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arxiv: 2407.17366 · v4 · submitted 2024-07-24 · 🧮 math.CA · math.RT

Automorphisms of the DAHA of type check{C₁}C₁ and non-symmetric Askey-Wilson functions

Tom H. Koornwinder , Marta Mazzocco This is my paper

Pith reviewed 2026-05-23 22:34 UTC · model grok-4.3

classification 🧮 math.CA math.RT
keywords double affine Hecke algebraAskey-Wilson polynomialsautomorphismsnon-symmetric functionsCherednik kernelparameter symmetriesbasic representation
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The pith

An automorphism t4 of the DAHA of type check C1 C1 maps Askey-Wilson polynomials to functions via a parameter change.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies automorphisms of the double affine Hecke algebra of type check C1 C1 that act simply on the generators and on the Askey-Wilson parameters. The symmetry t4 in particular sends the parameters (a,b,c,d) to (a,b, q d inverse, q c inverse) and maps the polynomials to functions. The non-symmetric Askey-Wilson function is defined as the rank one case of Stokman's Cherednik kernel for BC_n, from which the authors derive an explicit expression as the sum of one symmetric term and one anti-symmetric term. A sympathetic reader would care because this links algebraic automorphisms directly to the distinction between polynomial and function versions in the Askey-Wilson family. If the mapping holds, it supplies an algebraic route to relations that mix the symmetric and non-symmetric cases.

Core claim

The central claim is that the automorphism t4 of the DAHA sends Askey-Wilson polynomials to functions, and that the non-symmetric Askey-Wilson function, taken as the rank one case of Stokman's Cherednik kernel for BC_n, equals the sum of a symmetric term and an anti-symmetric term.

What carries the argument

The automorphism t4, which transforms the Askey-Wilson parameters by sending (a,b,c,d) to (a,b,q d^{-1},q c^{-1}), together with the rank-one Cherednik kernel taken as the definition of the non-symmetric Askey-Wilson function.

If this is right

  • t4 acts on the basic representation by converting polynomials into functions while preserving the algebraic relations.
  • The non-symmetric function decomposes explicitly into a symmetric part plus an anti-symmetric part.
  • Repeated application of t4 and other automorphisms generates further relations among Askey-Wilson objects at different parameter values.
  • The same algebraic mechanism applies to the symmetric Askey-Wilson polynomials under the induced parameter map.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same symmetry might produce analogous mappings when the construction is lifted to higher-rank DAHAs.
  • The decomposition into symmetric and anti-symmetric terms could be used to derive new contiguous relations or q-difference equations satisfied by the functions.
  • Applying the automorphism to the kernel itself might yield a direct algebraic proof of the decomposition without case-by-case computation.

Load-bearing premise

The rank one case of Stokman's Cherednik kernel for BC_n serves as the definition of the non-symmetric Askey-Wilson function.

What would settle it

An explicit closed-form evaluation of the rank-one Cherednik kernel that fails to reproduce the expected orthogonality or transformation properties of non-symmetric Askey-Wilson functions under the parameter change induced by t4.

read the original abstract

In this paper we consider the automorphisms of the double affine Hecke algebra (DAHA) of type $\check{C_1}C_1$ which have a relatively simple action on the generators and on the parameters, notably a symmetry $t_4$ which sends the Askey-Wilson parameters $(a,b,c,d)$ to $(a,b,qd^{-1},qc^{-1})$. We study how these symmetries act on the basic representation and on the symmetric and non-symmetric Askey-Wilson (AW) polynomials and functions. Interestingly $t_4$ maps AW polynomials to functions. We take the rank one case of Stokman's Cherednik kernel for $BC_n$ as the definition of the non-symmetric Askey--Wilson function. From it we derive an expression as a sum of a symmetric and an anti-symmetric term.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper examines automorphisms of the double affine Hecke algebra (DAHA) of type check-C1 C1, focusing on those with simple action on generators and parameters, especially the symmetry t4 mapping Askey-Wilson parameters (a,b,c,d) to (a,b,q d^{-1}, q c^{-1}). It studies the action on the basic representation and on symmetric/non-symmetric Askey-Wilson polynomials and functions, observing that t4 maps AW polynomials to functions. The non-symmetric AW function is defined as the rank-one case of Stokman's Cherednik kernel for BC_n, from which an expression as a sum of a symmetric term and an anti-symmetric term is derived.

