Introduction Results Background Proofs

Painlev´eequations, Cherednik algebras and q-Askey scheme

Marta Mazzocco

University of Loughborough

13 December 2014

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme the solutions w(z; c1, c2) satisfy

1 Painlev´e–Kowalevski property: w(z; c1, c2) have no critical points that depend on c1, c2.

2 For generic c1, c2, w(z; c1, c2) are new functions, Painlev´e Transcendents.

Introduction Results Background Proofs

Painlev´eequations are non linear second order ODE

d2w  dw  = F z, w, , z C, dz2 dz ∈

F (z, w, wz ) is rational in z, w, wz

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme 1 Painlev´e–Kowalevski property: w(z; c1, c2) have no critical points that depend on c1, c2.

2 For generic c1, c2, w(z; c1, c2) are new functions, Painlev´e Transcendents.

Introduction Results Background Proofs

Painlev´eequations are non linear second order ODE

d2w  dw  = F z, w, , z C, dz2 dz ∈

F (z, w, wz ) is rational in z, w, wz

the solutions w(z; c1, c2) satisfy

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme 2 For generic c1, c2, w(z; c1, c2) are new functions, Painlev´e Transcendents.

Introduction Results Background Proofs

Painlev´eequations are non linear second order ODE

d2w  dw  = F z, w, , z C, dz2 dz ∈

F (z, w, wz ) is rational in z, w, wz

the solutions w(z; c1, c2) satisfy

1 Painlev´e–Kowalevski property: w(z; c1, c2) have no critical points that depend on c1, c2.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs

Painlev´eequations are non linear second order ODE

d2w  dw  = F z, w, , z C, dz2 dz ∈

F (z, w, wz ) is rational in z, w, wz

the solutions w(z; c1, c2) satisfy

1 Painlev´e–Kowalevski property: w(z; c1, c2) have no critical points that depend on c1, c2.

2 For generic c1, c2, w(z; c1, c2) are new functions, Painlev´e Transcendents.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs

d2w d2w = 6w 2 + z, = 2w 3 + zw + α dz2 dz2 d2w 1 dw 2 1 dw αw 2 + β δ = + + γw 3 + dz2 w dz − z dz z w d2w 1 dw 2 3 β = + w 3 + 4zw 2 + 2(z2 α)w + dz2 2w dz 2 − w d2w  1 1  dw 2 1 dw γw = + + + dz2 2w w 1 dz − z dz z − (w 1)2  β  δw(w + 1) + − αw + + z2 w w 1 − 2     d w 1 1 1 1 2 1 1 1 = + + w + + wz + dz2 2 w w 1 w z z − z z 1 w z − − − − w(w 1)(w z)  z z 1 z(z 1)  + − − α + β + γ − + δ − z2(z 1)2 w 2 (w 1)2 (w z)2 − − −

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Let lN (π) =length of the longest subsequence of π.

l (π) 2√N  Z ∞  lim Prob( N t) = exp (t z)w 2(z) dz −1/6 N→∞ N ≤ − t −

w(z) satisfies PII. (Baik, Deift, Johansson 1999). Fredholm determinants are expressed via special solutions of Painlev´eequations. Tracy, Widom, Adler, van Moerbeke, Its, Bleher, Borodin, Forrester Only special solutions appear in this way

Introduction Results Background Proofs Painlev´etranscendents in Random Matrix Theory:

Example: (SN , PN ) group of permutations π of 1, 2,..., N with 1 uniform probability distribution PN (π) = N! .

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Fredholm determinants are expressed via special solutions of Painlev´eequations. Tracy, Widom, Adler, van Moerbeke, Its, Bleher, Borodin, Forrester Only special solutions appear in this way

Introduction Results Background Proofs Painlev´etranscendents in Random Matrix Theory:

Example: (SN , PN ) group of permutations π of 1, 2,..., N with 1 uniform probability distribution PN (π) = N! . Let lN (π) =length of the longest subsequence of π.

l (π) 2√N  Z ∞  lim Prob( N t) = exp (t z)w 2(z) dz −1/6 N→∞ N ≤ − t −

w(z) satisfies PII. (Baik, Deift, Johansson 1999).

