Difference analogues for conformal towers and the Ding-Iohara algebra

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540206
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Global analysis
• Research Institution: The University of Tokyo
• Principal Investigator: SHIRAISHI Junichi 東京大学, 数理(科)学研究科(研究院), 准教授 (20272536)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: Macdonald多項式 / Askey-Wilson多項式 / Koornwinder多項式 / Ding-Iohara代数 / topological vertex / Laumon空間

Outline of Final Research Achievements

A quantum group approach to the refined topological vertex of Iqbal-Kozcaz-Vafa is presented in terms of the Ding-Iohara algebra (joint work with H. Awata and B. Feigin).
A four-fold series expression is obtained for the Askey-Wilson polynomial. It gives us an explicit formula for the Koornwinder polynomials in one row case. Lassalle's conjectures for the Macdonald polynomials for types B,C,D are recovered, hence proved, as special cases of the formula for the Koornwinder polynomials (joint work with A. Hoshino and M. Noumi).
A geometric construction is presented for the Macdonald polynomials of A-type as the Euler characteristic of the twisted de Rham complex for the Laumon space. The Macdonald polynomials of other types can be considered in some other algebro-geometric setting, and a conjecture aout a geometric construction is given (joint work with A. Braverman and M. Finkelberg).

Free Research Field

量子可積分系