Bivariate Continuous q-Hermite Polynomials and Deformed Quantum Serre Relations
aa r X i v : . [ m a t h . QA ] N ov BIVARIATE CONTINUOUS q -HERMITE POLYNOMIALSAND DEFORMED QUANTUM SERRE RELATIONS W. RILEY CASPER, STEFAN KOLB, AND MILEN YAKIMOV
To Nicol´as Andruskiewitsch on his 60th birthday, with admiration
Abstract.
We introduce bivariate versions of the continuous q -Hermite poly-nomials. We obtain algebraic properties for them (generating function, explicitexpressions in terms of the univariate ones, backward difference equations andrecurrence relations) and analytic properties (determining the orthogonalitymeasure). We find a direct link between bivariate continuous q -Hermite poly-nomials and the star product method of [KY19] for quantum symmetric pairsto establish deformed quantum Serre relations for quasi-split quantum sym-metric pairs of Kac-Moody type. We prove that these defining relations areobtained from the usual quantum Serre relations by replacing all monomialsby multivariate orthogonal polynomials. Introduction
Quantum groups have played a key role in many areas of mathematics andmathematical physics since their introduction by Drinfeld [Dri87] and Jimbo [Jim85]in the 1980s. In the late 1990s Andruskiewitsch and Schneider initiated a powerfulprogram for classifying pointed Hopf algebras [AS02] which lead to far reachinggeneralizations of quantum groups, namely Drinfeld doubles of pre-Nichols algebras.Quantum symmetric pairs in the above frameworks have become the subjectof intense research. The general construction of quantum symmetric pairs in thesetting of quantized enveloping algebras of finite dimensional semisimple Lie alge-bras was given by Letzter [Let99]. The Kac–Moody setting was treated in [Kol14].Quantum symmetric pairs in the setting of Drinfeld doubles of pre-Nichols algebraswere defined in [KY19]. In those settings the quantum symmetric pairs havingIwasawa decompositions were characterized in [Let97, Let99, Kol14, KY19].Quantum symmetric pair coideal subalgebras B c depend on a set of parameters c = ( c i ) i ∈ I , and are defined in terms of generators B i for i ∈ I in the ambientHopf algebra. The generators B i satisfy deformed quantum Serre relations, see[Let03, Section 7], [Kol14, Section 7]. One of the outstanding problems in the areaof quantum symmetric pairs is to determine explicit, conceptual formulas for thedeformed quantum Serre relations. The goal of this paper is the following: Metatheorem.
The deformed quantum Serre relations for a quantum symmet-ric pair coideal subalgebra are obtained from the usual quantum Serre relations byreplacing all monomials by multivariate orthogonal polynomials . Mathematics Subject Classification.
Primary: 17B37, Secondary: 53C35, 16T05, 17B67.
Key words and phrases.
Quantum symmetric pairs, bivariate continuous q -Hermitepolynomials. While in the present paper we only establish the Metatheorem in the so calledquasi-split Kac–Moody setting, we expect that this phenomenon holds in full gen-erality. Our proof is based on a result from [KY19] that (quantum) symmetricpair coideal subalgebras are isomorphic to star products on partial bosonizations ofpre-Nichols algebras .In [Let03] Letzter developed a method to obtain the deformed quantum Serrerelations from coproducts. She applied her method to obtain explicit relations forall quantum symmetric pairs of finite type. Letzter’s method was extended to theKac-Moody setting in [Kol14] and applied in the case of Cartan matrices ( a ij ) i,j ∈ I with | a ij | ≤
3. Recall that quantum symmetric pairs depend on an involutivediagram automorphism τ : I → I . In the case τ ( i ) = j = i the correspondingdeformed quantum Serre relations were explicitly determined by Letzter’s methodin [BK15, Theorem 3.6]. In the case τ ( i ) = i = j Letzter’s method gets substantiallyharder. Nonetheless, recently, de Clercq used Letzter’s method to produce involvedcombinatorial formulas for deformed quantum Serre relations in the Kac–Moodycase for τ ( i ) = i = j [dC19]. However, the connection to orthogonal polynomials isnot immediate.In the quasi-split Kac–Moody setting the deformed quantum Serre relations inthe case τ ( i ) = i = j were first derived in [CLW18] in terms of so called ı dividedpowers. The ı divided powers are univariate polynomials and play an important rolein the theory of canonical bases for quantum symmetric pairs [BW18]. However, aninterpretation of ı divided powers in terms of orthogonal polynomials is not known.In the quasi-classical limit, formulas for the corresponding deformed Serre relationswere recently obtained by Stokman in [Sto19].In the present paper we give an explicit expression of deformed quantum Serrerelations for quasi-split quantum symmetric pairs in terms of bivariate continuous q -Hermite polynomials. Our proofs are shorter than those in previous approaches andare based on a direct relation between the star products of [KY19] and multivariateorthogonal polynomials.The classical Hermite polynomials are the polynomials given by the recurrencerelation H n +1 ( x ) = 2 xH n ( x ) − nH n − ( x ) , where H ( x ) = 1. They have two types of q -analogs, the continuous and discrete q -Hermite polynomials [KLS10, § q -Hermite polynomials[KLS10, § H n +1 ( x ; q ) = 2 xH n ( x ; q ) − (1 − q n ) H n − ( x ; q ) , where H ( x ; q ) = 1. They appear in a number of diverse situations. For instance,recently Borodin and Corwin used them in the study of the dynamic asymmetricsimple exclusion process [BC20]. Motivated by Itˆo’s complex bivariate orthogo-nal Hermite polynomials [Itˆo52], Ismail and Zhang [IZ17] defined and studied twoversions of bivariate q -Hermite polynomials H m,n ( x, y | q ) (without additional pa-rameters). They satisfy H m, ( x, y ) = x m , H ,n ( x, y ) = y n .In this paper we define and study a completely different (two-parameter) familyof bivariate continuous q -Hermite polynomials H m,n ( z , z ; q, r ). They satisfy H m, ( x, y ; q, r ) = H m ( x ; q ) and H ,n ( x, y ; q, r ) = H n ( y ; q ) IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 3 and are recursively defined by H m +1 ,n ( x, y ; q, r ) = 2 xH m,n ( x, y ; q, r ) − (1 − q m ) H m − ,n ( x, y ; q, r ) − q m (1 − q n ) rH m,n − ( x, y ; q, r ) . We establish algebraic and analytic properties of these polynomials. On the alge-braic side, we prove that they are explicitly given by H m,n ( x, y ; q, r ) = min( m,n ) X k =0 ( − k q ( k )( q ; q ) m ( q ; q ) n r k ( q ; q ) m − k ( q ; q ) n − k ( q ; q ) k H m − k ( x ; q ) H n − k ( y ; q ) , (1.1)and in particular, they are symmetric with respect to x and y : H m,n ( x, y ; q, r ) = H n,m ( y, x ; q, r ). We show that their generating function is given by ∞ X m,n =0 H m,n ( x, y ; q, r )( q ; q ) m ( q ; q ) n s m t n = ( rst ; q ) ∞ | ( se iθ , te iφ ; q ) ∞ | · We derive an operator formulation and a backward difference equation for thesepolynomials (see Theorem 2.7). On the analytic side we prove that they are or-thogonal with respect to the measure | ( e i ( α + β ) /r ; q ) ∞ | p (1 − x )(1 − y ) dxdy on [0 , × [0 , , where x = cos(2 α ) , y = cos(2 β ) . We believe that these polynomials will find application outside the realm of Hopfalgebras and quantum symmetric pairs.With the above notation we can now express the deformed quantum Serre re-lations for quasi-split quantum symmetric pairs in the case τ ( i ) = i = j . For w ( x, y ) = P r,s b rs x r y s ∈ K [ x, y ], set z y w ( x, y ) = X r,s b rs x r zy s . The following theorem is derived from the algebraic properties of bivariate continu-ous q -Hermite polynomials. The theorem holds for general deformation parameters q including roots of unity. Theorem. (Corollaries 4.10, 4.14)
Let i, j ∈ I and τ ( i ) = i = j . Then thegenerators B i , B j of the quantum symmetric pair coideal subalgebra B c satisfy therelation − a ij X ℓ =0 ( − ℓ (cid:20) − a ij ℓ (cid:21) q i B j y w − a ij − ℓ,ℓ ( B i , B i ) = 0 , where w m,n ( x, y ) = (2 b i ) − m − n H m,n ( b i x, b i y ; q i , q a ij i ) and b i = ( q i − q − i ) c − / i q − / i .This relation can also be written as − a ij X ℓ =0 ( − ℓ (cid:20) − a ij ℓ (cid:21) q i w − a ij − ℓ ( B i ) B j v ℓ ( B i ) = 0 , where w m ( x ) = 1(2 b i ) m H m ( b i x ; q i ) , v m ( x ) = 1(2 b i ) m H m ( b i x ; q − i ) . The paper is organized as follows. Section 2 contains background material onmultivariate orthogonal polynomials and the statements of our results on bivariate
W. RILEY CASPER, STEFAN KOLB, AND MILEN YAKIMOV continuous q -Hermite polynomials. Section 3 contains the proof of these results.In Section 4 we recall the isomorphism theorem from [KY19] identifying quantumsymmetric pair coideal subalgebras with star products on partial bosonizations ofpre-Nichols algebras. Then we use the algebraic facts on the bivariate continuous q -Hermite polynomials to derive the defining relations of quantum symmetric paircoideal subalgebras of quantum groups in the quasi-split Kac–Moody case. Acknowledgements.
