 that for any parabolic subgroup P J ⊂ G , we canobtain a positive representation for U q ( g R ), parametrized by λ ∈ R | I \ J | , where thequiver D ( i ) is given by a truncated word i from the longest word i .31 .2 Minimal positive representations for U q ( sl ( n + 1 , R )) In fact, from the construction, it is clear that the regular action can be obtainedfrom the maximal positive representation on L ( B − > \ G > ) by ignoring the Lusztigcoordinates (2.30) projected out by P J on the left. In particular all the calculationshave been done previously in [13, 14].In terms of the polarization, it means that if u i corresponds to those coordinates,then given the maximal positive representations of U q ( sl ( n + 1 , R )), we obtain theparabolic positive representation by setting u i = 0 and e ( ± p i ) = 0 . (4.15)Notice that this is in general not a well-defined quotient since the algebraic relation e πbp i e − πbp i = 1 is violated. However, we will prove in Section 5 that in general thisprocedure allows us to construct arbitrary parabolic positive representations. Theorem 4.3.
We have a family of irreducible representations of U q ( sl ( n + 1 , R )) by positive essentially self-adjoint operators on L ( R n ) , parametrized by λ ∈ R .Proof. We consider the action on C ∞ ( P > J \ G > ) where J = I \ { n } . Followingthe previous strategy, we obtain the following parabolic positive representations P Jλ given as positive essentially self-adjoint operators on L ( R n ), parametrized by λ ∈ R as π Jλ ( e ) = [ u ] e ( − p ) ,π Jλ ( e i ) = [ u i − u i − ] e ( − p i ) , i ≥ ,π Jλ ( f i ) = [ − u i + u i +1 ] e (2 p i ) , i < n,π Jλ ( f n ) = [2 λ − u n ] e (2 p n ) ,π Jλ ( K i ) = e ( − n X j =1 a ij u j ) , i < n,π Jλ ( K n ) = e ( − n X j =1 a nj u j + 2 λ )= e ( − u n + u n − + 2 λ ) . We can show irreducibility by explicitly reconstructing the Weyl operators e ( u i )and e ( p i ) from the representations of U q ( g R ) as certain rational functions of thegenerators. Firstly, the action of the K i generator is given by e ( P a ij u j ) (togetherwith the constant e ( λ )), hence since the Cartan matrix is invertible, we obtain every e ( u i ) from certain (fractional) monomials of K i , which is well-defined since K i ispositive self-adjoint. Then from the expressions of e i we obtain the operators e ( p i )as a rational functions of the generators K i and e i .For the cluster algebraic realization, we have a homomorphism D q ( sl n +1 ) −→ X q
32f the Drinfeld’s double to the quantum torus algebra X q := X D ( i ) q associated tothe double quiver D ( i ) where i = ( n, n − , ..., , ,
1) such that explicitly e i X f i + X f i ,f i − , f i X f − i + X f − i ,f i , K i X f i ,f i − ,f − i , K ′ i X f − i ,f i ,f i . Here we redefined f := e . e f − f f f − f f f − f f ... ...... f n − f − n f n f n Figure 6: The quiver D ( i ) for i = ( n, n − ..., , , In this subsection, we investigate the non-simple generators stated in Example 2.22.33 roposition 4.4.
The non-simple generators can be given explicitly as follows: f i,j X f − i ,...,f − j − ,f i ,...,f j − + X f − i ,...,f − j − ,f i ,...,f j − , (4.16) e i,j X f i ,...,f j − + X f i ,...,f j − ,f i − , (4.17) for ≤ i < j ≤ n , where again we redefine f := e .Proof. We have f X f − + X f − ,f f X f − + X f − ,f f = T f := [ f , f ] q / q − q − = q / f f − q − / f f q − q − X f − ,f − ,f + X f − ,f ,f − ,f ,f Noticing that f ij = T i f i +1 ,j = [ f i +1 ,j , f i ] q / q − q − the expressions follow easily by induction.The calculations for the expression of e ij is completely analogous.From the explicit expression of the non-simple generators, we notice that thehomomorphism D q ( g ) −→ X q is in general not an embedding. One can solve for X i in multiple ways by taking ratios of the non-simple generators, hence there arelinear dependencies on the PBW basis of U q ( g ). In fact, from [10, Conjecture 2.30] ,the functional dimension of the representations should match only in the case whenthe parabolic subgroup P J is the minimal one, i.e. the Borel subgroup P J = B − .However, since all the non-simple generators are non-trivial, we can concludethat Theorem 4.5.
The universal R operator R = K Y α ∈ Φ + g b α ( e α ⊗ f α ) (4.18) is well-defined as unitary operators on tensor products P Jλ ⊗P Jλ ′ of parabolic positiverepresentations. Here g b is the quantum dilogarithm function and the product is over the naturalorder of Φ + for the standard reduced expression (2.57) of the longest word i , while K is the Cartan part of the R operator. See [15] for details.34 .4 Central characters In this subsection, we study the central characters of the minimal positive repre-sentations in detail.
Proposition 4.6.