Significance. If the central claims hold, the work provides explicit information on how DAHA automorphisms act on AW objects in the rank-one case, including a concrete decomposition of the non-symmetric function. This could be useful for further study of symmetries in orthogonal polynomials and special functions associated to DAHA of type check-C1 C1.

major comments (1)
  1. [Abstract] Abstract (definition paragraph): The paper adopts the rank-one case of Stokman's Cherednik kernel for BC_n as the definition of the non-symmetric Askey-Wilson function and then claims that t4 maps AW polynomials to functions while deriving the symmetric-plus-anti-symmetric decomposition. For these claims to be load-bearing, the kernel must be shown to satisfy the required intertwining relations with the DAHA generators under the specific parameter map of t4. The abstract supplies no indication that this compatibility is verified beyond the definition itself; if the kernel fails to commute appropriately in this specialization, both the mapping claim and the derived expression are at risk.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point where the abstract could better signal the technical content of the paper. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (definition paragraph): The paper adopts the rank-one case of Stokman's Cherednik kernel for BC_n as the definition of the non-symmetric Askey-Wilson function and then claims that t4 maps AW polynomials to functions while deriving the symmetric-plus-anti-symmetric decomposition. For these claims to be load-bearing, the kernel must be shown to satisfy the required intertwining relations with the DAHA generators under the specific parameter map of t4. The abstract supplies no indication that this compatibility is verified beyond the definition itself; if the kernel fails to commute appropriately in this specialization, both the mapping claim and the derived expression are at risk.

    Authors: The rank-one specialization of Stokman's Cherednik kernel is chosen precisely because it satisfies the required DAHA intertwining relations under the parameter map of t4; this is verified by direct computation on the generators in the body of the paper (Sections 3–4), which then justifies both the mapping statement and the symmetric-plus-anti-symmetric decomposition. The abstract is deliberately concise and therefore omits this verification step. We will revise the abstract to include a brief clause indicating that the compatibility is established in the text. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper adopts the rank-one case of Stokman's external Cherednik kernel for BC_n explicitly as the definition of the non-symmetric Askey-Wilson function, then derives the symmetric-plus-anti-symmetric decomposition and studies the t4 action on polynomials and functions. No quoted step reduces by the paper's own equations to a fitted input, self-definition, or self-citation chain; the central claims rest on the independent external kernel and prior DAHA results without the derivations collapsing to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, ad-hoc axioms, or invented entities are stated. Standard mathematical background (Hecke algebra relations, parameter conventions) is assumed but not detailed.

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 3 internal anchors

  1. [1]

    Askey and J

    R. Askey and J. A. Wilson, Some basic hypergeometric orthogonal polynomials that gen- eralize Jacobi polynomials, Mem. Amer. Math. Soc. 54, No. 319 (1985)

    work page 1985

  2. [2]

    Cherednik, Difference Macdonald–Mehta conjecture, Internat

    I. Cherednik, Difference Macdonald–Mehta conjecture, Internat. Math. Res. Notices (1997), No. 10, 449–467

    work page 1997

  3. [3]

    Cramp´ e , L

    N. Cramp´ e , L. Frappat , J. Gaboriaud, L. Poulain d’Andecy, E. Ragoucy and L. Vinet, The Askey–Wilson algebra and its avatars, J. Phys. A: Math. Theor., 54 (2021)

    work page 2021

  4. [4]

    B. A. Dubrovin and M. Mazzocco, Monodromy of certain Painlev´ e-VI transcendents and reflection group, Invent. Math. 141 (2000), 55–147

    work page 2000

  5. [5]

    Gasper and M

    G. Gasper and M. Rahman, Basic hypergeometric series, Second edition, Cambridge Uni- versity Press, Cambridge, 2004

    work page 2004

  6. [6]

    Haine and P

    L. Haine and P. Iliev, Askey–Wilson type functions, with bound states, Ramanujan J. 11 (2006), 285–329

    work page 2006

  7. [7]

    J. E. Humphreys, Introduction to Lie algebras and representation theory , Springer-Verlag, Third printing, 1990. 37

    work page 1990

  8. [8]

    M. E. H. Ismail and M. Rahman, The associated Askey–Wilson polynomials, Trans. Amer. Math. Soc. 328 (1991), 201–237

    work page 1991

  9. [9]

    E. G. Kalnins and W. Miller, Jr., Symmetry techniques for q-series: Askey–Wilson polyno- mials, Rocky Mountain J. Math. 19 (1989), 223–230

    work page 1989

  10. [10]

    Koelink and J

    E. Koelink and J. V. Stokman, The Askey–Wilson function transform, Internat. Math. Res. Notices (2001), No. 22, 1203–1227

    work page 2001

  11. [11]

    T. H. Koornwinder, The relationship between Zhedanov’s algebra AW(3) and the dou- ble affine Hecke algebra in the rank one case, SIGMA 3 (2007), 063, 15 pages; arXiv:math/0612730v4 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  12. [12]

    T. H. Koornwinder, Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra, SIGMA 4 (2008), 052, 17 pages; arXiv:0711.2320v3 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  13. [13]

    T. H. Koornwinder, Charting the q-Askey scheme. II. The q-Zhedanov scheme, Indag. Math. (N.S.) 34 (2023) 317–337,1419–1420; arXiv:2209.07995v3 [math.CA]

    work page arXiv 2023

  14. [14]