We are grateful to the referee for the detailed commentswhich helped us to improve the exposition. The research of W.R.C. was supportedby a 2018 AMS-Simons Travel Grant. The research of M.Y. was supported by NSFgrant DMS-1901830 and Bulgarian Science Fund grant DN02/05.2.
Orthogonal polynomials
In this section, we provide a brief review of the theory of orthogonal polyno-mials in a single variable and introduce the Hermite and continuous q -Hermitepolynomials as examples. We then recall the definition of multivariate orthogonalpolynomials and introduce a bivariate analog of the continuous q -Hermite polyno-mials.2.1. Classical Orthogonal Polynomials.
A sequence of orthogonal polynomialson the real line is a sequence p ( x ) , p ( x ) , . . . of complex-valued polynomials withdeg p n ( x ) = n for all n ≥
0, which satisfy the orthogonality condition Z R p m ( x ) p n ( x ) dµ ( x ) = δ m,n c n for some positive Borel measure µ on R and sequence of positive constants { c n } ∞ n =0 .An elementary argument shows that any sequence of orthogonal polynomialsautomatically satisfies a three-term recursion relation of the form(2.1) xp n ( x ) = α n p n +1 ( x ) + β n p n ( x ) + γ n p n − ( x ) , for some sequence of constants { α n } ∞ n =0 , { β n } ∞ n =0 and { γ n } ∞ n =1 , with p − ( x ) := 0.The values are related to the moments of µ ( x ) and the leading coefficients of the p n ( x )’s. Conversely, for any sequences { α n } ∞ n =0 , { β n } ∞ n =0 and { γ n } ∞ n =1 with β n real and α n γ n positive, the sequence of polynomials defined recursively by (2.1)will be orthogonal polynomials for some Borel measure µ ( x ). This result is knownas Favard’s theorem and is a consequence of the spectral theorem applied to thesemi-infinite Jacobi matrix defined by the three-term recursion relation [Fav35].The most fundamental examples of orthogonal polynomials are the classical or-thogonal polynomials of Hermite, Laguerre, and Jacobi. These polynomials satisfythe additional property that they are eigenfunctions of a second-order differentialequation in the variable x , i.e. a ( x ) p ′′ n ( x ) + a ( x ) p ′ n ( x ) + a ( x ) p n ( x ) = λ n p n ( x )for some functions a ( x ) , a ( x ) and a ( x ) and sequence of complex numbers { λ n } ∞ n =0 .Each sequence of classical orthogonal polynomials satisfies a Rodrigues-type recur-rence relation and has a nice generating function formula. For example considerthe classical Hermite polynomials H n ( x ) defined by(2.2) H n ( x ) = n ! ⌊ n/ ⌋ X m =0 ( − m m !( n − m )! (2 x ) n − m . IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 5 Example 2.1.
The Hermite polynomials H n ( x ) have the following properties [KLS10] : • orthogonality relation: Z R H m ( x ) H n ( x ) e − x dx = √ π n n ! δ m,n • three-term recursion relation: xH n ( x ) = 12 H n +1 ( x ) + nH n − ( x ) • second-order differential equation: H ′′ n ( x ) − xH ′ n ( x ) = − nH n ( x ) . • generating function: e xt − t = ∞ X n =0 H n ( x ) n ! t n • Rodrigues-type recurrence relation: H n ( x ) = ( − n e x (cid:18) ddx (cid:19) n · e − x The classical orthogonal polynomials naturally generalize when we replace thedifferential operator with a second-order difference or q -difference operator, in whichwe obtain the various families obtained from the Askey and q -Askey scheme, such asthe Wilson, Racah, Hahn, Meixner, Meixner-Pollaczek, Krawtchouk, and Charlierpolynomials and their q -analogues. As before, each such sequence of orthogonalpolynomials satisfies a three-term recursion relation, a differential, difference, or q -difference equation, a Rodrigues-type recurrence relation, and has a nice generatingfunction formula. In this paper, we will be particularly interested in the continuous q -Hermite polynomials H n ( x ; q ) defined for x = cos( θ ) by(2.3) H n ( x ; q ) := n X k =0 ( q ; q ) n ( q ; q ) k ( q ; q ) n − k e i ( n − k ) θ = e inθ φ (cid:20) q − n , − ; q, q n e − iθ (cid:21) . Here ( a ; q ) n is the q -Pochhammer symbol and φ (cid:20) a,b − ; q, z (cid:21) is the q -hypergeometricfunction defined respectively by( a ; q ) n = n − Y k =0 (1 − aq k ) and φ (cid:20) a, b − ; q, z (cid:21) = ∞ X n =0 ( a ; q ) n ( b ; q ) n ( q ; q ) n ( − n q ( n ) z n . We also have the infinite q -Pochhammer symbol ( a ; q ) ∞ = lim n →∞ ( a ; q ) n whichhas useful series expansion we will rely on later in this paper(2.4) ( a ; q ) ∞ = ∞ Y k =0 (1 − aq k ) = ∞ X n =0 ( − n q ( n )( q ; q ) n a n . Note that in terms of the Chebyshev polynomials of the first kind T n ( x ), thismay be rewritten as H n ( x ; q ) := n X k =0 ( q ; q ) n ( q ; q ) k ( q ; q ) n − k T | n − k | ( x ) , so that in particular H n ( x ; q ) is a polynomial in x of degree n for all n . W. RILEY CASPER, STEFAN KOLB, AND MILEN YAKIMOV
Example 2.2.
The continuous q -Hermite polynomials H n ( x ; q ) satisfy the follow-ing properties [KLS10] : • orthogonality relation: (2.5) Z − H m ( x ; q ) H n ( x ; q ) | ( e iθ ; q ) ∞ | √ − x dx = 2 πδ m,n ( q n +1 ; q ) ∞ . • three-term recursion relation: (2.6) 2 xH n ( x ; q ) = H n +1 ( x ; q ) + (1 − q n ) H n − ( x ; q ) . • (forward) q -difference equation: (2.7) D q H n ( x ; q ) = 2 q − ( n − / (1 − q n )1 − q H n − ( x ; q ) . • generating function: (2.8) ∞ X n =0 H n ( x ; q )( q ; q ) n s n = 1 | ( se iθ ; q ) ∞ | . • Rodrigues-type recurrence relation: (2.9) H n ( x ; q ) = (cid:18) q − (cid:19) n q n ( n − √ − x | ( e iθ ; q ) ∞ | ( D q ) n · | ( e iθ ; q ) ∞ | √ − x Remark . In the q -difference equation above, D q is the q -difference operatorfound in [KLS10, Equation 1.16.4], given by D q f ( x ) = δ q f ( x ) δ q x , x = cos( θ )where here δ q f ( e iθ ) = f ( q / e iθ ) − f ( q − / e iθ ) , so that in particular δ q x = − q − / (1 − q )( e iθ − e − iθ ) for x = cos θ .2.2. Multivariate orthogonal polynomials.
The theory of multivariate orthog-onal polynomials on R d is considerably more complicated than the single variablesituation and far less complete. Even so, the basics of the theory remain the same aslong as the definitions are taken appropriately. Some useful introductory referencesare [DX14, Xu05].For simplicity, we will adopt the vector notation ~x = ( x , . . . , x d ) and ~n =( n , . . . , n d ) and will write | ~n | to mean n + · · · + n d . We will also use the monomialnotation x ~n for the product x n x n . . . x n d d . Definition 2.4.