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Only special solutions appear in this way

Introduction Results Background Proofs Painlev´etranscendents in Random Matrix Theory:

Example: (SN , PN ) group of permutations π of 1, 2,..., N with 1 uniform probability distribution PN (π) = N! . Let lN (π) =length of the longest subsequence of π.

l (π) 2√N  Z ∞  lim Prob( N t) = exp (t z)w 2(z) dz −1/6 N→∞ N ≤ − t −

w(z) satisfies PII. (Baik, Deift, Johansson 1999). Fredholm determinants are expressed via special solutions of Painlev´eequations. Tracy, Widom, Adler, van Moerbeke, Its, Bleher, Borodin, Forrester

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Painlev´etranscendents in Random Matrix Theory:

Example: (SN , PN ) group of permutations π of 1, 2,..., N with 1 uniform probability distribution PN (π) = N! . Let lN (π) =length of the longest subsequence of π.

l (π) 2√N  Z ∞  lim Prob( N t) = exp (t z)w 2(z) dz −1/6 N→∞ N ≤ − t −

w(z) satisfies PII. (Baik, Deift, Johansson 1999). Fredholm determinants are expressed via special solutions of Painlev´eequations. Tracy, Widom, Adler, van Moerbeke, Its, Bleher, Borodin, Forrester Only special solutions appear in this way

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme To each Painlev´eequation we associate a cubic surface called monodromy manifold. The monodromy manifold admits a Poisson bracket and can be quantised. These quantum algebras admit representations on the space of Laurent polynomials. The eigenspaces in this representation are spanned by q-orthogonal polynomials (q-Askey scheme).

Introduction Results Background Proofs Today: Quantisation and representation theory

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme The monodromy manifold admits a Poisson bracket and can be quantised. These quantum algebras admit representations on the space of Laurent polynomials. The eigenspaces in this representation are spanned by q-orthogonal polynomials (q-Askey scheme).

Introduction Results Background Proofs Today: Quantisation and representation theory

To each Painlev´eequation we associate a cubic surface called monodromy manifold.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme These quantum algebras admit representations on the space of Laurent polynomials. The eigenspaces in this representation are spanned by q-orthogonal polynomials (q-Askey scheme).

Introduction Results Background Proofs Today: Quantisation and representation theory

To each Painlev´eequation we associate a cubic surface called monodromy manifold. The monodromy manifold admits a Poisson bracket and can be quantised.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme The eigenspaces in this representation are spanned by q-orthogonal polynomials (q-Askey scheme).

Introduction Results Background Proofs Today: Quantisation and representation theory

To each Painlev´eequation we associate a cubic surface called monodromy manifold. The monodromy manifold admits a Poisson bracket and can be quantised. These quantum algebras admit representations on the space of Laurent polynomials.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Today: Quantisation and representation theory

To each Painlev´eequation we associate a cubic surface called monodromy manifold. The monodromy manifold admits a Poisson bracket and can be quantised. These quantum algebras admit representations on the space of Laurent polynomials. The eigenspaces in this representation are spanned by q-orthogonal polynomials (q-Askey scheme).

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs q-Askey scheme Koekoek, Lesky, Swarttouw 2010

Askey-Wilson q-Racha

Connuous Connuous Big Dual q-Hahn q-Hahn q-Jacobi q-Hahn Dual q-Hahn

Al-Salam q-Meixner Connuous Big Lile Quantum Affine Dual q-Meixner q-Krawtchouk Chihara Pollaczek q-Jacobi q-Laguerre q-Jacobi q-Krawtchouk q-Krawtchouk q-Krawtchouk

Connuous Connuous Lile Al-Salam Al-Salam q-Laguerre q-Bessel q-Charlier big q-Hermite q-Laguerre q-Laguerre Carlitz I Carlitz II

Connuous Seltjes Discrete Discrete q-Hermite Wigert q-Hermite I q-Hermite II

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs

Painlev´eequations and q-Askey polynomials M.M. arXiv:1307.6140

Askey-Wilson q-Racha

Connuous Connuous Big Dual q-Hahn q-Hahn q-Jacobi q-Hahn Dual q-Hahn

Al-Salam q-Meixner Connuous Big Lile Quantum Affine Dual q-Meixner q-Krawtchouk Chihara Pollaczek q-Jacobi q-Laguerre q-Jacobi q-Krawtchouk q-Krawtchouk q-Krawtchouk

Connuous Connuous Lile Al-Salam Al-Salam q-Laguerre q-Bessel q-Charlier big q-Hermite q-Laguerre q-Laguerre Carlitz I Carlitz II

Connuous Seltjes Discrete Discrete q-Hermite Wigert q-Hermite I q-Hermite II

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Confluence Painlev´eequations,