The rank of X q is n , with center generated by K i K ′ i = X f − i ,f i ,f i ,f i ,f i − ,f − i , i = 1 , ..., n and C := X f ,f ,...,f n = n Y k =0 X f k . Therefore the parabolic positive representation of U q ( sl ( n + 1 , R )), where wequotient out by the relation h K i K ′ i = 1 i , is parametrized by the central element C acting as a scalar π λ ( C ) = e ( − λ ) . (4.19)Let us consider the simply-connected form by adjoining fractional powers of theCartan generators K i and define U scq ( g ) := U q ( g )[ K ± h i ] i ∈ I (4.20)where h is the Coxeter number. When g = sl n +1 , h = n +1. Positive representationsof U q ( g ) naturally extends to positive representations of U scq ( g ).The center of U scq ( g ) is generated by the Casimirs C k for k = 1 , ..., rank( g ) = n .Since the parabolic positive representation is irreducible, they act on P Jλ as scalars,and we can calculate directly their eigenvalues using the tools of virtual highestweight vectors developed in [16]. Recall that each Casimir C k can be written in theform C k = C K + C E where C K ∈ U scq ( h ) only depends on (fractional powers of) the Cartan generators K i , while C E ∈ U scq ( g ) consists of expressions with E i acting on the right.Explicitly, if V k is the k -th fundamental representation of U q ( g ) and ( −→ W j ), j =1 , ..., dim V k = (cid:18) n + 1 k (cid:19) are the weights of the weight spaces, then [16] C k = ( − n dim V k X j =1 q P ni =1 W ji n Y i =1 K W ji i + C E (4.21)In particular, the first Casimir, for the ( n + 1)-dimensional standard represen-tation V , has the form C = ( − n n X j =1 q n +2 − j j Y k =1 K − kn +1 k n Y k = j +1 K n +2 − kn +1 k + C E xample 4.7. [16] When g = sl , the positive Casimirs are given by C := K ( q K K + K − K + q − K − K − − q K f e − q − K − f e + f e ) C := K − ( q K K + K K − + q − K − K − − q − K − f e − q K f e + f e )where K = K K − and e ij = T i ( e j ), f ij = T i ( f j ) are the non-simple generators(cf. Definition 2.20).The idea is that under the (irreducible) parabolic positive representations, theseoperators acts by certain scalars. Hence their action on test functions on L ( R n ) isjust multiplication by the same scalars. But when we consider only the algebraicrelation, then these operators make sense on generalized distributions F also, inthe sense that h X · F , f i := hF , X · f i , X ∈ U q ( g R ) , f ∈ W , (4.22)where the test functions f are rapidly decreasing and entire. Hence we can findthe eigenvalues by acting C k on certain distributions that is annihilated by the E i generators, i.e. these distributions behave like highest weight vector, despite notliving in the Hilbert space.Here we use the complex delta functionals, formally viewed as evaluation at acomplex value. We consider the functional F ( u , ..., u n ) := n Y k =1 δ ( u k − √− kγ b ) (4.23)where γ b := b − b − . (4.24)Then by direct calculation we obtain e i · F = 0 ,K i · F = F , i = 1 , ..., n − ,K n · F = e (2 λ − √− n + 1) γ b ) F . Hence one can interpret this parabolic positive representations as the “ n -th funda-mental representations” due to the fact that the Cartan generators H i acts triviallyfor i = 1 , ..., n − C k are positive and lie inside a connected region on the positivequadrant bounded by the discriminant variety in R n [16]. Theorem 4.8.
The spectrum of the Casimirs C k acting on the parabolic positiverepresentations are real-valued, and lie outside the positive spectrum of the positiveCasimirs of the maximal positive representations. This is the adjoint of the ones presented in [16]. roof. Since the generators K i , i = 1 , ..., n − π Jλ ( C ) = ( − n n X j =1 q n +2 − j π ( K n ) n +1 + ( − n q − n π ( K n ) − nn +1 = ( − n ( q n + · · · + q − n +2 ) e (cid:18) n + 1 (2 λ − i ( n + 1) γ b ) (cid:19) + ( − n q − n e (cid:18) − nn + 1 (2 λ − i ( n + 1) γ b ) (cid:19) = ( − n +1 ( q n − + q n − + · · · + q − n ) t + t − n = ( − n +1 [ n ] q t + t − n where t := e ( 4 λn + 1 ) > e ( inb − ) = e πin = ( − n .Recall that the region for the spectrum of the positive Casimirs is bounded bythe discriminant of the polynomial P ( x ) = x n + C ( λ ) x n − + · · · + C n ( λ ) x + 1where P ( x ) = Q ( x + e L i ) and L i is the exponential terms of the first Casimir C ( λ ) = π ( C ) =: X e ( L i )In particular, in the parabolic positive representations, we have P ( x ) = n − Y k =1 ( x + ( − n +1 q n − − k t )( x + t − n )so that this polynomial always has ( n + 1) distinct roots, exactly one of which isreal, when t > q = 1. Hence the discriminant is either strictly positive ornegative when t varies, and lie on one side of the positive spectrum.Recall that the spectrum of the first positive Casimir has π λ ( C ) ≥ n + 1.We split into cases. Assume [ n ] q >
0. Then when n is even and t is sufficientlylarge, π Jλ ( C ) can take negative values, hence it lies outside of the positive spectrum.When n is odd, π Jλ ( C ) achieve a minimal value of( n + 1)[ n ] nn +1 q n nn +1 < n + 1when t = (cid:18) n [ n ] q (cid:19) n +1 since [ n ] q < n . Hence again π ( C ) lies outside the positivespectrum.The case for [ n ] q < n switched.37s a corollary, we see that the parabolic positive representation behaves slightlydifferently with respect to the parameter λ : Corollary 4.9.
The representations P Jλ and P J − λ are not unitarily equivalent.Proof. From the explicit expression in the proof above, we see that the spectrumof π Jλ ( C ) is not invariant under the exchange λ ←→ − λ . U q ( b sl n +1 ) Finally, we observe that one can “wrap around” the minimal quiver in type A to construct a family of positive representations for the affine quantum groups U q ( b sl n +1 )!Given the quiver associated to the minimal positive representations of type A n +1 above (with I = { , ..., n } ), we form a new quiver b D by taking the root index i (mod n + 1), f ǫi f ǫi , i ∈ Z / ( n + 1) Z . (4.26)More precisely, we identify on the original quiver (indexed from 0 to n ) the vertex e with f n , and adding new dashed arrows between f ± with f ± n , forming a closedloop with Z n +1 symmetry. See Figure 7.The new quiver b D consists of 3( n + 1) vertices. Then it is straightforward tosee that that Proposition 4.10.