A sequence of orthogonal polynomials in d variables is a sequence p ~n ( ~x ) of polynomials in variables x , . . . , x d such that(a) for all m the polynomials { p ~n ( ~x ) : | ~n | ≤ m } define a basis for the space ofpolynomials of total degree at most m (b) there exists a positive Borel measure µ on R d with finite moments R R | x ~n | dµ ( ~x ) < ∞ satisfying Z R d p ~m ( ~x ) p ~n ( ~x ) dµ ( ~x ) = 0 for | ~m | 6 = | ~n | . IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 7 In other words, polynomials of different total degrees are orthogonal, but differentpolynomials with the same total degree may not be. In particular, one may haveto perform a change of basis e p ~n ( x, y ) = X | ~m | = | ~n | a ~m p ~m ( ~x ) , to get a sequence of orthogonal polynomials satisfying the more intuitive orthogo-nality condition(2.10) Z R e p ~m ( ~x ) e p ~n ( ~x ) dµ ( ~x ) = 0 when ~m = ~n. to apply for all ~m and ~n .A sequence of orthogonal polynomials again gives rise to a three-term recursionrelation, except that the summands are in terms of the total degree and can in-volve multiple polynomials with the same total degree. Specifically, there will existconstants α ~n,~m,j , β ~n,~m,j , γ ~n,~m,j such that for all j = 1 , . . . , dx j p ~n ( ~x ) = X | ~m | = | ~n | +1 α ~n,~m,j p ~m ( ~x ) + X | ~m | = | ~n | β ~n,~m,j p ~m ( ~x ) + X | ~m | = | ~n |− γ ~n,~m,j p ~m ( ~x ) . An analog of Favard’s theorem has also been proved [Xu93], i.e. for sufficientlynice sequences of constants, the sequence of polynomials defined by the three-termrecursion relation will be orthogonal with respect to some measure µ on R d . Asmentioned above, we can then change our basis so that the orthogonal polynomialssatisfy the simple orthogonality condition (2.10), but this in turn will completelychange the original recurrence relations and the new orthogonal polynomials maylose other desirable properties such as having monomial leading coefficients.In the next section, we will construct two dimensional analogs of the continu-ous q -Hermite polynomials defined above, which we will hereafter refer to as thebivariate continuous q -Hermite polynomials(2.11) H m,n ( x, y ; q, r ) = min( m,n ) X k =0 ( − k q ( k )( q ; q ) m ( q ; q ) n r k ( q ; q ) m − k ( q ; q ) n − k ( q ; q ) k H m − k ( x ; q ) H n − k ( y ; q ) . Note that H m, ( x, y ; q, r ) = H m ( x ; q ) and H ,n ( x, y ; q, r ) = H n ( y ; q ) . Moreover H m,n ( x, y ; q,
0) = H m ( x ; q ) H n ( y ; q ) , so H m,n ( x, y ; q, r ) may be thought of as a deformation of the family of orthogonalpolynomials H m ( x ; q ) H n ( y ; q ) with deformation parameter r . Remark . Our bivariate continuous q -Hermite polynomials are very differentfrom those constructed by Ismail and Zhang [IZ17], which were motivated by thecomplex bivariate orthogonal Hermite polynomials introduced by Itˆo [Itˆo52]. Remark . We do not define the bivariate continuous q -Hermite polynomials with(2.11). Instead, we define them in the next section in terms of a symmetry conditionand a three-term recursion relation reminiscent of the recursion relation for the onevariable case. We then prove that the resulting sequence satisfies (2.11). W. RILEY CASPER, STEFAN KOLB, AND MILEN YAKIMOV
In the next section we will prove several important properties of these poly-nomials, including orthogonality, recurrence relations, q -difference equations, anda generating function formulation. We summarize these properties here for theconvenience of the reader. Theorem 2.7.
The bivariate continuous q -Hermite polynomials H m,n ( x, y ; q, r ) satisfy the following properties. • Orthogonality relation: Z − Z − H m,n ( x, y ; q, r ) H e n, e m ( x, y ; q, r ) | ( e i ( α + β ) /r ; q ) ∞ | p (1 − x )(1 − y ) dxdy = c m,n δ m, e m δ n, e n , where here x = cos(2 α ) , y = cos(2 β ) and c m,n = 2 π ( q ; q ) ∞ X i + k + ℓ = mj + k + ℓ = n ( − k q ( k )( q ; q ) m ( q ; q ) n ( q ; q ) i ( q ; q ) j ( q ; q ) k ( q ; q ) ℓ r m + n . • Three-term recursion relations: xH m,n ( x, y ; q, r ) = H m +1 ,n ( x, y ; q, r ) + (1 − q m ) H m − ,n ( x, y ; q, r )+ q m (1 − q n ) rH m,n − ( x, y ; q, r ) , yH m,n ( x, y ; q, r ) = H m,n +1 ( x, y ; q, r ) + (1 − q n ) H m,n − ( x, y ; q, r )+ q n (1 − q m ) rH m − ,n ( x, y ; q, r ) . • Generating function: ∞ X m,n =0 H m,n ( x, y ; q, r )( q ; q ) m ( q ; q ) n s m t n = ( rst ; q ) ∞ | ( se iθ , te iφ ; q ) ∞ | ·• Operator formulation: H m,n ( x, y ; q, r ) = 1 (cid:16) − q m + n − (cid:0) − q (cid:1) rD q,x D q,y ; q (cid:17) ∞ · H m ( x ; q ) H n ( y ; q ) . • Additional relations: D q,x · H m,n ( x, y ; q, r ) = 2 q − m − (1 − q m )1 − q H m − ,n ( x, y ; q, √ qr ) ,D q,y · H m,n ( x, y ; q, r ) = 2 q − n − (1 − q n )1 − q H m,n − ( x, y ; q, √ qr ) . Generating function, recursion relations and orthogonality
In this section we define the bivariate continuous q -Hermite polynomials andprove the properties stated in Theorem 2.7. Excepting the initial definition of thebivariate continuous q -Hermite polynomials below, we will write H m,n ( x, y ) in placeof H m,n ( x, y ; q, r ) throughout this section for sake of brevity. IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 9 The bivariate continuous q -Hermite polynomials. As a two-dimensionalanalog of the continuous q -Hermite polynomials, we consider the following sequenceof bivariate polynomials. Definition 3.1.
