Dualities in the q-askey scheme and degenerate DAHA

Transcription

1 Dualities in the q-askey scheme and degenerate DAHA Based on M. Mazzocco Nonlinearity 16 and T. Koornwinder-M. Mazzocco arxiv:

2 Outline Askey Wilson polynomials and their properties Zhedanov algebra q-askey scheme and its geometric interpretation duality Non-symmetric AW polynomials, DAHA and degenerations. Outlook

3 Askey-Wilson polynomials Definition R n [z; a, b, c, d q] := n k=0 (q n, q n 1, abcd, az, az 1 ; q) k (ab, ac, ad; q) k (a 1,...,a r ; q) k := (a 1 ; q) k (a 2 ; q) k (a r ; q) k, (a; q) k := Π k 1 j=0 (1 aqj ) Example R 2 [z; a,...,d; q] =1 q k (1 + q)( 1 + abcdq)(a z)( 1 + az) + ( 1 + ab)( 1 + ac)( 1 + ad)qz + (abcdq 1)(abcdq2 1)(a z)(aq z)(az 1)(aqz 1) (ab 1)(ac 1)(ad 1)q(abq 1)(acq 1)(adq 1)z 2

4 Properties of the Askey-Wilson polynomials Symmetric: R n [z; a, b, c, d q] =R n [1/z; a, b, c, d q]. Orthogonal q-difference equation L z ( Rn [z] ) =(q n + abcdq n 1 ) R n [z], L z ( f [z] ) :=A(z) ( f [qz] f [z] ) + A(1/z) ( f [q 1 z] f [z] ), A(z) = (1 az)(1 bz)(1 cz)(1 dz) (1 z 2 )(1 qz 2 ) Three term recursion relation M n ( Rn [z] ) =(z + z 1 ) R n [z], M n ( gn ) := An g n+1 +(a + a 1 A n C n )g n + C n g n 1.

5 Zhedanov algebra The operators L z and ( z + 1 ) z generate the Zhedanov algebra: ( ) (K 0 f )[z] :=L z f [z], (K1 f )[z] :=(z + z 1 )f [z], (q + q 1 )K 1 K 0 K 1 K1 2 K 0 K 0 K1 2 = BK 1 + C 0 K 0 + D 0, (q + q 1 )K 0 K 1 K 0 K0 2 K 1 K 1 K0 2 = BK 0 + C 1 K 1 + D 1. B, C 0, C 1, D 0, D 1 constants: C 0 := (q q 1 ) 2 and B =(1 q 1 ) 2 (abc + abd + acd + bcd +(a + b + c + d)q), C 1 = q 1 (q q 1 ) 2 abcd, D 0..., D 1 =...

6 q-askey scheme Take degenerations of Askey Wilson polynomials to produce other families of polynomials such that Example Symmetric Laurent or standard polynomials Orthogonal q-difference equation Three-term recursion relation. d = 0isagooddegeneration: p n [z; a, b, c q] := n (q n,q n 1,az,az 1 ;q) k k=0 (ab,ac;q) q k. k a = 0isnotagooddegeneration.

7 q-askey scheme

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9 Underlying mechanism L z, ( z + 1 z ) Zhedanov algebra

10 Underlying mechanism L z, ( z + 1 z ) Zhedanov algebra q 1 [Oblomkov 04] Degenerations correspond to chewing-gum moves [M. M. Nonlinearity 16]

11 Chewing-gum moves [Chekhov-M. M. Nonlinearity 17] Hooking holes: Pinching two sides of the same hole:

12 Bordered cusped character variety Standard case Cusped case Fundamental group: π 1 (Σ g,s ) Representations: Hom ( π 1 (Σ g,s ) SL 2 (C) ) Character variety: Hom ( π 1 (Σ g,s ) SL 2 (C) ) / SL2 (C) dim(c g,s )=6g 6 + 3s Fundamental groupoid of arcs: π a (Σ g,s,n ) Representations: Hom ( π a (Σ g,s,n ), SL 2 (C) ) Cusped Character Variety: Hom ( π a (Σ g,s,n ), SL 2 (C) ) / nj=1 U j dim(c g,s,n )=6g 6 + 3s + 2n

13 Geometric q-askey scheme

14 Duality The operators L z, z + z 1 act on Sym[z]. They generate the Zhedanov algebra L z, ( z + 1 z ) on Askey-Wilson polynomials L z ( Rn [z] ) =(q n + abcdq n 1 ) R n [z] and (z + z 1 ) R n [z] =M n ( Rn [z] ) Duality: L z, (z + z 1 ) M n, Λ n [Noumi-Stokman 04]. R n [ a 1 q m ; a, b, c, d q ] = R m [ã 1 q n ; ã, b, c, d q ] ã =(q 1 abcd) 1 2 b = ab/ã, c = ac/ã, d = ad/ã.

15 q-askey scheme Askey Wilson Big q-jacobi Continuous dual q Hahn Big q Laguerre Little q Jacobi Al Salam Chihara Little q Laguerre Continuous big q Hermite Continuous q Hermite

16 q-askey scheme Askey Wilson Big q-jacobi Continuous dual q Hahn Big q Laguerre Little q Jacobi Al Salam Chihara Little q Laguerre Continuous big q Hermite Continuous q Hermite

17 Non symmetric Askey-Wilson polynomials E n [z] :=R n [z; a, b, c, d q] q 1 n (1 q n )(1 q n 1 cd) (1 qab)(1 ab)(1 ac)(1 ad) az 1 (1 az)(1 bz)r n 1 [z; qa, qb, c, d q] n 0, E n [z] :=R n [z; a, b, c, d q] q1 n (1 q n ab)(1 q n 1 abcd) (1 qab)(1 ab)(1 ac)(1 ad) b 1 z 1 (1 az)(1 bz)r n 1 [z; qa, qb, c, d q] n 1, [Koornwinder 07]

18 Properties of the non-symmetric Askey-Wilson polynomials They form a basis in the space of Laurent polynomials. Orthogonal q-difference equation YE n = q n 1 abcd E n (n = 0, 1, 2,...), YE n = q n E n (n = 1, 2,...). Three term recursion relation M n ( En [z] ) = z 1 E n [z]. [Koornwinder 07]

19 DAHA Z, Y and T generate the DAHA of type Č1C 1 : (T + ab)(t + 1) =0, (T 1 Y + q 1 cd)(t 1 Y + 1) =0, (az 1 T 1 + 1)(bZ 1 T 1 + 1) =0, (c + qz 1 T 1 Y )(d + qz 1 T 1 Y )=0, ZZ 1 = 1 = Z 1 Z. [Sahi 99] Can we degenerate DAHA as well?

20 Underlying mechanism L z, R n Zhedanov algebra DAHA Č1C 1 q 1

21 Underlying mechanism L z, R n Zhedanov algebra DAHA Č1C 1 q 1

22 Underlying mechanism L z, R n Zhedanov algebra q 1 DAHA Č1C 1 q 1 Hol(, Σ 0,4 ) Degenerations of DAHA were constructed in M.M. 16. Chewing-gum moves correspond to confluences of poles in [Chekhov,M.M, Rubtsov 18]

23 Outlook R.h.s. of q Askey scheme. Macdonald polynomials. The cusped character variety carries a cluster algebra structure - what role does this play in the theory?