Confluence monodromy manifolds, Rubtsov and M.M. arXiv:1212.6723 Confluence Askey Wilson algebra,

Confluence Cherednik algebra seven new algebras. M.M. ⇒ arXiv:1307.6140

Introduction Results Background Proofs

Other results Introduction Cherednik and its spherical sub-algebra PVI and Cherednik algebra Proofs for PVI The other Painlev´eequations Other results

Riemann$$ MonodromyThe$ quan7sa7on monodromy$ Askey group$ of PVICerednik admits$$ a natural quantisation to PVI$ Hilbert$ manifold$the CherednikWilson$ algebra ofalgebra$ type Cˇ C . H 1 1 The monodromy manifold of PVI quantises to the spherical subalgebra of . H 7 new algebras as confluences of such that their spherical H subalgebra is the quantisation of the monodromy manifold of each Painlev´eequation.

M.M. arXiv:1307.6140

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme

Marta Mazzocco Confluence of the Painlev´eequations, Cherednik algebras and q-Askey scheme Confluence monodromy manifolds, Rubtsov and M.M. arXiv:1212.6723 Confluence Askey Wilson algebra,

Confluence Cherednik algebra seven new algebras. M.M. ⇒ arXiv:1307.6140

Introduction Results Background Proofs

Other results Introduction Cherednik and its spherical sub-algebra PVI and Cherednik algebra Proofs for PVI The other Painlev´eequations Other results

Riemann$$ MonodromyThe$ quan7sa7on monodromy$ Askey group$ of PVICerednik admits$$ a natural quantisation to PVI$ Hilbert$ manifold$the CherednikWilson$ algebra ofalgebra$ type Cˇ C . H 1 1 The monodromy manifold of PVI quantises to the spherical Confluence Painlev´eequations,subalgebra of . H 7 new algebras as confluences of such that their spherical H subalgebra is the quantisation of the monodromy manifold of each Painlev´eequation.

M.M. arXiv:1307.6140

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme

Marta Mazzocco Confluence of the Painlev´eequations, Cherednik algebras and q-Askey scheme Confluence Askey Wilson algebra,

Confluence Cherednik algebra seven new algebras. M.M. ⇒ arXiv:1307.6140

Introduction Results Background Proofs

Other results Introduction Cherednik and its spherical sub-algebra PVI and Cherednik algebra Proofs for PVI The other Painlev´eequations Other results

Riemann$$ MonodromyThe$ quan7sa7on monodromy$ Askey group$ of PVICerednik admits$$ a natural quantisation to PVI$ Hilbert$ manifold$the CherednikWilson$ algebra ofalgebra$ type Cˇ C . H 1 1 The monodromy manifold of PVI quantises to the spherical Confluence Painlev´eequations,subalgebra of . H Confluence monodromy manifolds, Rubtsov and M.M. arXiv:1212.6723 7 new algebras as confluences of such that their spherical H subalgebra is the quantisation of the monodromy manifold of each Painlev´eequation.

M.M. arXiv:1307.6140

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme

Marta Mazzocco Confluence of the Painlev´eequations, Cherednik algebras and q-Askey scheme Confluence Cherednik algebra seven new algebras. M.M. ⇒ arXiv:1307.6140

Introduction Results Background Proofs

Other results Introduction Cherednik and its spherical sub-algebra PVI and Cherednik algebra Proofs for PVI The other Painlev´eequations Other results

Riemann$$ MonodromyThe$ quan7sa7on monodromy$ Askey group$ of PVICerednik admits$$ a natural quantisation to PVI$ Hilbert$ manifold$the CherednikWilson$ algebra ofalgebra$ type Cˇ C . H 1 1 The monodromy manifold of PVI quantises to the spherical Confluence Painlev´eequations,subalgebra of . H Confluence monodromy manifolds, Rubtsov and M.M. arXiv:1212.6723 7 new algebras as confluences of such that their spherical Confluence Askey Wilson algebra, H subalgebra is the quantisation of the monodromy manifold of each Painlev´eequation.