The bivariate continuous q -Hermite polynomials are the uniquesequence of orthogonal polynomials H m,n ( x, y ; q, r ) defined for all integers m, n ≥ satisfying the symmetry condition (3.1) H m,n ( x, y ; q, r ) = H n,m ( y, x ; q, r ) as well as the three-term recursion relation xH m,n ( x, y ; q, r ) = H m +1 ,n ( x, y ; q, r )(3.2) + (1 − q m ) H m − ,n ( x, y ; q, r )+ q m (1 − q n ) rH m,n − ( x, y ; q, r ) , with H , ( x, y ; q, r ) = 1 and H − , ( x, y ; q, r ) = H , − ( x, y ; q, r ) = 0 . Note in particular H m,n ( x, y ; q, r ) is a polynomial of bidegree ( m, n ), and that H m, ( x, y ; q, r ) = H m ( x ; q ).Mimicking the generating function in the single-variable case (2.8), we considerthe function ψ (cid:18) x, ys, t (cid:19) := ∞ X m,n =0 ψ m,n (cid:18) x, ys, t (cid:19) , where ψ m,n (cid:18) x, ys, t (cid:19) := H m,n ( x, y ) s m t n ( q ; q ) m ( q ; q ) n . Note that the recursion relation above tells us2 xψ m,n (cid:18) x, ys, t (cid:19) = 1 s (cid:18) ψ m +1 ,n (cid:18) x, ys, t (cid:19) − ψ m +1 ,n (cid:18) x, yqs, t (cid:19)(cid:19) + sψ m − ,n (cid:18) x, ys, t (cid:19) + rtψ m,n − (cid:18) x, yqs, t (cid:19) . Summing this, we find2 xψ (cid:18) x, ys, t (cid:19) = 1 s (cid:18) ψ (cid:18) x, ys, t (cid:19) − ψ (cid:18) x, yqs, t (cid:19)(cid:19) + sψ (cid:18) x, ys, t (cid:19) + rtψ (cid:18) x, yqs, t (cid:19) , which simplifies to the homogeneous q -difference equation ψ (cid:18) x, yqs, t (cid:19) = (cid:18) xs − s − rst − (cid:19) ψ (cid:18) x, ys, t (cid:19) . Factoring (cid:18) xs − s − rst − (cid:19) = (1 − ( x + √ x − s )(1 − ( x − √ x − s )1 − rst and using the fact that ( qx ; q ) ∞ (1 − x ) = ( x ; q ) ∞ , we see that the general solutionof this q -difference equation is ψ (cid:18) x, ys, t (cid:19) = ( rst ; q ) ∞ (( x + √ x − s ; q ) ∞ (( x − √ x − s ; q ) ∞ ψ (cid:18) x, y , t (cid:19) = ( rst ; q ) ∞ | ( e iθ s ; q ) ∞ | ψ (cid:18) x, y , t (cid:19) , for x = cos( θ ) . Finally by symmetry and the choice that H , ( x, y ) = 1, or alternatively by using(2.8), we obtain a generating function formula for the polynomials H m,n ( x, y ) with x = cos( θ ) and y = cos( φ )(3.3) ψ (cid:18) x, ys, t (cid:19) := ∞ X m,n =0 H m,n ( x, y )( q ; q ) m ( q ; q ) n s m t n = ( rst ; q ) ∞ | ( se iθ , te iφ ; q ) ∞ | . The generating function equation also allows us to express our bivariate con-tinuous q -Hermite polynomials in terms of the continuous q -Hermite polynomialsin a single variable. By applying (2.8) along with the series expansion for the q -Pochhammer symbol (2.4) for ( rst ; q ) ∞ we see( rst ; q ) ∞ | ( se iθ , te iφ ; q ) ∞ | = ∞ X k,m,n =0 ( − k q ( k ) r k s m + k t m + k ( q ; q ) k ( q ; q ) m ( q ; q ) n H m ( x ) H n ( y ) . Comparing similar powers of s and t , we find(3.4) H m,n ( x, y ) = min( m,n ) X k =0 ( − k q ( k )( q ; q ) m ( q ; q ) n r k ( q ; q ) m − k ( q ; q ) n − k ( q ; q ) k H m − k ( x ; q ) H n − k ( y ; q ) . Orthogonality.
By Xu’s extension of Favard’s theorem [Xu93], we expect asequence of multivariate polynomials with n variables which satisfies a sufficientlynice three-term recursion relation to be orthogonal with respect to some innerproduct defined by a measure on R n . This is indeed the case for the bivariatecontinuous q -Hermite polynomials we defined, as we prove in the following theorem. Theorem 3.2.
The bivariate continuous q -Hermite polynomials satisfy the orthog-onality relation Z − Z − H m,n ( u, v ; q, r ) H e n, e m ( u, v ; q, r ) | ( e i ( α + β ) /r ; q ) ∞ | p (1 − u )(1 − v ) dudv = c m,n δ m, e m δ n, e n , where here u = cos(2 α ) , v = cos(2 β ) and (3.5) c m,n = 2 π ( q ; q ) ∞ X i + k + ℓ = mj + k + ℓ = n ( − k q ( k )( q ; q ) m ( q ; q ) n ( q ; q ) i ( q ; q ) j ( q ; q ) k ( q ; q ) ℓ r m + n . Remark . Note that ( α, β ) → (cos(2 α ) , cos(2 β )) defines a fourfold cover fromthe diamond region D with vertices (0 , , ( π/ , π/ , ( π/ , − π/
2) and ( π,
0) tothe triangular T with vertices ( − , − , ( − , , and (1 , θ = α + β and φ = α − β maps D to the square region [0 , π ] . Thus if I m,n, e m, e n isthe integral in Theorem 3.2, we have by symmetry I m,n, e m, e n = 2 ZZ T H m,n ( u, v ; q, r ) H e n, e m ( u, v ; q, r ) | ( e i ( α + β ) /r ; q ) ∞ | p (1 − u )(1 − v ) dudv = 2 ZZ D H m,n ( u, v ; q, r ) H e n, e m ( u, v ; q, r ) | ( e i ( α + β ) /r ; q ) ∞ | dαdβ = Z π Z π H m,n ( u, v ; q, r ) H e n, e m ( u, v ; q, r ) | ( e iθ /r ; q ) ∞ | dθdφ. IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 11 where here u = cos(2 α ), v = cos(2 β ), θ = α + β and φ = α − β . Thus theorthogonality expression above is equivalent to(3.6) Z π Z π H m,n ( u, v ; q, r ) H e n, e m ( u, v ; q, r ) | ( e iθ /r ; q ) ∞ | dθdφ = c m,n δ m, e m δ n, e n , for u = cos( θ + φ ) and v = cos( θ − φ ). Remark . If we define a new sequence of polynomials e H m,n ( x, y ) = (cid:26) H m,n ( x, y ) − H n,m ( x, y ) if m < n,H m,n ( x, y ) + H n,m ( x, y ) if m ≥ n then the new sequence satisfies the more intuitive orthogonality statement that Z π Z π e H m,n ( u, v ; q, r ) e H e m, e n ( u, v ; q, r ) | ( e iθ /r ; q ) ∞ | dθdφ = e c m,n δ m, e m δ n, e n for some constants e c m,n >
0, but will no longer have monomial leading coefficients.
Proof.
To prove Theorem 3.2, we will use the generating function formula for thebivariate continuous q -Hermite polynomials to deduce an orthogonality condition.We will also make use of the Askey–Wilson integral [AAR99, Theorem 10.8.1](3.7) Z π − π | ( e iθ ; q ) ∞ | | ( ae iθ , be iθ , ce iθ , de iθ ; q ) ∞ | dθ = 4 π ( abcd ; q ) ∞ ( ab, ac, ad, bc, bd, cd, q ; q ) ∞ . However, we require this integral in a slightly modified form. Note that the function f ( z ) = | ( e iz /r ; q ) ∞ | | ( ae iz , be iz , ce iz , de iz ; q ) ∞ | is, 2 π -periodic and holomorphic on the domain Im( z ) > ln max( | a | , | b | , | c | , | d | ).Therefore by the Cauchy residue theorem, as long as r > max( | a | , | b | , | c | , | d | ) wehave no poles in the rectangle [ − π, π ] × [ − ( i/
2) ln( r ) ,
0] and so Z π − π f ( θ ) dθ = Z π − π f ( θ − ( i/
2) ln r ) dθ + i Z (1 /
2) ln r f ( π − ix ) dx − i Z (1 /
2) ln r f ( − π − ix ) dx = Z π − π f ( θ − ( i/
2) ln r ) dx. Consequently for r > max( | a | , | b | , | c | , | d | ) we see that(3.8) Z π − π | ( e iθ /r ; q ) ∞ | | ( ae iθ , be iθ , ce iθ , de iθ ; q ) ∞ | dθ = 4 π ( abcdr ; q ) ∞ ( abr, acr, adr, bcr, bdr, cdr, q ; q ) ∞ . Using this, assume max( | s | , | t | , | e s | , | e t | ) < r and consider the integral I = I ( s, t, e s, e t ; q, r ) = Z π Z π ψ (cid:18) u, vs, t (cid:19) ψ (cid:18) v, u e s, e t (cid:19) | ( e iθ /r ; q ) ∞ | dθdφ. We calculate I = Z π Z π ( rst, r e s e t ; q ) ∞ | ( e iθ /r ; q ) ∞ | | ( e i ( θ + φ ) s, e i ( θ − φ ) t, e i ( θ − φ ) e s, e i ( θ + φ ) e t ; q ) ∞ | dθdφ = 12 Z π Z π − π ( rst, r e s e t ; q ) ∞ | ( e iθ /r ; q ) ∞ | | ( e i ( θ + φ ) s, e i ( θ − φ ) t, e i ( θ − φ ) e s, e i ( θ + φ ) e t ; q ) ∞ | dθdφ = 2 π ( q ; q ) ∞ Z π ( rst, r e s e t, r s e st e t ; q ) ∞ ( rst, rs e s, rs e te iφ , r e ste − iφ , rt e t, r e s e t ; q ) ∞ dφ = 2 π ( r s e st e t ; q ) ∞ ( rs e s, rt e t, q ; q ) ∞ Z π rs e te iφ , r e ste − iφ ; q ) ∞ dφ = 2 π ( r s e st e t ; q ) ∞ ( rs e s, rt e t, q ; q ) ∞ ∞ X m,n =0 Z π ( rs e t ) m ( r e st ) n ( q ; q ) m ( q ; q ) n e iφ ( m − n ) dφ = 2 π ( r s e st e t ; q ) ∞ ( rs e s, rt e t, q ; q ) ∞ ∞ X n =0 ( r s e t e st ) n ( q ; q ) n = 2 π ( q ; q ) ∞ X i,j,k,ℓ ( − k q ( k )( rs e s ) i + k + ℓ ( rt e t ) j + k + ℓ ( q ; q ) i ( q ; q ) j ( q ; q ) k ( q ; q ) ℓ = ∞ X m,n =0 c m,n ( q ; q ) m ( q ; q ) n ( s e s ) m ( t e t ) n , where the c m,n ’s are given by (3.5). Furthermore, using the explicit series expressionfor ψ (cid:0) x,ys,t (cid:1) in terms of the continuous bivariate q -Hermite polynomials, we see I = ∞ X m,n =0 ∞ X e m, e n =0 s m t n e s e m e t e n ( q ; q ) m ( q ; q ) n ( q ; q ) e m ( q ; q ) e n I m,n, e m, e n for I m,n, e m, e n = Z π Z π H m,n ( u, v ) H e m, e n ( v, u ) | ( e iθ /r ; q ) ∞ | dθdφ. Combining this with our previous expression for I , we find that I m,n, e m, e n = δ m, e m δ n, e n c m,n . We prove below that the constants c m,n are positive when r and q are real. Thisproves the statement of Theorem 3.2. (cid:3) Complex interpretation.