The assignments e i X f i + X f i ,f i − , f i X f − i + X f − i ,f i , K i X f i ,f i − ,f − i , K ′ i X f − i ,f i ,f i , with i ∈ Z / ( n + 1) Z gives a homomorphism of D q ( b sl n +1 ) , the Drinfeld’s double of(the Borel part of the) affine quantum group, onto the quantum torus algebra X b D q associated to the quiver b D .In particular, a polarization of X b D q gives a representation of U q ( b sl ( n + 1 , R )) aspositive self-adjoint operators on L ( R n ) . In the special case of n = 1, we have a degenerate quiver with no dashed arrowsbetween the frozen nodes, see Figure 8. The homomorphism defined by e X + X , , K X , , , e X + X , , K X , , , f X + X , , K ′ X , , , f X + X , , K ′ X , , f − f f f − f f f − f f ... ...... f n − f − n f n f n = ⇒ f − f f f − f f f − f f ... ...... f n − f − n f n f n Figure 7: Construction of the quiver b D .satisfies the defining relation for U q ( b sl ). In particular the Serre relation ( a = − X i X j − [3] q X i X j X i + [3] q X i X j X i − X j X i = 0 , i = j (4.27)where X = e , f . Therefore it gives an irreducible representation of U q ( b sl ) realizedas positive operators on L ( R )!We can check directly that in general the quiver b D has rank 2 n , with 3( n + 1) −
39 2 34 5 6Figure 8: The quiver for U q ( b sl (2 , R )).2 n = n + 3 central characters. The center is generated by n + 1 elements K i K ′ i , i = 0 , ..., n, (4.28)as well as 3 central elements given by D ǫ := n Y i =0 X f ei , ǫ = − , , . (4.29)Note that they are not independent: n Y i =0 K i K ′ i = Y ǫ ∈{− , , } D ǫ (4.30)in order to produce the correct rank of the quiver.Therefore a choice of polarization of the quantum torus algebra provides thepositive representation acting on the same space P Jλ ≃ L ( R n ) as the minimalpositive representation of U q ( sl n +1 ), parametrized by two characters.Recall the simply-connected form U scq ( sl n +1 ) defined in (4.20) which containsthe fractional power K n +1 i := q H in +1 . In the positive setting, we have the followinghomomorphism modified from [19] U q ( b sl n +1 ) −→ U scq ( sl n +1 )sending e e ( µ ) Kf ,n , e i e i , i > , f e ( − µ ) K − e ,n , f i f i , i > , K n Y k =1 K − k , K i K i , i > , where µ ∈ R and K := K n − n +1 · · · K − nn +1 n . (4.31)40he action of K i naturally extends to the action of K on P Jλ , and the evaluationmodule [19], denoted by P µλ , obtained by evaluating the above expression on P Jλ iswell-defined. Since K commutes with e ,n and f ,n , the evaluation module P µλ isrealized by positive operators on L ( R n ). Remark 4.11.
In the original evaluation module [19] on the ( n + 1)-dimensionalfundamental representation π of U q ( sl n +1 ), the Cartan factor is defined to be π ( K ) = q E + E n +1 ,n +1 (4.32)where E ij is the elementary matrix with ( i, j )-th entry equals 1. Recall that underthe fundamental representation, π ( H k ) = E kk − E k +1 ,k +1 . (4.33)It is then straightforward to show that n X k =1 n + 1 − kn + 1 π ( H k ) + 2 n + 1 I = E + E n +1 ,n +1 . (4.34)Letting K i = q H i we obtain the formal expression for K (together with the constant q n +1 which can be absorbed into µ ), which makes sense when evaluating on P Jλ . Theorem 4.12.
The positive representation π of U q ( b sl n +1 ) defined by Proposition4.10 is unitarily equivalent to the evaluation module P µλ under the minimal positiverepresentations of U q ( sl n +1 ) , where the central characters are determined by theaction of the central elements of X q : e ( − λ ) := π ( D ) , (4.35) e ( µ ) := π ( D n +1 D ) . (4.36) Proof.
Note that the full subquiver obtained by removing the frozen vertices { f ± } is exactly the same as the minimal quiver D ( i ) for U q ( sl n +1 ). Therefore the repre-sentation of the subgroup h e i , f i , K i i i =1 ,...,n is exactly the minimal positive repre-sentations of U q ( sl n +1 ) with central character π ( D ) = e ( − λ ). Therefore it sufficesto show that the action of the 0-th generators coincides with the required expressionof the evaluation homomorphism.Consider the generator f , which is represented on X b D q by f = X f − + X f − ,f = X f − (1 + qX f ) . On the other hand, e ,n is given explicitly on X b D q by e ,n = X f ,f ,...,f n + X f ,f ,...,f n ,f = X f ,f ,...,f n (1 + qX f ) . Recall (cf. Notation 3.17) that we use bold face to denote elements in U q , while Roman scriptto denote elements in X q . f = X f − X − f ,f ,...,f n = X f − X f D − e ,n , where the two X terms commute. Hence we only need to show that the factor X f − X f represents a multiple of K .In fact, by solving the system of linear equations on the powers of X i , there isa unique solution expressed in terms of K i , K ′ i and D ǫ by X f − ,f = n Y k =1 K − n +1 − kn +1 k ( K ′ k ) − kn +1 D nn +1 D − D . When we specify K ′ k = K − k in the quotient, we see that the expression coincideswith π ( f ) = π ( D nn +1 D − ) π ( K − e ,n ) . In particular, noting that π ( D − D D ) = 1, the other central character is given by e ( − µ ) = π ( D nn +1 D − ) = π ( D − n +1 D − )as required.By considering ( D D ) − e , the calculations for the e generator is completelyanalogous. Finally, note that in X q , D − D D = K K · · · K n = K ′ K ′ · · · K ′ n . Under the positive representations, π ( D − D D ) = 1, hence we obtain π ( K ) = π ( n Y k =1 K − k ) ,π ( K ′ ) = π ( n Y k =1 K ′ k − ) = π ( n Y k =1 K k )as desired. In this section, we construct a family of positive representations of U q ( g R ) by meansof quantizing the regular representations over the totally positive partial flag variety P > J \ G > for any parabolic subgroup P J , generalizing the construction in theprevious section. 42 .1 The Main Theorem Let g be a simple Lie algebra with root index I and Weyl group W . Let J ⊂ I and let W J ⊂ W be the corresponding Weyl subgroup. Let w ∈ W be the longestelement, and w J ∈ W J the longest element of W J ⊂ W , with a choice of reducedwords i and i J respectively. Definition 5.1.
We define the unique element w ∈ W such that w = w J w. (5.1)Let i be a reduced word of w .By Lemma 2.3, we know that l ( w ) = l ( w ) − l ( w J ) = N − N J (5.2)is the codimension of the parabolic subgroup P J in G . Note that we have Q ( i ) = Q ( i J ) ∗ Q ( i ) (5.3)and the symplectic double D ( i J ) (Definition 3.9) is naturally a full subquiver of thedouble D ( i ), up to the dashed arrows of its frozen vertices.The Main Theorem of the paper is the following result. Theorem 5.2.