M.M. arXiv:1307.6140

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme

Marta Mazzocco Confluence of the Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs

Other results Introduction Cherednik and its spherical sub-algebra PVI and Cherednik algebra Proofs for PVI The other Painlev´eequations Other results

Riemann$$ MonodromyThe$ quan7sa7on monodromy$ Askey group$ of PVICerednik admits$$ a natural quantisation to PVI$ Hilbert$ manifold$the CherednikWilson$ algebra ofalgebra$ type Cˇ C . H 1 1 The monodromy manifold of PVI quantises to the spherical Confluence Painlev´eequations,subalgebra of . H Confluence monodromy manifolds, Rubtsov and M.M. arXiv:1212.6723 7 new algebras as confluences of such that their spherical Confluence Askey Wilson algebra, H subalgebra is the quantisation of the monodromy manifold of each Confluence Cherednik algebra seven new algebras. M.M. Painlev´eequation.⇒ arXiv:1307.6140

M.M. arXiv:1307.6140

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme

Marta Mazzocco Confluence of the Painlev´eequations, Cherednik algebras and q-Askey scheme Askey-Wilson polynomials:

 q−n, qn−1abcd, az, az−1  pn (z; a, b, c, d q) := ϕ ; q, q | 4 3 ab, ac, ad n+1 −n n−1 −1 X (q , q abcd, az, az ; q)k = qk . (ab, ac, ad; q)k k=0

Introduction Results Background Proofs Askey Wilson polynomials

k−1 j q-Pochhammer symbol: (a; q)k := Π (1 a q ) j=0 − (a ,..., ar ; q)k := (a ; q)k (a ; q)k (ar ; q)k 1 1 2 ··· q-hypergeometric function:   ∞ a1, a2,..., ar X (a1, a2,..., ar ; q)k k r ϕr−1 ; q, x := x . b1,..., br−1 (b1,..., br−1; q)k k=0

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Askey Wilson polynomials

k−1 j q-Pochhammer symbol: (a; q)k := Π (1 a q ) j=0 − (a ,..., ar ; q)k := (a ; q)k (a ; q)k (ar ; q)k 1 1 2 ··· q-hypergeometric function:   ∞ a1, a2,..., ar X (a1, a2,..., ar ; q)k k r ϕr−1 ; q, x := x . b1,..., br−1 (b1,..., br−1; q)k k=0

Askey-Wilson polynomials:

 q−n, qn−1abcd, az, az−1  pn (z; a, b, c, d q) := ϕ ; q, q | 4 3 ab, ac, ad n+1 −n n−1 −1 X (q , q abcd, az, az ; q)k = qk . (ab, ac, ad; q)k k=0

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Open question: what algebras control all other elements of the q-Askey scheme?

Introduction Results Background Proofs Facts about Askey Wilson polynomials

They are special cases of Macdonald polynomials. Their algebraic structure is controlled by the Cherednik algebra of type Cˇ1C1. They contain all other basic hypergeometric polinomyals as degenerations.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Facts about Askey Wilson polynomials

They are special cases of Macdonald polynomials. Their algebraic structure is controlled by the Cherednik algebra of type Cˇ1C1. They contain all other basic hypergeometric polinomyals as degenerations. Open question: what algebras control all other elements of the q-Askey scheme?

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Quantum algebra results

For each Painlev´eequation, a new quantum algebra and its representation theory is introduced. These algebras control some elements of the q-Askey scheme. PAINLEVEEQUATIONS,CHEREDNIKALGEBRASANDQ-ASKEYSCHEME´ 7

Askey–Wilson ZVI

V Big q-JacobiZ Continuous dual q–Haun

Big q–Laguerre Al Salam–Chihara ZIV ZIII

Little q–Laguerre Continuous big q–Hermite D7 II Z ZIII

Little q–Laguerre with a =0 Continuous q–Hermite D8 I Z ZIII

Figure 1. The confluence scheme for the Zhedanov algebras and the polynomials in the q-Askey scheme M.M. arXiv:1307.6140 Since most results obtained in this paper are proved without relying onthe theory of Painlev´eequations,Marta Mazzocco we organise the paperPainlev´eequations, as follows: in Section Cherednik 2, we algebras and q-Askey scheme recall some background material on the theory of the Cherednik algebra of type Cˇ1C1 and its representation theory. In Section 3, we prove Theorem 1.1.InSec- tion 4, we explain how to derive our confluent Cherednik algebras and give some equivalent presentations for the algebras , , . In Section 5, we embed HV HIV HIII V , IV , III, II, I into Mat(2, Tq). In Section 6, we discuss the spherical H H H H H D7 D8 sub–algebras of V , IV , III, III, III, and produce a set of elements that sat- isfy a cubic relationH H whichH in theH semiclassicalH limit coincide with the monodromy manifolds of the corresponding Painlev´eequations. In Section 7, weprovethateach spherical sub-algebra is isomorphic to the corresponding confluent Zhedanov algebra and show that the latter act as symmetries of some elements of the q-Askey scheme. Finally, in Section 8, we recall a few basic facts about the isomonodromic deforma- tion equations associated to the sixth Painlev´eequation and show how Cherednik algebra of type Cˇ1C1 appears naturally as quantisation of the monodromy group associated to the sixth Painlev´eequation.