The swapping of the variables u and v in theinner product expression of Theorem 3.2 is somewhat startling! However, it isquite natural when viewed in terms of an inner product on the 2-torus T = { ( z, w ) ∈ C : | z | = | w | = 1 } . To see what we mean specifically, consider the complex functions θ m,n ( z, w ) definedon T by θ m,n ( z, w ) = min( m,n ) X k =0 ( − k q ( k )( q ; q ) m ( q ; q ) n ( q ; q ) m − k ( q ; q ) n − k ( q ; q ) k r k z m − k w n − k × φ (cid:20) q k − m , − ; q, q m − k z − (cid:21) φ (cid:20) q k − n , − ; q, q n − k w − (cid:21)(cid:19) . IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 13 These are complex bivariate trigonometric polynomials on T . Note in particular θ m,n ( e iθ , e iφ ) = H m,n ( x, y ) , for x = cos( θ ) and y = cos( φ ) . With this in mind the inner product expression of Theorem 3.2 becomes(3.9) 14
Z Z T θ m,n ( zw, zw ) θ e m, e n ( zw, zw ) | ( z /r ; q ) ∞ | d | z | d | w | = c m,n δ m, e m δ n, e n . In this way, we can see that the inner product expression from Theorem 3.2 isactually a Hermitian inner product on L ( T ). In particular, when r and q are realthe coefficients c m,n are necessarily positive as they are Hermitian inner productsof polynomials with respect to an absolutely continuous positive measure whosesupport contains a dense open subset of T .3.4. Additional properties.
Equation (3.4) combined with the q -difference equa-tion for the Hermite polynomials (2.7) immediately tells us a simple operator iden-tity relating the polynomials H m,n ( x, y ) to H m ( x ; q ) H n ( y ; q ). Specifically, we canwrite H m,n ( x, y ) as a certain differential operator of infinite order acting on theproduct H m ( x ; q ) H n ( x ; q ), namely(3.10) H m,n ( x, y ; q, r ) = 1 (cid:16) − q m + n − (cid:0) − q (cid:1) rD q,x D q,y ; q (cid:17) ∞ · H m ( x ; q ) H n ( y ; q ) . To see this, note that D kq,x H m ( x ; q ) = 2 k (1 − q ) k q − ( mk − ( k ) − k ) ( q ; q ) m ( q ; q ) m − k H m − k ( x ; q ) , so that D kq,x D kq,y H m ( x ; q ) H n ( y ; q )= 2 k (1 − q ) k q − ( m + n k − ( k ) − k ) ( q ; q ) m ( q ; q ) n ( q ; q ) m − k ( q ; q ) n − k H m − k ( x ; q ) H n − k ( x ; q ) . Therefore H m,n ( x, y ) = min( m,n ) X k =0 (cid:16) − q m + n − (cid:0) − q (cid:1) rD q,x D q,y (cid:17) k ( q ; q ) k · H m ( x ; q ) H n ( y ; q )= ∞ X k =0 (cid:16) − q m + n − (cid:0) − q (cid:1) rD q,x D q,y (cid:17) k ( q ; q ) k · H m ( x ; q ) H n ( y ; q )= 1 (cid:16) − q m + n − (cid:0) − q (cid:1) rD q,x D q,y ; q (cid:17) ∞ · H m ( x ; q ) H n ( y ; q ) . Note that the second equality is due to the fact that D kq,x D kq,y H m ( x ; q ) H n ( y ; q ) iszero for k > min( m, n ), so all the additional terms appearing in the sum are justzero.The generating function formula (3.3) along with the operator formula (3.10)combined with properties of the continuous q -Hermite polynomials in the single-variable case, immediately guarantee certain nice recurrence relations for the bi-variate continuous q -Hermite polynomials. We list some of these in the next propo-sition. Proposition 3.5.
The bivariate continuous q -Hermite polynomials satisfy the fol-lowing equations D q,x · H m,n ( x, y ; q, r ) = 2 q − m − (1 − q m )1 − q H m − ,n ( x, y ; q, √ qr ) , (3.11) D q,y · H m,n ( x, y ; q, r ) = 2 q − n − (1 − q n )1 − q H m,n − ( x, y ; q, √ qr ) . (3.12) Proof.
The forward difference equations follow from the operator relation (3.10).In detail, define L k ( q, r ) = 1 (cid:16) − q k/ − (cid:0) − q (cid:1) rD q,x D q,y ; q (cid:17) ∞ and notice that L k ( q, r ) = L k − ( q, √ qr ). Therefore by (3.10) D q,x · H m,n ( x, y ) = D q,x L m + n ( q, r ) · H m ( x ; q ) H n ( y ; q )= q − m − (cid:18) − q (cid:19) (1 − q m ) L m + n ( q, r ) · H m − ( x ; q ) H n ( y ; q )= q − m − (cid:18) − q (cid:19) (1 − q m ) L m + n − ( q, √ qr ) · H m − ( x ; q ) H n ( y ; q )= q − m − (cid:18) − q (cid:19) (1 − q m ) H m − ,n ( x, y ; q, √ qr ) . The proof of the other difference equation is similar. (cid:3) Defining relations for quantum symmetric pairs
We now explain how bivariate continuous q -Hermite polynomials appear in thetheory of quantum symmetric pairs.4.1. Quasi-split quantum symmetric pairs.
