SIGMA 10 (2014), 021, 17 pages      arXiv:1305.7097      https://doi.org/10.3842/SIGMA.2014.021
Contribution to the Special Issue in honor of Anatol Kirillov and Tetsuji Miwa

Commutative Families of the Elliptic Macdonald Operator

Yosuke Saito
Mathematical Institute of Tohoku University, Sendai, Japan

Received October 01, 2013, in final form February 25, 2014; Published online March 11, 2014

Abstract
In the paper [J. Math. Phys. 50 (2009), 095215, 42 pages], Feigin, Hashizume, Hoshino, Shiraishi, and Yanagida constructed two families of commuting operators which contain the Macdonald operator (commutative families of the Macdonald operator). They used the Ding-Iohara-Miki algebra and the trigonometric Feigin-Odesskii algebra. In the previous paper [arXiv:1301.4912], the present author constructed the elliptic Ding-Iohara-Miki algebra and the free field realization of the elliptic Macdonald operator. In this paper, we show that by using the elliptic Ding-Iohara-Miki algebra and the elliptic Feigin-Odesskii algebra, we can construct commutative families of the elliptic Macdonald operator. In Appendix, we will show a relation between the elliptic Macdonald operator and its kernel function by the free field realization.

Key words: elliptic Ding-Iohara-Miki algebra; free field realization; elliptic Macdonald operator.

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References

1. Ding J., Iohara K., Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys. 41 (1997), 181-193.
2. Feigin B., Hashizume K., Hoshino A., Shiraishi J., Yanagida S., A commutative algebra on degenerate CP¹ and Macdonald polynomials, J. Math. Phys. 50 (2009), 095215, 42 pages, arXiv:0904.2291.
3. Feigin B., Hoshino A., Shibahara J., Shiraishi J., Yanagida S., Kernel function and quantum algebras, arXiv:1002.2485.
4. Feigin B., Odesskii A., A family of elliptic algebras, Int. Math. Res. Not. 1997 (1997), no. 11, 531-539.
5. Komori Y., Noumi M., Shiraishi J., Kernel functions for difference operators of Ruijsenaars type and their applications, SIGMA 5 (2009), 054, 40 pages, arXiv:0812.0279.
6. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
7. Miki K., A (q,γ) analog of the W_1+∞ algebra, J. Math. Phys. 48 (2007), 123520, 35 pages.
8. Saito Y., Elliptic Ding-Iohara algebra and the free field realization of the elliptic Macdonald operator, arXiv:1301.4912.