2. Notation and background on the Cherednik algebra of type Cˇ1C1 In this section we recall some background material on the theory of the Cherednik algebra of type Cˇ1C1, a few useful facts about its basic representation and about Introduction Results Background Proofs ˇ Cherednik algebra of type C1C1 Cherednik ’92, Sahi ’99

Algebra generated by V0, V1, Vˇ0, Vˇ1:

(V k )(V + k−1) = 0 0 − 0 0 0 (V k )(V + k−1) = 0 1 − 1 1 1 (Vˇ u )(Vˇ + u−1) = 0 0 − 0 0 0 (Vˇ u )(Vˇ + u−1) = 0 1 − 1 1 1 −1/2 Vˇ1V1V0Vˇ0 = q ,

k0, k1, u0, u1 scalars.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Representation theory:

Cherednik algebra on Laurent polynomials

h 2 1  1  i  1  1 z u1z − + k1− z z u1− +k1− V [f ] = u1 k1 f 1
u1 k1 f (z) 1 z2−1 z − z2−1

z2 √  1  h 1  √  1 i qk0− + q u0− z z k0− + q u0− V [f ] = k0 u0 f q
k0 u0 f (z) 0 z2−1 z − z2−1

1+Vˇ1 The operator 1+u2 acts as symmetrizer representation of the 1 ⇒ spherical sub–algebra on symmetric Laurent polynomials.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme −1
X1 = e Vˇ1V1 + (Vˇ1V1) , −1
X2 = e Vˇ1V0 + (Vˇ1V0) , 1/2 −1/2 −1
X3 = e q V1V0 + q (V1V0)

q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 1 2 − 2 1 − 3 − 3 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 2 3 − 3 2 − 1 − 1 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 3 1 − 1 3 − 2 − 2

1 2 −1 2 2 1 − 1 1 q 2 X X X qX q X qX +q 2 ω X +q 2 ω X +q 2 ω X = ω e. 2 1 3− 2 − 1 − 3 2 2 1 1 3 3 4

Zhedanov ’91, Oblomkov ’04

Introduction Results Background Proofs Spherical sub-algebra:

1+Vˇ1 generated by: e = 2 , 1+u1

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 1 2 − 2 1 − 3 − 3 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 2 3 − 3 2 − 1 − 1 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 3 1 − 1 3 − 2 − 2

1 2 −1 2 2 1 − 1 1 q 2 X X X qX q X qX +q 2 ω X +q 2 ω X +q 2 ω X = ω e. 2 1 3− 2 − 1 − 3 2 2 1 1 3 3 4

Zhedanov ’91, Oblomkov ’04

Introduction Results Background Proofs Spherical sub-algebra:

1+Vˇ1 generated by: e = 2 , 1+u1 −1
X1 = e Vˇ1V1 + (Vˇ1V1) , −1
X2 = e Vˇ1V0 + (Vˇ1V0) , 1/2 −1/2 −1
X3 = e q V1V0 + q (V1V0)

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Spherical sub-algebra:

1+Vˇ1 generated by: e = 2 , 1+u1 −1
X1 = e Vˇ1V1 + (Vˇ1V1) , −1
X2 = e Vˇ1V0 + (Vˇ1V0) , 1/2 −1/2 −1
X3 = e q V1V0 + q (V1V0)

q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 1 2 − 2 1 − 3 − 3 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 2 3 − 3 2 − 1 − 1 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 3 1 − 1 3 − 2 − 2

1 2 −1 2 2 1 − 1 1 q 2 X X X qX q X qX +q 2 ω X +q 2 ω X +q 2 ω X = ω e. 2 1 3− 2 − 1 − 3 2 2 1 1 3 3 4

Zhedanov ’91, Oblomkov ’04

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Spherical sub–algebra and Askey Wilson polynomials

