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 $\mathcal{A}(N)$ 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 $\mathcal{A}(N)$, which proves the conjecture that the Poincaré–Birkhoff–Witt-type vectors of the algebra $\mathcal{A}(N)$ form a basis in its representation space.