No abstract

Abstract. We introduce a unital associative algebra A over degenerate CP 1 . We show that A is a commutative algebra and whose Poincaré series is given by the number of partitions. Thereby we can regard A as a smooth degeneration limit of the elliptic algebra introduced by one of the authors and Odesskii [FO]. 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 A and the Heisenberg representation of the commutative family studied by one of the authors [S2]. It is found that the Ding-Iohara algebra [DI]

We study the representation theory of the Ding-Iohara algebra U to find qanalogues of the Alday-Gaiotto-Tachikawa (AGT) relations. We introduce the endomorphism T (u, v) of the Ding-Iohara algebra, having two parameters u and v. We define the vertex operator Φ(w) by specifying the permutation relations with the Ding-Iohara generators x ± (z) and ψ ± (z) in terms of T (u, v). For the level one representation, all the matrix elements of the vertex operators with respect to the Macdonald polynomials are factorized and written in terms of the Nekrasov factors for the K-theoretic partition functions as in the AGT relations. For higher levels m = 2, 3, . . ., we present some conjectures, which imply the existence of the q-analogues of the AGT relations.

We shall study some moduli spaces of Bridgeland's semi-stable objects on abelian surfaces and K3 surfaces with Picard number 1. Under some conditions, we show that the moduli spaces are isomorphic to the moduli spaces of Gieseker semi-stable sheaves. We also study the ample cone of the moduli spaces.

We give an explicit formula of Whittaker vector for Virasoro algebra in terms of the Jack symmetric functions. Our fundamental tools are the Feigin-Fuchs bosonization and the split expression of the Calogero-Sutherland model given by Awata, Matsuo, Odake and Shiraishi.