1 X (pn(z, a, b, c, d q)) = (z + )pn(z, a, b, c, d q), 1 | z | −n n−1 X (pn(z, a, b, c, d q)) = (q + abcdq )(pn(z, a, b, c, d q)) . 2 | | Koornwinder ’07

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Recap:

Cherednik algebra of type Cˇ1C1:

−1 (V0 k0)(V0 + k0 ) = 0 − −1 Represented on (V1 k1)(V1 + k1 ) = 0 ˇ − ˇ −1 Laurent polynomials (V0 u0)(V0 + u0 ) = 0 ˇ − ˇ −1 Non-symmetric (V1 u1)(V1 + u1 ) = 0 − −1/2 Askey-Wilson (Koornwinder 07) Vˇ1V1V0Vˇ0 = q .

Spherical subalgebra (Zhedanov):

q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e, 1 2 − 2 1 − 3 − 3 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e, 2 3 − 1 3 − 1 − 1 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e. 3 1 − 1 3 − 2 − 2 Representation spanned by Askey-Wilson polynomials

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme This means that the monodromy data of the linear system dY = A(λ; z, w, wz )Y dλ are locally constant along solutions of the Painlev´eequation. The monodromy data play the role of initial conditions

Introduction Results Background Proofs

All Painlev´eequations are isomonodromic deformation equations (Jimbo-Miwa 1980)

dB dA = [A, B] dλ − dz A = A(λ; z, w, wz ), B = B(λ; z, w, wz ) sl . ∈ 2

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs

All Painlev´eequations are isomonodromic deformation equations (Jimbo-Miwa 1980)

dB dA = [A, B] dλ − dz A = A(λ; z, w, wz ), B = B(λ; z, w, wz ) sl . ∈ 2 This means that the monodromy data of the linear system dY = A(λ; z, w, wz )Y dλ are locally constant along solutions of the Painlev´eequation. The monodromy data play the role of initial conditions

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs PVI monodromy data

M1, M2, M3, M∞ SL(2, C) s.t. ∈ eigenv(Mi ) = exp( i pi ), i = 1, 2, 3, , ± ∞ M∞M1M2M3 = 1. Jimbo, Miwa ’81

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Cherednik algebra as quantisation of the monodromy group

(M ei p3 )(M e−i p3 ) = 0, (V k )(V + k−1) = 0, 3 − 3 − 0 − 0 0 0 (M ei p2 )(M e−i p2 ) = 0, (V k )(V + k−1) = 0, 2 − 2 − 1 − 1 1 1 (M ei p1 )(M e−i p1 ) = 0, (Vˇ u )(Vˇ + u−1) = 0, 1 − 1 − 1 − 1 1 1 i p∞ −i p∞ −1 (M∞ e )(M∞ e ) = 0( Vˇ u )(Vˇ + u ) = 0, − − 0 − 0 0 0 −1/2 M∞M3M2M1 = 1. Vˇ1V1V0Vˇ0 = q .

There is a natural quantisation which works.

M iVˇ , M iV , M iV , M∞ iVˇ . 1 → 1 2 → 1 3 → 0 → 0 − p1 − p3 − p2 −S −S −S u = ie 2 , k = ie 2 , k = ie 2 , u = ie 1 2 3 . 1 − 0 − 1 − 0 −

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Cherednik algebra as quantisation of the monodromy group

(M ei p3 )(M e−i p3 ) = 0, (V k )(V + k−1) = 0, 3 − 3 − 0 − 0 0 0 (M ei p2 )(M e−i p2 ) = 0, (V k )(V + k−1) = 0, 2 − 2 − 1 − 1 1 1 (M ei p1 )(M e−i p1 ) = 0, (Vˇ u )(Vˇ + u−1) = 0, 1 − 1 − 1 − 1 1 1 i p∞ −i p∞ −1 (M∞ e )(M∞ e ) = 0( Vˇ u )(Vˇ + u ) = 0, − − 0 − 0 0 0 −1/2 M∞M3M2M1 = 1. Vˇ1V1V0Vˇ0 = q .

There is a natural quantisation which works.