Let D ( i ) be the symplectic double of the basic quiver Q ( i ) associatedto the reduced word i of the element w , and let X D ( i ) q be the associated quantumtorus algebra.Then there is a homomorphism D q ( g ) −→ X D ( i ) q and the image are universally Laurent polynomials in the cluster mutation class of X D ( i ) q .A polarization of X D ( i ) q induces a family of irreducible representations P Jλ of U q ( g R ) parametrized by the scalars λ ∈ R | I \ J | , acting as positive essentially self-adjoint operators on L ( R l ( w ) ) . Definition 5.3.
We call P Jλ ≃ L ( R l ( w ) ) (5.4)the parabolic positive representations of U q ( g R ). Remark 5.4.
As noted before, when J = ∅ , the parabolic positive representationscoincides with the usual (maximal) positive representations constructed previouslyin [9, 13, 14].The name naturally comes from the following consequence of Theorem 5.2.43 orollary 5.5. The parabolic positive representations P Jλ is obtained as certaintwisted quantization of the parabolic induction, by ignoring the variables u i corre-sponding (under the Mellin transform) to the Lusztig coordinates of M > of theLanglands decomposition (2.29) of P > J in the principal series representations. More explicitly, a (symplectic double) polarization of D ( i ) can naturally beobtained from the group-like polarization (Definition 3.16) of D ( i ) by setting thecorresponding Weyl operators e ( u i ) = 1 and e ( ± p i ) = 0, as well as the parameter λ j = 0 for j ∈ J . The non-trivial conclusion of Theorem 5.2 is that such reductionactually gives a representation of U q ( g R ). Remark 5.6.
The truncating procedure above is tight. In general we do not obtaina representation of U q ( g R ) by arbitrarily truncating any full subquiver Q ( i ) from Q ( i ) and ignoring the corresponding variables. Recall that we have the Heisenberg double relations (Corollary 3.20), which is aspecial case of the following more general construction:
Definition 5.7.
Let { e ± i , f ± i , K ± i , K ′ i ± } i ∈ I satisfy the following relations:[ e + i , f + j ] q i − q − i = δ ij K ′ i + + ω ij K + i , (5.5)[ e − i , f − j ] q i − q − i = − δ ij K i − − ω ij K ′ i − (5.6)for some scalars ω ij ∈ C and (within each ± ) the other standard quantum grouprelations (2.36)–(2.41) (i.e. except (2.39)) of U q ( g ).We call the algebra H ± q,ω ( g ) := h e ± i , f ± i , K ± i , K ′± i i (5.7)the generalized Heisenberg double of g . Remark 5.8.
When ω ij ≡
0, the above relations are the standard relations for theHeisenberg double [20].Let i be a reduced word. Assume h e Ri , f Ri , K Ri , K ′ Ri i is a homomorphic imageof H + q,ω ( g ) in X Q ( i ) q . Then by symmetry, we have elements { e Li , f Li , K Li , K ′ iL } inthe quantum torus algebra X Q ( i op ) q obtained by replacing the X i variables in the R elements with X ± − i as follows: e Li := e Ri | X i X − − i , (5.8) f Li := e Ri | X i X − − i , (5.9) K Li := e Ri | X i X − i , (5.10) K ′ iL := e Ri | X i X − i , (5.11)44here by abuse of notation, we denote by X − i ∈ X Q ( i op ) q the corresponding oppositevariable of X i ∈ X Q ( i ) q , i.e. interchanging the labels { f − ji , e i } ←→ { f ji , e i } . Lemma 5.9.
Define in X Q ( i op ) q the elements b e iL := q − i K Li e Li , (5.12) b f iL := q − i K ′ iL f Li . (5.13) Then h b e iL , b f iL , K Li , K ′ iL i is a homomorphic image of H − q,ω ( g ) in X Q ( i op ) q .Proof. By definition and symmetry, we see that[ e Li , f Lj ] q i − q − i = − δ ij ( K ′ iL ) − − ω ij ( K Li ) − (5.14)and other quantum group relations are satisfied. Then we have the commutationrelation [ b e iL , b f jL ] q i − q − i = K Li K ′ jL [ e Li , f Lj ] q i − q − i = K Li K ′ jL ( − δ ij ( K ′ iL ) − − ω ij ( K iL ) − )= − δ ij K Li − ω ij K ′ jL while all other quantum group relations, including the Serre relation, remains thesame. Proposition 5.10.
The elements in X Q ( i op ) q ⊗ X Q ( i ) q defined by e i = e Ri + b e iL K Ri (5.15) f i = b f iL + K ′ iL f Ri (5.16) K i = K Li K Ri (5.17) K ′ i = K ′ iL K ′ iR (5.18) satisfy all the quantum group relations of D q ( g ) .Furthermore, they are elements of the amalgamation X D ( i ) q if the elements K i , K ′ i , e Ri , b f iL ∈ X D ( i ) q . (5.19)Here by abuse of notation, the L generators are elements of X Q ( i op ) q ⊗ R generators are elements of 1 ⊗ X Q ( i ) q . 45 roof. First observe that the L generators and R generators commute by definition.The q -commutation relation of K i with e j , f j also follows from definition.By definition the L and R generators satisfy the Serre relations. Since theexpression above can actually be interpreted as a coproduct of the form∆( e ) = 1 ⊗ e + b e ⊗ K , ∆( f ) = b f ⊗ K ′ ⊗ f , so the Serre relations of e i and f i follow from the standard algebraic manipulationsof the coproduct.To show the remaining commutation relations, first we observe that e Ri com-mutes with b f jL and b e iL K Ri commutes with K ′ jL f Rj for any i, j ∈ I . So it sufficesto look at the other cross terms.We have [ e Ri , K ′ jL f Rj ] q i − q − i = K ′ jL [ e Ri , f Rj ] q i − q − i = K ′ jL ( δ ij K ′ iR + ω ij K Ri )= δ ij K ′ i + ω ij K ′ jL K Ri , [ b e iL K Ri , b f jL ] q i − q − i = K Ri [ b e iL , b f jL ] q i − q − i = K Ri ( − δ ij K Li − ω ij K ′ jL )= − δ ij K i − ω ij K ′ jL K Ri . Adding together, we obtain the required quantum relations.The second statement follows from definition. For example, e Ri ∈ X D ( i ) q isequivalent to the fact that it does not involve variables of the amalgamated vertices,so does e Li by definition of symmetry, and hence both belong to X D ( i ) q . Furthermore, b e iL K Ri = q − i K Li e Li K Ri = q − i K i e Li ∈ X D ( i ) q therefore e i ∈ X D ( i ) q as required. Similar argument works for the f i variables. Recall that the (maximal) positive representations can be decomposed into itsHeisenberg double counterpart (Corollary 3.20) as e i = e + i + K + i e − i ,f i = f − i + K ′ i − f + i ,K i = K + i K − i ,K ′ i = K ′ i + K ′ i − , { e ± i , f ± i , K ± i , K ′ i ± } is an embedding of the Heisenberg dou-ble H ± q ( g ) := H ± q, ( g ) into X Q ( i ) q or X Q ( i op ) q . Note that it is consistent with thedecomposition of Proposition 5.10 for ω ij ≡ e + i = e Ri , e − i = b e iL ,f + i = f Ri , f − i = b f iL ,K + i = K Ri , K − i = K Li ,K ′ i + = K ′ iR , K ′ i − = K ′ iL . Definition 5.11.