Let g be a symmetrizable Kac-Moody algebra with generalized Cartan matrix ( a ij ) i,j ∈ I where I is a finite set.Let { d i | i ∈ I } be a set of relatively prime positive integers such that the matrix( d i a ij ) is symmetric. Let Π = { α i | i ∈ I } be the set of simple roots for g and let Q = Z Π be the root lattice. Consider the symmetric bilinear form ( · , · ) : Q × Q → Z defined by ( α i , α j ) = d i a ij for all i, j ∈ I . Let g ′ = [ g , g ] be the derived subalgebra of g . We now recall the definition of the corresponding quantized enveloping algebra.Let K be a field of characteristic zero and let q ∈ K × such that q d i = 1 forall i ∈ I . Recall the symmetric q -numbers, q -factorials and q -binomial coefficientsdefined by[ n ] q = q n − q − n q − q − , [ n ] ! q = [ n ] q [ n − q · · · [2] q [1] q , (cid:20) nm (cid:21) q = [ n ] ! q [ n − m ] ! q [ m ] ! q for any m, n ∈ N with m ≤ n , see for instance in [Lus94, 1.3.3]. We abbreviate q i = q d i for any i ∈ I . For any i, j ∈ I let S ij ( x, y ) denote the noncommutativepolynomial in variables x, y given by S ij ( x, y ) = − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i x − a ij − n yx n . IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 15 Define U q ( g ′ ) to be the K -algebra with generators E i , F i , K ± i for i ∈ I and definingrelations K i K j = K j K i , K i E j = q − ( α i ,α j ) E j K i , K i F j = q − ( α i ,α j ) F j K i ,E i F j − F j E i = δ ij K i − K − i q i − q − i , (4.1) S ij ( E i , E j ) = S ij ( F i , F j ) = 0(4.2)for all i, j ∈ I . The relations (4.2) are known as the quantum Serre relations. If q is not a root of unity, then U q ( g ′ ) is the quantized universal enveloping algebraof g ′ for the deformation parameter q as defined in [Lus94]. If q is a root of unity,then U q ( g ′ ) is the big quantum group of g ′ at q , defined and studied by De Conciniand Kac [DK90]. In either case U q ( g ′ ) is a Hopf algebra with coproduct ∆ definedfor all i ∈ I by ∆ ( E i ) = E i ⊗ K i ⊗ E i , ∆ ( F i ) = F i ⊗ K − i + 1 ⊗ F i , ∆ ( K i ) = K i ⊗ K i . Let τ : I → I be a bijection such that a τ ( i ) τ ( j ) = a ij for all i, j ∈ I . The diagramautomorphism τ gives rise to a Lie algebra automorphism τ : g ′ → g ′ denoted bythe same symbol. Let ω : g ′ → g ′ be the Chevalley involution as defined in [Kac90,(1.3.4)]. Consider the involutive Lie algebra automorphism θ = τ ◦ ω of g ′ and let k ′ = { x ∈ g ′ | θ ( x ) = x } denote the corresponding pointwise fixed Lie subalgebra.The theory of quantum symmetric pairs provides quantum group analogs of theuniversal enveloping algebra U ( k ′ ) as coideal subalgebras of U q ( g ′ ). More precisely,let H θ ⊂ U q ( g ′ ) denote the Hopf subalgebra generated by the elements K i K − τ ( i ) forall i ∈ I . Let c = ( c i ) i ∈ I ∈ ( K × ) I be a family of parameters such that c i = c τ ( i ) for all i ∈ I with a iτ ( i ) = 0.(4.3)We define B c to be the subalgebra of U q ( g ′ ) generated by H θ and the elements B i = F i − c i E τ ( i ) K − i for all i ∈ I .(4.4)By definition the coproduct ∆ of U q ( g ′ ) satisfies ∆ ( B i ) = B i ⊗ K − i + 1 ⊗ F i − c i K τ ( i ) K − i ⊗ E τ ( i ) K − i and hence B c is a right coideal subalgebra of U q ( g ′ ), that is ∆ ( B c ) ⊆ B c ⊗ U q ( g ′ ).We call B c a quasi-split quantum symmetric pair coideal subalgebra of U q ( g ). Remark . The condition (4.3) on the parameters c guarantees that the sub-algebra B c has many desirable properties, see [Kol14, (5.9)], [KY19, Proposition3.1]. Remark . For q not a root of unity, quantum symmetric pairs of Kac-Moodytype were defined in [Kol14] depending on a pair ( X, τ ) where τ : I → I is a diagramautomorphism and X is a subset of I satisfying the admissibility conditions givenin [Kol14, Definition 2.3]. Following [CLW18] we call a quantum symmetric pairquasi-split if X = ∅ . In the present paper we only consider quasi-split quantumsymmetric pairs.The definition of quantum symmetric pairs in [Kol14] involves a second param-eter family s = ( s i ) i ∈ I . The corresponding coideal subalgebras B c , s are isomorphicas algebras for all s under a map which maps generators to generators, see [Kol14,Theorem 7.1]. In the present paper we are only concerned with the defining rela-tions of B c , s and we hence restrict to the case s i = 0 for all i ∈ I . The ∗ -product on H θ ⋉ U − . We now recall a method devised in [KY19] todescribe the algebra B c in terms of generators and relations. Let U − denote thesubalgebra of U q ( g ′ ) generated by all F i for i ∈ I . The algebra U − is Q -graded with U −− µ = span K { F i . . . F i m | P mj =1 α i j = µ } for all µ ∈ Q + = N Π, and U −− µ = { } otherwise. For any i ∈ I let ∂ Ri , ∂ Li : U − → U − denote the linear maps uniquelydetermined by the property that ∂ Ri ( F j ) = ∂ Li ( F j ) = δ ij for all j ∈ I and ∂ Ri ( f g ) = q ( α i ,ν ) ∂ Ri ( f ) g + f ∂ Ri ( g )(4.5) ∂ Li ( f g ) = ∂ Li ( f ) g + q ( α i ,µ ) f ∂ Li ( g ) , (4.6)for all f ∈ U −− µ , g ∈ U −− ν . Consider the semidirect product H θ ⋉ U − which is thesubalgebra of U q ( g ′ ) generated by H θ and U − . The algebra B c is a deformation of H θ ⋉ U − . Theorem 4.3. (1) [KY19, Theorem 4.7, Lemma 5.2]
There exists an associativeproduct ∗ on H θ ⋉ U − which is uniquely determined by the following properties: h ∗ g = hg, g ∗ h = gh for all h ∈ H θ , g ∈ U − , (4.7) F i ∗ g = F i g − c i q ( α i ,α τ ( i ) ) q i − q − i K τ ( i ) K − i ∂ Lτ ( i ) ( g ) for all i ∈ I , g ∈ U − . (4.8) (2) [KY19, Corollary 5.8] There is a uniquely determined isomorphism of algebras ψ : B c → ( H θ ⋉ U − , ∗ ) such that ψ ( h ) = h for all h ∈ H θ and ψ ( B i ) = F i for all i ∈ I .Remark . Property (4.8) in Theorem 4.3 can be replaced by the property g ∗ F i = gF i − c τ ( i ) q ( α i ,α τ ( i ) ) q i − q − i ∂ Rτ ( i ) ( g ) K i K − τ ( i ) for all i ∈ I , g ∈ U − .(4.9)The resulting algebra structure on H θ ⋉ U − coincides with the algebra structureobtained in Theorem 4.3.(1). Remark . The coefficient in (4.8) differs from the corresponding coefficient in[KY19, (4.25)]. This is due to the fact that we follow standard conventions (4.1)while [KY19] works with E i F j − F j E i = δ ij ( K i − K − i ). Moreover, our conventionfor the coefficient c i differs from [KY19] by a sign. The conventions in the presentpaper follow [Kol14] but we additionally allow q to be a root of unity.Set V − = L i ∈ I K F i and let T ( V − ) denote the corresponding tensor algebra.By [KY19] the first part of the above theorem also holds when U − is replaced by T ( V − ). More precisely, there exists an associative product ⊛ on H θ ⋉ T ( V − ) whichis uniquely determined by (4.7) and (4.8) or (4.9) for all h ∈ H θ , g ∈ T ( V − ), i ∈ I with ∗ replaced by ⊛ . By construction, the canonical projection gives rise to analgebra homomorphism η : ( H θ ⋉ T ( V − ) , ⊛ ) → ( H θ ⋉ U − , ∗ )of deformed algebras. Proposition 4.6. [KY19, Proposition 5.9]
The kernel of the algebra homomor-phism η is generated by the quantum Serre polynomials S ij ( F i , F j ) ∈ T ( V − ) for i, j ∈ I . IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 17 Proposition 4.6 and the second part of Theorem 4.3 together provide an effectivemethod to obtain the defining relations for the algebra B c . Indeed, the algebra H θ ⋉ T ( V − ) is generated over H θ by the elements F i for i ∈ I subject only to therelations K j K − τ ( j ) F i = q − ( α j − α τ ( j ) ,α i ) F i K j K − τ ( j ) . The additional relations in B c areobtained by rewriting the quantum Serre polynomials S ij ( F i , F j ) in terms of thedeformed product ⊛ on T ( V − ).For any noncommutative polynomial r ( x , . . . , x n ) = P J a J x j . . . x j m in n variables with coefficients a J = a ( j ,...,j m ) ∈ H θ and any elements u , . . . , u n ∈ H θ ⋉ T ( V − ) we write r ( u ⊛ , . . . ⊛ , u n ) = X J a J u j ⊛ · · · ⊛ u j m . (4.10)If τ ( i ) = { i, j } then (4.8) implies that S ij ( F i , F j ) = S ij ( F i ⊛ , F j ). Hence it remainsto consider the two cases τ ( i ) = i and τ ( i ) = j .4.3. Deformed quantum Serre relations for τ ( i ) = i . All through this sectionwe fix i, j ∈ I with τ ( i ) = i = j . In this case (4.8) and (4.9) for ⊛ become F i ⊛ g = F i g + c∂ Li ( g ) , g ⊛ F = gF i + c∂ Ri ( g )(4.11)where c = − c i q i q i − q − i . For any polynomial w ( x, y ) = P r,s b rs x r y s ∈ K [ x, y ] and any u , u , u ∈ H θ ⋉ T ( V − ) set u y w ( u ⊛ , u ) = X r,s b rs u ⊛ r ⊛ u ⊛ u ⊛ s . (4.12) Lemma 4.7.
