Singular Vectors of the Ding-Iohara-Miki Algebra

We review properties of generalized Macdonald functions arising from the AGT correspondence. In particular, we explain a coincidence between generalized Macdonald functions and singular vectors of a certain algebra A(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal A}(N)$\end{document} obtained using the level-(N, 0) representation (horizontal representation) of the Ding-Iohara-Miki algebra. Moreover, we give a factored formula for the Kac determinant of A(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal A}(N)$\end{document}, which proves the conjecture that the Poincaré-Birkhoff-Witt-type vectors of the algebra A(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal A}(N)$\end{document} form a basis in its representation space.