Let J ⊂ I . The double Dynkin involution of i ∈ I is defined tobe the unique index i ∗∗ ∈ I such that w s i = s i ∗ w = s i ∗ w J w = w J s i ∗∗ w. (5.20)Equivalently, we can compute it using i ∗∗ := ( i ∗ W ) ∗ WJ , (5.21)where we take the Dynkin involution first with respect to the whole group W , thenwith respect to the subgroup W J . Here by convention i ∗ WJ := i, if i / ∈ J. (5.22)Now to construct the parabolic positive representations, the main observationis the decomposition of the Heisenberg double generators. Recall that Q ( i ) = Q ( i J ) ∗ Q ( i ). Remark 5.12.
Due to the rule in Definition 3.6, we observe that the extra vertices e i of Q ( i ) corresponds to the labeling e i ∗∗ in Q ( i J ). Lemma 5.13 (Decomposition Lemma) . Let J ⊂ I . The embedding H + q ( g ) ֒ → X Q ( i ) q ⊂ X Q ( i J ) q ⊗ X Q ( i ) q (5.23) can be decomposed into the form e + i = e i + K i e Ji ∗∗ , (5.24) f + i = f Ji + K ′ iJ f i , (5.25) K + i = K Ji ∗∗ K i , (5.26) K ′ i + = K ′ iJ K ′ i , (5.27) where e Ji = f Ji = 0 and K Ji = K ′ iJ = 1 if i / ∈ J , such that • X Ji ∈ X Q ( i J ) q ⊗ and X i ∈ ⊗ X Q ( i ) q for X = e, f, K, K ′ , hence they commutewith each other. { e Ji , f Ji , K Ji , K ′ iJ } forms a copy of the embedding of H + q ( g J ) in X Q ( i J ) q where g J is the Lie subalgebra of g corresponding to the root index J ⊂ I . • K i , K ′ i are monomials that q -commutes with e j , f j as in (2.36) – (2.38) . Theorem 5.2 is now a direct consequence of the following properties:
Proposition 5.14. h e i , f i , K i , K ′ i i is the image of the generalized Heisenberg double H + q,ω ( g ) for some parameters ω ij .Proof. By the same argument as in the proof of Proposition 5.10, the generatorssatisfy the q -commutation relations and the Serre relations. It suffices to considerthe commutation between e i and f j .The statement is trivial if either j / ∈ J or i ∗∗ / ∈ J . Hence assume j ∈ J and i ∗∗ ∈ J . Then same as the proof of Proposition 5.10 before, only the cross termsmatter. We have δ ij K ′ j + = [ e + i , f + j ] q i − q − i = [ e i , K ′ Jj f j ] q i − q − i + [ K i e Ji ∗∗ , f Jj ] q i − q − i = K ′ jJ [ e i , f j ] q i − q − i + K i [ e Ji ∗∗ , f Jj ] q i − q − i = K ′ jJ [ e i , f j ] q i − q − i + δ i ∗∗ j K i K ′ jJ . Therefore [ e i , f j ] q i − q − i = δ ij K ′ j − δ i ∗∗ j K i , which is the required relations for the generalized Heisenberg double H + q,ω with ω ij := (cid:26) j / ∈ J or i ∗∗ / ∈ J,δ i ∗∗ j otherwise. (5.28) Proof of Theorem 5.2.
Let { e Ri , f Ri , K Ri , K ′ iR } := { e i , f i , K i , K ′ i } . (5.29)Then by construction they satisfy the condition (5.19) of Proposition 5.10, hencewe can combine with the opposite copy to obtain a homomorphism of D q ( g ) onto X D ( i ) q .In particular, under the group-like polarization, truncating the subquiver Q ( i J )by killing the variables corresponding to the Lusztig coordinates of M > provides48he required polarization for the quantum tous algebra X D ( i ) q . This amounts tosetting e Ji , f Ji K Ji , K ′ Ji P Jλ , then it naturallyinduces an invariant subspace of the maximal positive representation P λ . Since P λ is irreducible, the parabolic positive repesentation P Jλ is also irreducible.The fact that the homomorphism sends D q ( g ) to the universally Laurent poly-nomials follows from the proof of the Decomposition Lemma in the next section.Pictorially, if the positive representations are represented by the e i and f i -paths,then the parabolic positive representation is obtained by appropriately contractingthe paths. See Section 6 for explicit examples. It remains to prove the Decomposition Lemma 5.13. We need to use the explicitconstruction of the positive representations, which involve understanding the com-binatorics of the Coxeter moves of the reduced words.Let w ∈ W with reduced word i = ( i , ..., i M ). Let C rs denote the Coxeter movesinvolving position r < s of i , namely, it is either of the form( ..., i |{z} r , j |{z} s , .... ) ( ..., j, i, .... ) , (5.30)( ..., i |{z} r , j, i |{z} s , .... ) ( ..., j, i, j, .... ) , (5.31)( ..., i |{z} r , j, i, j |{z} s , .... ) ( ..., j, i, j, i, .... ) , (5.32)where s = r + 1 , r + 2 or r + 3. We do not need to consider type G .Recall that a Coxeter move i i ′ corresponds to a (sequence) of cluster muta-tions that transforms the quiver Q ( i ) −→ Q ( i ′ ) , and maps the corresponding embedding of U q ( g ) generators to the other quantumtorus algebra. Lemma 5.15.