For any m, n ∈ N there exists a uniquely determined polynomial w m,n ( x, y ) = P r,s b rs x r y s ∈ K [ x, y ] such that F mi F j F ni = F j y w m,n ( F i ⊛ , F i ) . Proof.
By (4.11) the noncommutative monomial F mi F j F ni can be written as a non-commutative polynomial with respect to the product ⊛ on T ( V − ). This polyno-mial is homogeneous of degree one in F j and hence can be written in the form F j y w m,n ( F i ⊛ , F i ) for some polynomial w m,n ( x, y ) as in the lemma. The poly-nomial w m,n ( x, y ) is uniquely determined because the subalgebra of ( T ( V − ) , ⊛ )generated by F i , F j is a free algebra. (cid:3) It remains to determine the polynomials w m,n ( x, y ) in the above Lemma. To thisend observe that ∂ Li ( F ni ) = ∂ Ri ( F ni ) = ( n ) q i F n − i where we use the non-symmetricquantum integer ( n ) p defined by ( n ) p = 1 + p + · · · + p n − for any p ∈ K . Hencethe first equation in (4.11) and (4.6) imply that F i ⊛ ( F mi F j F ni ) = F m +1 i F j F ni + c ( m ) q i F m − i F j F ni + cq m + a ij i ( n ) q i F mi F j F n − i for m, n ∈ N \ { } . In view of Lemma 4.7 the above formula implies that thepolynomials w m,n ( x, y ) satisfy the recursion xw m,n ( x, y ) = w m +1 ,n ( x, y ) + c ( m ) q i w m − ,n ( x, y )(4.13) + cq m + a ij i ( n ) q i w m,n − ( x, y ) for all m, n ∈ N \ { } . Similarly, using the second equation in (4.11) and (4.5) weobtain yw m,n ( x, y ) = w m,n +1 ( x, y ) + c ( n ) q i w m,n − ( x, y )(4.14) + cq n + a ij i ( m ) q i w m − ,n ( x, y ) . The recursions (4.13), (4.14) also hold for m = 0 or n = 0 if we set w − ,t ( x, y ) = w s, − ( x, y ) = 0 for all s, t ∈ N . Moreover, w , ( x, y ) = 1 as F j y F j . Thesymmetry of the recursions (4.13) and (4.14) implies that w m,n ( x, y ) = w n,m ( y, x ) for all m, n ∈ N .(4.15)Recall the bivariate q -Hermite polynomials H m,n ( x, y ; q, r ) from Section 3.1 whichdepend on two parameters q, r . The recursions (4.13), (4.14) imply that up torescaling, w m,n ( x, y ) coincides with H m,n ( x, y ; q i , q a ij i ). Proposition 4.8.
The polynomials w m,n ( x, y ) are given by w m,n ( x, y ) = H m,n ( b i x, b i y ; q i , q a ij i )(2 b i ) m + n (4.16) where b i = ( q i − q − i ) c − / i q − / i .Remark . The factor b i may lie in a quadratic extension of the field K . However,the right hand side of (4.16) is still a well-defined polynomial in K [ x, y ] because if x i y j appears in H m,n ( x, y ) with nonzero coefficient then i + j ≡ m + n mod 2. Proof of Proposition 4.8.
For a square root b of a nonzero element in K and m, n ∈ N define a new polynomial r m,n ( x, y ) ∈ K [ x, y ] by r m,n ( x, y ) = (2 b ) m + n w m,n ( b − x, b − y ) . The recursion (4.13) for w m,n is equivalent to2 xr m,n ( x, y ) = r m +1 ,n ( x, y ) + 4 cb q mi − q i − r m − ,n ( x, y )+ 4 cb q m + a ij i q ni − q i − r m,n − ( x, y ) . Recall that c = − c i q i q i − q − i and hence the above recursion can be rewritten as2 xr m,n ( x, y ) = r m +1 ,n ( x, y ) + 4 c i q i b ( q i − q − i ) (1 − q mi ) r m − ,n ( x, y )+ 4 c i q i b ( q i − q − i ) q a ij i q mi (1 − q ni ) r m,n − ( x, y ) . For b = ( q i − q − i ) c − / i q − / i the above recursion coincides with the recursion (3.2)for H m,n ( x, y ; q i , q a ij i ). Moreover, r , ( x, y ) = 1 and r m,n ( x, y ) = r n,m ( y, x ) for all m, n ∈ N by (4.15). Hence r m,n ( x, y ) = H m,n ( x, y ; q i , q a ij i ) for this choice of b . (cid:3) For any polynomial w ( x, y ) = P r,s b rs x r y s ∈ K [ x, y ] and any u , u , u ∈ B c set u y w ( u , u ) = X r,s b rs u r u u s ∈ B c , IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 19 in analogy to the notation (4.12). Combining Theorem 4.3.(2), Proposition 4.6,Lemma 4.7 and Proposition 4.8 we are now able to write down the deformed quan-tum Serre relations satisfied by the generators B i , B j of B c . Recall that in thissection we always assume that i = τ ( i ) = j . Corollary 4.10.
The generators B i , B j of B c satisfy the relation − a ij X ℓ =0 ( − ℓ (cid:20) − a ij ℓ (cid:21) q i B j y w − a ij − ℓ,ℓ ( B i , B i ) = 0(4.17) where the polynomial w m,n ( x, y ) ∈ K [ x, y ] is given by (4.16) . With b i as in Proposition 4.8 we define univariate polynomials w m ( x ) ∈ K [ x ] by w m ( x ) = w m, ( x, y ) = 1(2 b i ) m H m ( b i x ; q i )(4.18)for all m ∈ N . By (4.13) the polynomials w m ( x ) satisfy the recursion w m +1 ( x ) = xw m ( x ) − c ( m ) q i w m − ( x ) . (4.19)Note that the polynomials w m ( x ) depend on a choice of i ∈ I , but we do notmake this explicit in the notation. Example 4.11.
For small values of m the polynomials w m ( x ) are given by w ( x ) = 1 , w ( x ) = x, w ( x ) = x − c, w ( x ) = x − (1 + (2) q i ) cx,w ( x ) = x − c ((1 + (2) q i + (3) q i ) x + c (3) q i . Remark . Define w ( n ) ( x ) ∈ K [ x ] by w ( n ) ( x ) = w n ( x )[ n ] ! q i = H n ( b i x ; q i )(2 b i ) n [ n ] ! q i . In terms of the divided powers w ( n ) ( x ) the recursion (4.19) can be rewritten as[ m ] q i w ( m ) ( x ) = xw ( m − ( x ) − cq m − i w ( m − ( x ) . (4.20)Interestingly, this recursion appeared for non quasi-split quantum symmetric pairsin [BW18, (5.8)].Using Equations (4.16) and (3.4) we can express w m,n ( x, y ) in terms of theunivariate polynomials w m ( x ). Additionally using the relations( q ; q ) k = ( − k q k ( k +1) / ( q − q − ) k [ k ] ! q , ( q ; q ) n ( q ; q ) m ( q ; q ) n − m = q m ( n − m ) (cid:20) nm (cid:21) q for n ≥ m , we obtain w m,n ( x, y ) = min( m,n ) X k =0 ( − k c k q k ( m + n + a ij − − k ( k +1)2 i (cid:20) mk (cid:21) q i (cid:20) nk (cid:21) q i [ k ] ! q i w m − k ( x ) w n − k ( y ) . With this relation we calculate − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i w − a ij − n,n ( x, y ) = − a ij X n =0 ( − n [1 − a ij ] ! q i ·· min(1 − a ij − n,n ) X k =0 ( − k c k q − k ( k +1) / i [1 − a ij − n − k ] ! q i [ n − k ] ! q i [ k ] ! q i w − a ij − n − k ( x ) w n − k ( y ) . Setting ℓ = n + k and m = k we obtain − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i w − a ij − n,n ( x, y )(4.21)= − a ij X ℓ =0 ( − ℓ (cid:20) − a ij ℓ (cid:21) q i w − a ij − ℓ ( x ) · ⌊ ℓ/ ⌋ X m =0 c m q − m ( m +1) / i (cid:20) ℓ m (cid:21) q i [2 m ] ! q i [ m ] ! q i w ℓ − m ( y ) . In view of Equation (4.21) it is natural to consider a second family of polynomials v n ( x ) ∈ K [ x ] defined for all n ∈ N by v n ( x ) = ⌊ n/ ⌋ X k =0 c k q − k ( k +1) / i (cid:20) n k (cid:21) q i [2 k ] ! q i [ k ] ! q i w n − k ( x )The polynomials v n ( x ) can also be interpreted in terms of continuous q -Hermitepolynomials. The proof of the first part of the following proposition is adaptedfrom a similar calculation in the proof of [BW18, Lemma 5.10]. Proposition 4.13.