M iVˇ , M iV , M iV , M∞ iVˇ . 1 → 1 2 → 1 3 → 0 → 0 − p1 − p3 − p2 −S −S −S u = ie 2 , k = ie 2 , k = ie 2 , u = ie 1 2 3 . 1 − 0 − 1 − 0 −

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 1 2 − 2 1 − 3 − 3 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 2 3 − 3 2 − 1 − 1 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 3 1 − 1 3 − 2 − 2

1 2 −1 2 2 1 − 1 1 q 2 X X X qX q X qX +q 2 ω X +q 2 ω X +q 2 ω X = ω e. 2 1 3− 2 − 1 − 3 2 2 1 1 3 3 4

Introduction Results Background Proofs

Monodromy manifold:

x1 = Tr(M2M3), x2 = Tr(M2M3), x3 = Tr(M3M1),

2 2 2 x1x2x3 + x1 + x2 + x3 + ω1x1 + ω2x2 + ω3x3 = ω4 Poisson bracket:

x , x = x x + 2x + ω , x , x = x x + 2x + ω , { 1 2} 1 2 3 3 { 2 3} 2 3 1 1 x , x = x x + 2x + ω . { 3 1} 1 3 2 2

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs

Monodromy manifold:

x1 = Tr(M2M3), x2 = Tr(M2M3), x3 = Tr(M3M1),

2 2 2 x1x2x3 + x1 + x2 + x3 + ω1x1 + ω2x2 + ω3x3 = ω4 Poisson bracket:

x , x = x x + 2x + ω , x , x = x x + 2x + ω , { 1 2} 1 2 3 3 { 2 3} 2 3 1 1 x , x = x x + 2x + ω . { 3 1} 1 3 2 2

q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 1 2 − 2 1 − 3 − 3 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 2 3 − 3 2 − 1 − 1 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e 3 1 − 1 3 − 2 − 2

1 2 −1 2 2 1 − 1 1 q 2 X X X qX q X qX +q 2 ω X +q 2 ω X +q 2 ω X = ω e. 2 1 3− 2 − 1 − 3 2 2 1 1 3 3 4

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Monodromy manifolds for the Painlev´eequations

2 2 2 PVI x1x2x3 + x1 + x2 + x3 + ω1x1 + ω2x2 + ω3x3 = ω4

2 2 PV x1x2x3 + x1 + x2 + ω1x1 + ω2x2 + ω3x3 = ω4

2 PIV x1x2x3 + x1 + ω1x1 + ω2x2 + ω2x3 + 1 = ω4

PIII x x x + x2 + x2 + ω x + ω x = ω 1 1 2 3 1 2 1 1 2 2 1 −

PII x1x2x3 + x1 + x2 + x3 = ω4

PI x1x2x3 + x1 + x2 + 1 = 0

Saito and van der Put Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Example PV: ϕ := x x x + x2 + x2 + ω x + ω x + ω x ω , 1 2 3 1 2 1 1 2 2 3 3 − 4 x , x = x x + ω , x , x = x x + 2x + ω , { 1 2} 1 2 3 { 2 3} 2 3 1 1 x , x = x x + 2x + ω . { 3 1} 1 3 2 2 Quantisation (Teichm¨ullertheory and cluster algebra)

(L. Chekhov and M.M. J.Phys A 2010).

Introduction Results Background Proofs Poisson structure on the monodromy manifolds

Cubic surface Mϕ := C[x1, x2, x3]/hϕ(x1,x2,x3)=0i Poisson bracket: ∂ϕ ∂ϕ ∂ϕ x1, x2 = , x2, x3 = , x3, x1 = . { } ∂x3 { } ∂x1 { } ∂x2

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Quantisation (Teichm¨ullertheory and cluster algebra)

(L. Chekhov and M.M. J.Phys A 2010).

Introduction Results Background Proofs Poisson structure on the monodromy manifolds

Cubic surface Mϕ := C[x1, x2, x3]/hϕ(x1,x2,x3)=0i Poisson bracket: ∂ϕ ∂ϕ ∂ϕ x1, x2 = , x2, x3 = , x3, x1 = . { } ∂x3 { } ∂x1 { } ∂x2

Example PV: ϕ := x x x + x2 + x2 + ω x + ω x + ω x ω , 1 2 3 1 2 1 1 2 2 3 3 − 4 x , x = x x + ω , x , x = x x + 2x + ω , { 1 2} 1 2 3 { 2 3} 2 3 1 1 x , x = x x + 2x + ω . { 3 1} 1 3 2 2

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Poisson structure on the monodromy manifolds

Cubic surface Mϕ := C[x1, x2, x3]/hϕ(x1,x2,x3)=0i Poisson bracket: ∂ϕ ∂ϕ ∂ϕ x1, x2 = , x2, x3 = , x3, x1 = . { } ∂x3 { } ∂x1 { } ∂x2

Example PV: ϕ := x x x + x2 + x2 + ω x + ω x + ω x ω , 1 2 3 1 2 1 1 2 2 3 3 − 4 x , x = x x + ω , x , x = x x + 2x + ω , { 1 2} 1 2 3 { 2 3} 2 3 1 1 x , x = x x + 2x + ω . { 3 1} 1 3 2 2 Quantisation (Teichm¨ullertheory and cluster algebra)

(L. Chekhov and M.M. J.Phys A 2010).