Let i, j ∈ I . If l ( s i ws j ) = l ( w ) , (5.33) then there is a sequence of Coxeter moves that brings the reduced word i of w whichbegins with i , to a reduced word i ′ which ends with j : i = ( i, .... ) i ′ = ( ..., j ) , (5.34) such that the sequence of Coxeter moves is of the form ( C r ′ ,s ′ −→ · · · −→ C r ′ m ′ ,s ′ m ′ ) −→ ( C r ,s −→ · · · −→ C r m ,s m ) (5.35)49 here r < s = r < s = r < · · · = r m < s m = M, (5.36) i.e. the sequence of Coxeter moves in the second portion consists of a chain ofmoves which begins with the first letter, increases in indices consecutively, and endswith the last letter.Proof. We prove this by induction. It is trivial when l ( w ) = 1 , , l ( s i ws j ) = l ( w ), we cannot apply Coxeter moves to bring i into the word i ′ = ( i, ..., j )hence at some point, we must need to apply a Coxeter move C ,s to change the firstindex.After applying this move, the reduced word becomes either of the form( k, i, ... |{z} i ) or ( k, i, k, ... |{z} i ) or ( k, i, k, i, ... |{z} i )for some k = i ∈ I . It is then easy to see that the element w s corresponding to thetruncated word i s satisfies the assumption of the Lemma: l ( s i w s j ) = l ( w ) , l ( s k w s j ) = l ( w ) , l ( s i w s j ) = l ( w ) , because otherwise the assumption on i will be violated. Therefore by inductionthere exists a sequence of Coxeter moves of the form( C ) −→ ( C )that brings i s to ( ..., j ), where ( C ) consists of a sequence of Coxeter moves thatdoes not move the first letter of i s , and ( C ) consists of a sequence of Coxeter movesthat increases in index consecutively.Now note that ( C ) actually consists of moves of the original word i with index > s , hence in particular, we can do these moves first, apply C ,s , and followed bythe sequence of moves ( C ). The moves C ,s −→ ( C ) gives the second portion ofthe Coxeter moves that increases in index consecutively as required. We are now ready to complete the proof of the Decomposition Lemma 5.13 andhence the Main Theorem.
Proof of Lemma 5.13.
For the K ′ i and f i generators it follows from Remark 3.19 bythe explicit embedding, since the embedding into X Q ( i ) q = X Q ( i J ) ∗ Q ( i ) q obviously50an be decomposed. We have f + i = X f i + X f i ,f i + · · · + X f i ,...,f nJi − i | {z } f Ji + X f i ,...,f nJii + · · · + X f i ,...,f ni − i | {z } K ′ Ji f i ,K + i = X Jf i ,f i ,...,f nJii | {z } K ′ Ji X f nJii ,...,f nii | {z } K ′ i , where X f nJii = X Jf nJii ⊗ X f nJii corresponds to the cluster variable of the amalgamated notes at level i .Let us now focus on the e + i and K + i generators. Recall that if the word i endswith the index i N = i on the right, then e + i = X f i ,K + i = X f i ,e i are simply monomials, and for general word e + i , K + i are obtained by successivecluster transformations corresponding to the Coxeter moves that brings this i tothe required word.First, by appropriate cluster transformations on the Q ( i J ) subquiver, we canassume that the J portion of e + i is a single cluster variable. Due to the fact that w s i = w J s i ∗∗ w, the index of the J portion is given by the double Dynkin involution i ∗∗ .In the current setup, we need to do the Coxeter moves that bring i with i N = i to the form i ′ := ( i J , i ) where i J ends with i ∗∗ . By Lemma 5.15, the Coxeter movesstart by doing a straight decreasing sequence with consecutive indices, and ends atthe rightmost letter i ∗∗ of i J .By direct application of the quantum cluster mutation formula (cf. Definition2.33) and induction, the Coxeter move (5.30) amounts to permuting the index, themove (5.31) transforms e + i : X i + · · · + X i ,...,i j X i + · · · + X i ,...,i j ,i j +1 ,K + i : X i ,...,i j ,e i X i ,...,i j ,i j +1 ,e i while the move (5.32) may transform in two different ways depending on the longand short decorations: e + i : X i + · · · + X i ,...,i j X i + · · · + X i ,...,i j ,i j +1 + X i ,...,i j ,i j +2 ,K + i : X i ,...,i j ,e i X i ,...,i j ,i j +1 ,i j +2 ,e i , e + i : X i + · · · + X i ,...,i j X i + · · · + [2] q s X i ,...,i j ,i j +1 + X i ,...,i j ,i j +1 + X i ,...,i j ,i j +1 ,i j +2 ,K + i : X i ,...,i j ,e i X i ,...,i j ,i j +1 ,i j +2 ,e i , where [2] q s := q + q − . In both cases the right hand side are elements in thequantum torus algebra of the mutated quiver.Therefore, by induction after the decreasing chain of Coxeter moves, the gener-ators are of the form e + i = X i + · · · + X i ,...,i k − | {z } e i + X i ,...,i k ,K + i = X i ,...,i k ,e i in the final quantum torus algebra X Q ( i ′ ) q , where X i k = X Ji k ⊗ X i k corresponds to the cluster variable of the amalgamated notes at level i ∗∗ , so thatwe have the decomposition (also recall Remark 5.12) X i ,...,i k = X i ,...,i k | {z } K i X Ji k |{z} e Ji ∗∗ ,K + i = X i ,...,i k ,e i = X i ,...,i k | {z } K i X Ji k ,e i | {z } K Ji ∗∗ . The rest of the Coxeter moves correspond to cluster mutations that do notinvolve variables from X Q ( i J ) q , and hence keep the terms e Ji ∗∗ and K Ji ∗∗ invariant.Therefore we obtain the required decomposition of the quantum group generators.From the explicit cluster mutations above, we also see that there exists a quan-tum cluster mutations involving only variables from X D ( i ) q that brings e i and f i to a single variable having the same adjacency of the quiver of the maximal posi-tive representations. In particular they are all sinks, and hence it follows from [10,Proposition 13.11] that these generators are universally Laurent polynomials in thequantum cluster mutation class of X D ( i ) q . So far we have dealt with the positive representations of the split real quantumgroup U q ( g R ), but all the results extend naturally to its modular double .Let q = e πib , b i := p d i b, b s = √ db, (5.37)52here d = min i ∈ I ( d i ) is the minimum of the multipliers (2.3). Define q ∨ = e πib − s (5.38)Recall [14, 10] that the modular double is defined to be the algebra U qq ∨ ( g R ) = U q ( g R ) ⊗ U q ∨ ( L g R ) (5.39)where L g is the Langlands dual of g . Theorem 5.16.