The polynomials v n ( x ) satisfy v − ( x ) = 0 , v ( x ) = 1 and therecursion v m +1 ( x ) = xv m ( x ) + cq − i ( m ) q − i v m − ( x )(4.22) for all m ∈ N . The polynomial v m ( x ) is given by v m ( x ) = 1(2 b i ) m H m ( b i x ; q − i )(4.23) where as before b i = ( q i − q − i ) c − / i q − / i .Proof. A direct calculation using (4.19) gives xv m ( x ) + cq − i ( m ) q − i v m − ( x ) (4.19) = ⌊ m/ ⌋ X k =0 c k q − k ( k +1) / i (cid:20) m k (cid:21) q i [2 k ] ! q i [ k ] ! q i (cid:18) w m − k +1 ( x ) + c ( m − k ) q i w m − k − ( x ) (cid:19) + c [ m ] q i q − m − i ⌊ ( m − / ⌋ X k =0 c k q − k ( k +1) / i (cid:20) m − k (cid:21) q i [2 k ] ! q i [ k ] ! q i w m − − k ( x )= w m +1 ( x )+ ⌊ ( m +1) / ⌋ X k =1 c k q − k ( k +1) / i [ m ] ! q i [ m +1 − k ] ! q i [ k ] ! q i (cid:18) [ m +1 − k ] q i + [ k ] q i (cid:0) q m +1 − ki + q k − ( m +1) i ) (cid:19) w m +1 − k ( x )= v m +1 ( x )which proves the recursion (4.22). Using c = − c i q i q i − q − i the recursion (4.22) can berewritten as v m +1 ( x ) = xv m ( x ) + c i q i ( q i − q − i ) ( q − mi − v m − ( x ) . Now Equation (4.23) follows by comparison with the recursion (2.6). (cid:3)
IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 21 Combining Corollary 4.10 with Equation (4.21) we are able to express the de-formed quantum Serre relations in terms of univariate continuous q -Hermite poly-nomials. Corollary 4.14.
Let i, j ∈ I with τ ( i ) = i = j . The generators B i , B j of B c satisfythe relation − a ij X ℓ =0 ( − ℓ (cid:20) − a ij ℓ (cid:21) q i w − a ij − ℓ ( B i ) B j v ℓ ( B i ) = 0 . (4.24) where the polynomials w m ( x ) and v m ( x ) in K [ x ] are given by (4.18) and (4.23) ,respectively. A resummation shows that (4.24) can alternatively be written as − a ij X ℓ =0 ( − ℓ (cid:20) − a ij ℓ (cid:21) q i v − a ij − ℓ ( B i ) B j w ℓ ( B i ) = 0 . (4.25) Remark . To make the relation between the polynomials w m ( x ) and v m ( x )even clearer, we write w m ( x ; q i , c ) and v m ( x ; q i , c ) for the polynomials given by therecursions (4.19) and (4.22), respectively, with w − ( x ; q i , c ) = v − ( x ; q i , c ) = 0 and w ( x ; q i , c ) = v ( x ; q i , c ) = 1. Then we have v m ( x ; q i , c ) = w m ( x ; q − i , − q − i c ) . (4.26)for all m ∈ N . Remark . Corollary 4.14 can be used to calculate the deformed quantum Serrerelation satisfied by the generators B i , B j explicitly in the case τ ( i ) = i for smallvalues of − a ij . Using the expressions for w m ( x ) in Example 4.11 and (4.26) oneobtains − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i B − a ij − ni B j B ni = a ij = 0, − q i c i B j if a ij = − − [2] q i q i c i ( B i B j − B j B i ) if a ij = − − ([3] q i + 1) q i c i ( B i B j + B j B i )+[4] q i ([2] q i + 1) q i c i B i B j B i − [3] q i ( q i c i ) B j if a ij = − q not a root of unity. Remark . In [CLW18, Eq. (3.9)] the relation (4.24) is expressed in terms of so-called ı divided powers for q not a root of unity. Similarly to (4.24) and (4.25), the ı divided powers allow two equivalent expressions for the deformed quantum Serrerelation in the case τ ( i ) = i = j . It would be interesting to establish a relationbetween the ı divided powers of [CLW18] and the continuous q -Hermite polynomials. Remark . Assume that K = k ( q ) is a field of rational functions in a variable q over some field k of characteristic zero. Let : K → K denote the bar involutionsending a rational function g ( q ) ∈ K to g ( q ) = g ( q − ). The map extends to an involutive k -algebra automorphism : K [ x ] → K [ x ] by action on the coefficients.In this setting Equation (4.26) can be rewritten as v m ( x ) = w m ( x ) if c = − q − i c .In the case c = − q − i c the above formula and the equivalence of the relations (4.24)and (4.25) show that relation (4.24) is preserved under the k -linear map given by B i B i , B j B j and q q − . This provides the essential step in the proofthat the algebra B c has a bar-involution in the quasi-split case, as first observed in[CLW18, Proposition 3.7]. Note that the existence of the bar-involution on B c alsofollows from the general theory in [KY19] without the need to have a presentationof B c in terms of generators and relations. This will be discussed elsewhere.4.4. Deformed quantum Serre relations for τ ( i ) = j . The deformed quan-tum Serre relations for τ ( i ) = j were determined in [BK15, Theorem 3.6] basedon Letzter’s method [Let03] involving coproducts. In this subsection we offer analternative proof in the quasi-split case based on the star product method from[KY19]. Throughout we fix distinct i, j ∈ I with τ ( i ) = j . In this case formula(4.8) for ⊛ becomes F i ⊛ g = F i g + γ i K j K − i ∂ Lj ( g ) , F j ⊛ g = F j g + γ j K i K − j ∂ Li ( g )for all g ∈ U − , where γ i = − c i q aiji q i − q − i and γ j = − c j q aiji q i − q − i . Hence F ni = F ⊛ ni and F j ⊛ F ni = F j F ni + γ j q ( n − a ij − i ( n ) q i F n − i K i K − j . (4.27)By induction on m one moreover gets F ⊛ mi ⊛ F j F ni = F mi F j F ni + γ i q n (2 − a ij ) i ( m ) q i F m + n − i K j K − i . Inserting (4.27) into the above equation we obtain F mi F j F ni = F ⊛ mi ⊛ F j ⊛ F ⊛ ni − γ i q n (2 − a ij ) i ( m ) q i F ⊛ ( m + n − i K j K − i (4.28) − γ j q ( n − a ij − i ( n ) q i F ⊛ ( m + n − i K i K − j . Using the relation ℓ X n =0 ( − n (cid:20) ℓn (cid:21) q q n ( ℓ +1) = ( q ; q ) ℓ for all ℓ ∈ N (4.29)one shows that − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i q n (2 − a ij ) i (1 − a ij − n ) q i = − q − i ( q i ; q i ) − a ij q i − q − i , (4.30) − a ij X n =0 ( − n (cid:20) − a ij n (cid:21) q i q n ( a ij − i ( n ) q i = − q − i ( q − i ; q − i ) − a ij q i − q − i . (4.31)Recall the notation (4.10). The formulas (4.30), (4.31) and Equation (4.28) implythe relation S ij ( F i , F j ) = S ij ( F i ⊛ , F j ) + γ i q − i ( q i ; q i ) − a ij q i − q − i F ⊛ ( − a ij ) i K j K − i + γ j q − a ij i ( q − i ; q − i ) − a ij q i − q − i F ⊛ ( − a ij ) i K i K − j IVARIATE q -HERMITE POLYNOMIALS AND DEFORMED SERRE RELATIONS 23 in T ( V − ). Using again Theorem 4.3.(2) and Proposition 4.6 one obtains the fol-lowing result. Theorem 4.19.
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