    T. H. Koornwinder and F. Bouzeffour, Non-symmetric Askey–Wilson polynomials as vector- valued polynomials Appl. Anal. 90 (2011), no. 3-4, 731–746

    work page 2011

  15. [15]

    T. H. Koornwinder and M. Mazzocco, Dualities in the q-Askey scheme and degenerate DAHA, Studies Appl. Math. 141 (2018), 424–473

    work page 2018

  16. [16]

    Lisovyy and Y

    O. Lisovyy and Y. Tykhyy, Algebraic solutions of the sixth Painlev´ e equation, J. Geom. Phys. 85 (2014), 124–163

    work page 2014

  17. [17]

    Mazzocco, Confluences of the Painlev´ e equations, Cherednik algebras and q-Askey scheme, Nonlinearity 29 (2016), 2565–2608

    M. Mazzocco, Confluences of the Painlev´ e equations, Cherednik algebras and q-Askey scheme, Nonlinearity 29 (2016), 2565–2608

    work page 2016

  18. [18]

    Mazzocco, Embedding of the rank 1 DAHA into M at(2, Tq) and its automorphisms, in: Representation theory, special functions and Painlev´ e equations, RIMS 2015 , Adv

    M. Mazzocco, Embedding of the rank 1 DAHA into M at(2, Tq) and its automorphisms, in: Representation theory, special functions and Painlev´ e equations, RIMS 2015 , Adv. Stud. Pure Math. 76, pp. 449–468, Math. Soc. Japan, Tokyo, 2018

    work page 2015

  19. [19]

    Askey-Wilson polynomials: an affine Hecke algebraic approach

    M. Noumi and J. V. Stokman, Askey–Wilson polynomials: an affine Hecke algebraic ap- proach, in: Laredo Lectures on Orthogonal Polynomials and Special Functions, Adv. Theory Spec. Funct. Orthogonal Polynomials, Nova Sci. Publ., Hauppauge, NY (2004) 111–144; arXiv:math/0001033v1 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  20. [20]

    Oblomkov, Double affine Hecke algebras of rank 1 and affine cubic surfaces, Internat

    A. Oblomkov, Double affine Hecke algebras of rank 1 and affine cubic surfaces, Internat. Math. Res. Notices (2004), No. 18, 877–912

    work page 2004

  21. [21]

    Okamoto, Studies on the Painlev´ e equations

    K. Okamoto, Studies on the Painlev´ e equations. I. Sixth Painlev´ e equationPVI, Ann. Mat. Pura Appl. (4) 146 (1987), 337–381. 38

    work page 1987

  22. [22]

    Rahman, Askey–Wilson functions of the first and second kinds: series and integral representations of C2 n(x; β | q) + D2 n(x; β | q), J

    M. Rahman, Askey–Wilson functions of the first and second kinds: series and integral representations of C2 n(x; β | q) + D2 n(x; β | q), J. Math. Anal. Appl. 164 (1992), 263–284

    work page 1992

  23. [23]

    Rahman, A q-extension of a product formula of Watson, Quaest

    M. Rahman, A q-extension of a product formula of Watson, Quaest. Math. 22 (1999), 27–42

    work page 1999

  24. [24]

    Sahi, Non-symmetric Koornwinder polynomials and duality, Ann

    S. Sahi, Non-symmetric Koornwinder polynomials and duality, Ann. of Math. (2) 150, (1999), 267–282

    work page 1999

  25. [25]

    Sahi, Some properties of Koornwinder polynomials, in: q-Series from a contemporary perspective, Contemporary Math

    S. Sahi, Some properties of Koornwinder polynomials, in: q-Series from a contemporary perspective, Contemporary Math. 254 (2000), 395–411

    work page 2000

  26. [26]

    J. V. Stokman, An expansion formula for the Askey–Wilson function, J. Approx. Theory 114 (2002), 308–342

    work page 2002

  27. [27]

    J. V. Stokman, Difference Fourier transforms for nonreduced root systems, Selecta Math. (N.S.) 9 (2003), 409–494

    work page 2003

  28. [28]

    S. K. Suslov, The theory of difference analogues of special functions of hypergeometric type, Russian Math. Surveys 44 (1989) no. 2, 227–278

    work page 1989

  29. [29]

    S. K. Suslov, Some orthogonal very well poised 8ϕ7-functions. J. Phys. A 30 (1997), 5877– 5885

    work page 1997

  30. [30]

    S. K. Suslov, Some orthogonal very-well-poised 8ϕ7-functions that generalize Askey–Wilson polynomials, Ramanujan J. 5 (2001), 183–218

    work page 2001

  31. [31]

    Terwilliger, The universal Askey–Wilson algebra, SIGMA 7 (2011), 069, 24 pp

    P. Terwilliger, The universal Askey–Wilson algebra, SIGMA 7 (2011), 069, 24 pp. 39

    work page 2011