The parabolic positive representations is a representation of themodular double in the sense of [9, 13, 14]. Namely the generators { e ∨ i , f ∨ i , K ∨ i } of U q ∨ ( L g R ) acts by π λ ( e ∨ i ) = π λ ( e i ) b i (5.40) π λ ( f ∨ i ) = π λ ( f i ) b i (5.41) π λ ( K ∨ i ) = π λ ( K i ) b i (5.42) as positive self-adjoint operators on the same space P Jλ of the U q ( g R ) representation,and they commute weakly with the generators of U q ( g R ) up to a sign.Proof. Note that the right hand side makes sense via functional calculus since ourrepresentations are positive self-adjoint. As in the end of the proof of Lemma 2.29, itfollows from the fact that under the unitary transformation given by the a sequenceof quantum cluster mutations, the generators in X D ( i ) q becomes a single monomialwith polarization of the form π ( X i ) = e πbL i .Hence the modular double counterpart given by π ( X ∨ i ) := π ( X i ) b i = e πbb − i L i = e πb − d − i L i (5.43)provides the Langlands dual polarization of X D ( i ) q ∨ in the sense that the multipliers d i are inverted, or equivalently the long and short root decorations are interchanged. E We begin by illustrating the construction of parabolic positive representation usingtype E as an example, since it simultaneously captures the subquivers of type A n and D n , as well as a nontrivial double Dynkin involution i ∗∗ .In [17], using the labeling of the Dynkin diagram1 2 3 4 5053he positive representations of type E corresponding to the reduced word i = (3 43 034 230432 12340321 5432103243054321) (6.1)of w ∈ W , which comes from the embedding of Dynkin diagram A ⊂ A ⊂ A ⊂ D ⊂ D ⊂ E (6.2)is given in Figure 9. Recall that the quiver is nothing but the double D ( i ) = Q ( i op ) ∗ Q ( i ).The embedding of the f i generators are just the horizontal path at level i givenas telescoping sums: f i = X i + X i ,i + · · · + X i ,...,i ni − . (6.3)On the other hand, the embedding of the e i generators are represented by paths ofdifferent colors from right to left, which again represent telescoping sums. (Recallthat the telescoping sum does not include the last term of the paths, cf. Example3.21.) Finally, the generators K i and K ′ i are represented by monomials of the nodesalong the e i and f i -paths respectively.We shade the quivers to indicate the boundary of amalgamation of the fullsubquivers Q ( i J ) for the parabolic subgroups corresponding to the Dynkin chain(6.2) above, i.e. J are the different subsets { } ⊂ { , } ⊂ { , , } ⊂ { , , , } ⊂ { , , , , } . (6.4)(Note: the node { f } is not part of the green region corresponding to the A subquiver.)We can verify then that the e i -paths pass through the correct index of the fullsubquivers. For example, the e -path passes through the 0-th index of the A subquiver generated by the root index J = { , , } , where (2 ∗ W ) ∗ WJ = 4 ∗ WJ = 0.The double Dynkin involution actually partially explains the behavior of the e i -paths. It was mysterious to us previously the reason why the path goes up anddown across the whole quiver passing through different level. We can now see fromthe quiver diagram that in this case, in fact the e i -paths are forced to take theunique paths along the arrows (without tracing backward) that allow them to gothrough the correct levels as depicted by the double Dynkin involution.As another example, by considering the parabolic subgroup corresponds to the D subquiver corresponding to J = { , , , , } (i.e. the whole colored region),we truncate the quiver and obtain the e i and f i -paths representing the parabolicpositive representations on the quiver D ( i ) as indicated in Figure 10. This is a“minimal” representation in the sense that the P J has the smallest codimension in G . B n The construction obvious works for arbitrary types, including non-simply-lacedcase. We demonstrate the minimal parabolic positive representations in type B n .Recall that { } is the short root. 54 e f e f e f e f e f e f e e e e e e Figure 9: The E quiver, with the e i -paths (from right to left) in different colors. f e f e f e f e f e f e Figure 10: Parabolic positive representations of D ⊂ E , with the e i -paths (fromright to left) in different colors.When J = { , , ..., n − } we choose the longest reduced word to be i = (1212 32123 4321234 ... n n − ...n − n ) (6.5)so that i = ( n n − ... ... n − n ), and we obtain the parabolic positive repre-sentations as depicted in Figure 11 on the quiver D ( i ), where the e i and f i -pathsare shown in red and blue respectively. (In this quiver, the top row has multiplier d i = .)On the other hand, for J = { , , ..., n } , g J is of type A n − . Then the longest The same word used in [14]. f f f e e e e Figure 11: Parabolic positive representations of type B with J = { , , } . The e i and f i -paths are shown in red and blue respectively.reduced word i is of the form i = i J i A n , (6.6)where i J a longest word for w J , and it turns out i A n is also the standard longestword (2.57), i.e. we can take i = ( n, n − , n, ..., , , ..., n, , , , , , , ..., n, ... . (6.7)Then i = i A n and we obtain the parabolic positive representations on D ( i ) as inFigure 12. However, note that this is not the usual type A n full quiver since wehave extra multipliers in this Example. In particular the auxiliary quiver H ( i ) onlyconsists of a single node { e } .Here the red circles on a node k of the quiver indicates a modification of the e i -paths by doubling the nodes with a [2] q s factor as in the proof of Lemma 5.13: · · · + X ...,j + [2] q s X ...,j,k + X ...,j,k + X ...,j,k ,l + · · · . (6.8)As remark in Section 5.6, we still have the modular double transformation π ( e ∨ i ) := π ( e i ) b i (6.9) This is the reduced word used in [21]. f f f e e e e Figure 12: Parabolic positive representations of type B with J = { , , } . The e i -paths (from right to left) are shown in different colors.to reproduce the parabolic positive representations of U q ∨ ( g C n ) in type C n , wherethe quiver remains the same, but with the multipliers of each nodes changes ac-cordingly by d i d i . There are several natural questions arising from the construction of parabolic pos-itive representations that will be interesting to understand, some of which are mo-tivated from the simplest case of the minimal positive representations in Section4. During the construction, we have used the basic quiver Q ( i ) which is naturallyassociated to the Poisson structure of the partial configuration space Conf eu ( A )described in [10]. Therefore one should try to understand and possibly simplifythe proofs of the parabolic positive representations by quantizing the geometricalmethods using, perhaps, partial decorated G -local system, where the decorationsare provided by partial flags. However, it seems we do not have a simple associationof the partial quiver Q ( i ) to the triangles of triangulated surfaces since the frozendegree of each edge does not match. 