  For each r = (r_1, r_2,...,r_N) we construct a highest weight module M_r of
the Lie algebra W_{1+infty}. The highest weight vectors are specific
tau-functions of the N-th Gelfand--Dickey hierarchy. We show that these modules
are quasifinite and we give a complete description of the reducible ones
together with a formula for the singular vectors.
  This paper is the first of a series of papers (q-alg/9602010, q-alg/9602011,
q-alg/9602012) on the bispectral problem.
