Dualities in the $q$-Askey scheme and degenerate DAHA

Abstract

The Askey-Wilson polynomials are a four-parameter family of orthogonal symmetric Laurent polynomials $R_n[z]$ which are eigenfunctions of a second-order $q$-difference operator $L$, and of a second-order difference operator in the variable $n$ with eigenvalue $z+z^{-1}=2x$. Then $L$ and multiplication by $z+z^{-1}$ generate the Askey-Wilson (Zhedanov) algebra. A nice property of the Askey-Wilson polynomials is that the variables $z$ and $n$ occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the non-symmetric case and in the underlying algebraic structures: the Askey-Wilson algebra and the double affine Hecke algebra (DAHA). In this paper we follow the degeneration of the Askey-Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey-Wilson algebras, for the non-symmetric polynomials and for the (degenerate) DAHA and its representations.

arXiv comment46 pages, 1 figure. In version 2, we added some references, a section (Summary of other related work and further perspective) and some further material. Fixed a number of typos. Version 3 is the final version

References in corpus (10)

• The Analytic Theory of Matrix Orthogonal Polynomials
• Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems
• Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case. II. The Spherical Subalgebra
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• Duality of q-polynomials, orthogonal on countable sets of points
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• Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
• Non-Symmetric Basic Hypergeometric Polynomials and Representation Theory for Confluent Cherednik Algebras

Cited by in corpus (9)

• The Askey-Wilson algebra and its avatars
• Charting the $q$-Askey scheme
• Sklyanin-like algebras for ($q$-)linear grids and ($q$-)para-Krawtchouk polynomials
• Charting the $q$-Askey scheme. II. The $q$-Zhedanov scheme
• The Virasoro fusion kernel and Ruijsenaars' hypergeometric function
• Grassmann graphs, degenerate DAHA, and non-symmetric dual $q$-Hahn polynomials
• Meta Algebras and Biorthogonal Rational Functions: The Hahn Case
• Automorphisms of the DAHA of type $\check{C_1}{C}_1$ and non-symmetric Askey-Wilson functions
• Non-Symmetric Askey--Wilson Shift Operators