Determinant Formulae of Quasi-Finite Representation of W_{1+\infty}   Algebra at Lower Levels
  We calculate the Kac determinant for the quasi-finite representation of \Winf algebra up to level 8. It vanishes only when the central charge is integer. We give an algebraic construction of null states and propose the character formulae. The character of the Verma module is related to free fields in three dimensions which has rather exotic modular properties. 
H.Awata, M.Fukuma, Y.Matsuo, S.Odake