Commutative Families of the Elliptic Macdonald Operator

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.