Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces

Abstract. A one-parameter family of two-sided coideals in Uq(gl(n)) is defined and the corresponding algebras of infinitesimally right invariant functions on the quantum unitary group Uq(n) are studied. The Plancherel decomposition of these algebras with respect to the natural transitive Uq(n)-action is shown to be the same as in the case of a complex projective space. By computing the radial part of a suitable Casimir operator, we identify the zonal spherical functions (i.e. infinitesimally bi-invariant matrix coefficients of finite-dimensional irreducible representations) as Askey-Wilson polynomials containing two continuous and one discrete parameter. In certain limit cases, the zonal spherical functions are expressed as big and little q-Jacobi polynomials depending on one discrete parameter. IntroductionIn this paper, we study a family of two-sided coideals k (c,d) (c, d non-negative real numbers) in the quantized universal enveloping algebra U q (gl(n)). The coideals k (c,d) can be viewed as a q-analogue of the Lie subalgebra gl(n − 1) ⊕ gl(1) ⊂ gl(n). By considering the algebra of functions on the quantum unitary group U q (n) that are "infinitesimally" invariant with respect to the coideal k (c,d) , we obtain a family of quantum projective spaces CP n−1 q (c, d) endowed with a natural transitive action of the quantum unitary group U q (n). These quantum U q (n)-spaces were studied for the first time by Vaksman and Korogodsky [KV], who defined them in a "global" way by means of a q-analogue of the classical Hopf fibration S 2n−1 → CP n−1 . We also analyse the zonal spherical functions (infinitesimally left k (c,d) -invariant and right k (c ,d ) -invariant matrix coefficients) corresponding to finite-dimensional irreducible representations of U q (n). They are expressed in terms of a family of Askey-Wilson polynomials containing two continuous and one discrete parameter. We obtain this result by showing that the zonal spherical functions are eigenfunctions of a certain second-order q-difference operator which arises as the radial part of a suitable Casimir operator.

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“…The proof is similar to that of [N,Lemma 3.2]. Suppose that 0 = v ∈ V (λ) is k σ -fixed and write v as a sum v = µ∈P v µ of weight vectors.…”

Section: ])mentioning

confidence: 78%

A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials

Dijkhuizen

Noumi

1998

Trans. Amer. Math. Soc.

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“…For m = 2 this quantum algebra coincides with the twisted quantised enveloping algebra U tw (sp 2n ) [21,18]. The affine quantum algebra (6.43) coincides in the case of m = 1 (m = 2) with the twisted q-Yangian Y…”

Section: Quantisationmentioning

confidence: 90%

Poisson Algebras of Block-Upper-Triangular Bilinear Forms and Braid Group Action

Chekhov

Mazzocco

2013

Commun. Math. Phys.

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Abstract. In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on C N with the property that for any n, m ∈ N such that nm = N , the restriction of the Poisson algebra to the space of bilinear forms with block-upper-triangular matrix composed from blocks of size m × m is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-uppertriangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case m = 1 the quantum affine algebra is the twisted q-Yangian for on and for m = 2 is the twisted q-Yangian for sp 2n . We describe the quantum braid group action in these two examples and conjecture the form of this action for any m > 2. Finally, we give an R-matrix interpretation of our results and discuss the relation with Poisson-Lie groups.

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“…for Φ : U q (g) → C a spherical function (of type 0), which should be compared to [30,Lemma 5.1], see also [35,37].…”

Section: Where M 1 (Z) Is a Tridiagonal And N 1 (Z) Is A Diagonal Matmentioning

confidence: 99%

“…[28,[34][35][36][37][38] and references given there. When considering other quantum symmetric pairs in relation to matrix-valued spherical functions, the branching rule of a representation of the quantised universal enveloping algebra to a coideal subalgebra seems to be difficult.…”

Section: Introductionmentioning

confidence: 99%

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

2016

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Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (SU(2) × SU(2), diag) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey-Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey-Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials. B Pablo Román

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Macdonald's Symmetric Polynomials as Zonal Spherical Functions on Some Quantum Homogeneous Spaces

Cited by 165 publications

References 16 publications

A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials

A family of quantum projective spaces and related $q$-hypergeometric orthogonal polynomials

Poisson Algebras of Block-Upper-Triangular Bilinear Forms and Braid Group Action

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

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