^{1,a)}, K. Hashizume

^{2,b)}, A. Hoshino

^{3,c)}, J. Shiraishi

^{4,d)}and S. Yanagida

^{2,e)}

### Abstract

We introduce a unital associative algebra associated with degenerate . We show that is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby, we can regard as a smooth degeneration limit of the elliptic algebra introduced by Feigin and Odesskii [Int. Math. Res. Notices11, 531 (1997)]. Then we study the commutative family of the Macdonald difference operators acting on the space of symmetric functions. A canonical basis is proposed for this family by using and the Heisenberg representation of the commutative family studied by Shiraishi [ Commun. Math. Phys.263, 439 (2006)]. It is found that the Ding–Iohara algebra [Lett. Math. Phys.41, 183 (1997)] provides us with an algebraic framework for the free field construction. An elliptic deformation of our construction is discussed, showing connections with the Drinfeld quasi-Hopf twisting [Leningrad Math. J.1, 1419 (1990)] in the sence of Babelon-Bernard–Billey [Phys. Lett. B.375, 89 (1996)], the Ruijsenaars difference operator [Commun. Math. Phys.110, 191 (1987)], and the operator of Okounkov–Pandharipande [e-print arXiv:math-ph/0411210].

The authors thank Professor M. Noumi for valuable comments, discussion, and allowing to include his private result in Proposition 3.24. K.H. and S.Y. would like to express their gratitude to him for inviting them in this subject.

I. INTRODUCTION

A. Commutative algebra with three shift parameters on degenerate

B. Gordon filtration and intersection

C. Heisenberg representation of the Macdonald difference operators

D. Ding–Iohara algebra

E. Elliptic deformation

F. Plan of the paper

II. PROOF OF THEOREM 1.5

A. Construction of the algebra

B. Gordon filtration

C. Commutativity of

D. Derivations and the final step of the proof

III. FREE FIELD, WRONSKI RELATION, AND PROOF OF THEOREMS 1.10 AND 1.19

A. Preliminaries

B. Proofs of Propositions 1.16 and 1.18

C. Proof of Proposition 3.6

D. Wronski relation of the Macdonald difference operators

E. Triangularity and the proof of Theorems 1.10 and 1.19

F. Vertex operator and the Ding–Iohara algebra

G. Appendix: Remarks on the calculation in this section

1. Direct calculation of

2. Remark on the eigenvalues

IV. ELLIPTIC ALGEBRAS AND

A. Commuting elliptic algebra

B. Elliptic current

C. Intertwining operator and the Ruijsenaars difference operator

D. Conjecture about the eigenvalues of the operator

E. Elliptic bosons and the relation to the Okounkov–Pandharipande operator

### Key Topics

- Algebras
- 58.0
- Polynomials
- 26.0
- Difference operators
- 20.0
- Eigenvalues
- 12.0
- Rational functions
- 8.0

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