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs For d 0 (k 0, u u ): → 0 → 0 → 0 First confluent Cherednik algebra: 2 V0 + V0 = 0, Represented on −1 (V1 k1)(V1 + k ) = 0 Laurent polynomials − 1 ˇ 2 −1 ˇ V0 + u0 V0 = 0, Non-symmetric ˇ ˇ −1 (V1 u1)(V1 + u1 ) = 0 Continuous 1/2−ˇ ˇ −1 q V1V1V0 = V0 + u0 , dual q-Hahn 1/2 q Vˇ0Vˇ1V1 = V0 + 1. polynomials (M.M. arXiv:1409.4287) Spherical subalgebra (confluent Zhedanov): q−1/2X X q1/2X X = (q−1/2 q1/2)ω e, 1 2 − 2 1 − 3 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e, 2 3 − 1 3 − 1 − 1 q−1/2X X q1/2X X = (q−1 q)X + (q−1/2 q1/2)ω e. 3 1 − 1 3 − 2 − 2 Representation spanned by Continouos dual q-Hahn polynomials

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Outlook

Askey-Wilson q-Racha

Connuous Connuous Big Dual q-Hahn q-Hahn q-Jacobi q-Hahn Dual q-Hahn

Al-Salam q-Meixner Connuous Big Lile Quantum Affine Dual q-Meixner q-Krawtchouk Chihara Pollaczek q-Jacobi q-Laguerre q-Jacobi q-Krawtchouk q-Krawtchouk q-Krawtchouk

Connuous Connuous Lile Al-Salam Al-Salam q-Laguerre q-Bessel q-Charlier big q-Hermite q-Laguerre q-Laguerre Carlitz I Carlitz II

Connuous Seltjes Discrete Discrete q-Hermite Wigert q-Hermite I q-Hermite II

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Outlook

Askey-Wilson q-Racha

Connuous Connuous Big Dual q-Hahn q-Hahn q-Jacobi q-Hahn Dual q-Hahn

Al-Salam q-Meixner Connuous Big Lile Quantum Affine Dual q-Meixner q-Krawtchouk Chihara Pollaczek q-Jacobi q-Laguerre q-Jacobi q-Krawtchouk q-Krawtchouk q-Krawtchouk

Connuous Connuous Lile Al-Salam Al-Salam q-Laguerre q-Bessel q-Charlier big q-Hermite q-Laguerre q-Laguerre Carlitz I Carlitz II

Connuous Seltjes Discrete Discrete q-Hermite Wigert q-Hermite I q-Hermite II

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme cover the whole → q-Askey scheme. q-difference and elliptic Painlev´eequations may correspond to elliptic hypergeometric bi-orthogonal polynomials. Multivariable high order analogues of the Painlev´eequations confluence scheme for Macdonald polynomials →

Introduction Results Background Proofs Outlook

Additive discrete Painlev´eequations

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme q-difference and elliptic Painlev´eequations may correspond to elliptic hypergeometric bi-orthogonal polynomials. Multivariable high order analogues of the Painlev´eequations confluence scheme for Macdonald polynomials →

Introduction Results Background Proofs Outlook

Additive discrete Painlev´eequations cover the whole → q-Askey scheme.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Multivariable high order analogues of the Painlev´eequations confluence scheme for Macdonald polynomials →

Introduction Results Background Proofs Outlook

Additive discrete Painlev´eequations cover the whole → q-Askey scheme. q-difference and elliptic Painlev´eequations may correspond to elliptic hypergeometric bi-orthogonal polynomials.

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme Introduction Results Background Proofs Outlook

Additive discrete Painlev´eequations cover the whole → q-Askey scheme. q-difference and elliptic Painlev´eequations may correspond to elliptic hypergeometric bi-orthogonal polynomials. Multivariable high order analogues of the Painlev´eequations confluence scheme for Macdonald polynomials →

Marta Mazzocco Painlev´eequations, Cherednik algebras and q-Askey scheme