58 olynomial image and Lusztig’s braid group action We have seen that the homomorphism D q ( g ) −→ X D ( i ) q lies in the universally Laurent polynomials of the quantum cluster mutation classof X D ( i ) q . However from all calculations so far we observe that the generators e i , f i are in fact always polynomials , without any negative powers of X k involved. Conjecture 7.1.
The image of the generators of D q ( g ) in X D ( i ) q are polynomialsin the cluster variables for any cluster in the mutation class of the quantum torusalgebra. This assumption actually allows us to verify the Lusztig’s braid group action on D ( i ). Recall from Definition 2.20 that we have Lusztig’s braid group action T i ,which is known to be a cluster map [10, 21], i.e. represented by cluster transforma-tion on X D ( i ) q . In particular, the automorphism for the longest word i : T i := T i · · · T i N interchanges the action of the e i and f i generators as in Proposition 2.21. In termsof quiver, this means that there exists a sequence of mutations such that the ba-sic quiver is mirror reflected and the three frozen sides { f i , f n i i , e i } are cyclicallypermuted. Conjecture 7.2.
There exists a cluster map of X D ( i ) q for any parabolic positiverepresentations P Jλ that interchanges the action of e i and f i . Using the explicit formula (3.25), one can compute directly that the non-simplegenerators f α are always nonzero on the double X D ( i ) q . Hence a consequence ofthis conjecture is that Corollary 7.3.
The universal R operator (4.18) is well-defined on tensor products P Jλ ⊗ P J ′ λ ′ of parabolic positive representations as unitary transformation. For the parabolic positive representations, we have verified in a few simple casesthat indeed we can interchange the generators by cluster transformations. But ingeneral the mutation sequence for T i is very complicated and meshed up all thevariables, and it is unclear how it can be factorized in the parabolic case.For example, Ian Le previously constructed explicitly the mutation sequence forthe T i action in type E corresponding to the Coxeter words i := (12345670 · · · | {z }
15 copies ) op . This is not true without the double. For example if we take i = (1 ,
2) then f = 0 on X Q ( i ) q .
59y construction we know that the generators f i each involve 15 monomials on X Q ( i ) q . Using the mutation sequence, we observe that during the mutations, the f generator grows up to 825887337 terms before dropping down to 147249 terms atthe end giving the required action of the generator e . During the process all theexpressions involved are polynomials of the cluster variables as conjectured, and itwill be important to give a more combinatorial description of these polynomials. Casimir operators
In the minimal positive representations, we have studied in detail the central char-acters and the action of the Casimir operators, and show that they lie outside thespectrum of the positive Casimirs of the maximal case. It is then natural to proposethe following
Conjecture 7.4.
The spectrum of the Casimirs of arbitrary parabolic positive rep-resentations P Jλ of U q ( g R ) are all real-valued and disjoint in R n . The techniques using virtual highest weight vector should be generalized toarbitrary parabolic case, and in particular, for the maximal parabolic subgroupcorresponding to J = I \ { j } for a single index j , we expect again that the repre-sentations behave like the “fundamental representations” in the sense that K i actstrivially for i ∈ J on the virtual highest weight functionals.Understanding the spectrum of the Casimirs is an important step towards thedecomposition of the tensor product of positive representations. It is clear [17] thatthe parabolic positive representations of the tensor product P Jλ ⊗ P J ′ λ ′ can be constructed by amalgamating two copies D ( i ) ∗ D ( i ′ ) side by side by con-catenating the corresponding e i and f i -paths. It is then a natural question todecompose it into irreducible components, and together with the existence of theuniversal R operator in Conjecture 7.3, possibly provide us with another candidateof continuous braided tensor category .In the case of the usual (maximal) positive representations, the decompositionwas done in type A n in [26, 27] by a sequence of cluster mutations corresponding toflipping the triangulation on a two-punctured disk in order to simplify the Casimiroperators and compare them with the Hamiltonians of a Coxeter-Toda conformalfield theory. The same trick will not work here as pointed out above, because thequivers involved now are not naturally associated to any triangles of triangulationsof punctured Riemann surfaces. We remark that the decomposition in the othertypes for the usual positive representations is also still an open question. Modules for affine quantum groups
Finally, the construction of the evaluation modules of U q ( b sl n +1 ) in Section 4.5utilizes the symmetry of the minimal quiver D ( i ), and one can try to construct The number of monomial terms for the other generators are (1 , , , , , , A (1) n it is clear that one approach is to use the evaluation module to construct ahigher rank quiver from those of U q ( sl n +1 ), and it will be interesting to understandthe combinatorics behind and find natural generalization to other types. References [1] A. Berenstein, A. Zelevinsky,
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