An elliptic generalization of Cherednik-Macdonald-Mehta identities is introduced using Shiraishi functions, with an elliptic matrix model and a proof to first order in the elliptic parameter.
CMM formula as superintegrability property of unitary model
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abstract
A typical example of superintegrability is provided by expression of the Hopf link hyperpolynomial in an arbitrary representation through a pair of the Macdonald polynomials at special points. In the simpler case of the Hopf link HOMFLY-PT polynomial and a pair of the Schur functions, it is a relation in the unitary matrix model. We explain that the Cherednik-Mehta-Macdonald (CMM) identity for bilinear Macdonald residues with an elliptic weight function is nothing but a reformulation of these same formulas. Their lifting to arbitrary knots and links, even torus ones remains obscure.
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Type C∨C DAHA and Koornwinder systems mirror type-A Macdonald structures for Hamiltonians, recursions, evaluations and dualities, but lack a usable Noumi-Shiraishi-style universal series and SL(2,Z)-type twisting automorphisms.
Refined Chern-Simons theory universality is restricted to simply laced Lie groups, unlike the original which applies to all simple Lie groups.
citing papers explorer
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Elliptic Generalization of Cherednik-Macdonald-Mehta identities
An elliptic generalization of Cherednik-Macdonald-Mehta identities is introduced using Shiraishi functions, with an elliptic matrix model and a proof to first order in the elliptic parameter.
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Integrable systems inspired by DAHA and DIM algebra: type $C^\vee C$ versus type $A$
Type C∨C DAHA and Koornwinder systems mirror type-A Macdonald structures for Hamiltonians, recursions, evaluations and dualities, but lack a usable Noumi-Shiraishi-style universal series and SL(2,Z)-type twisting automorphisms.
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A note on universality in refined Chern-Simons theory
Refined Chern-Simons theory universality is restricted to simply laced Lie groups, unlike the original which applies to all simple Lie groups.